1Introduction

From the modelling point of view, the presence of external perturbations to otherwise autonomous evolutions is a very natural assumption. Let \(d \in \mathbb{N}\) and consider the ODE in \(\mathbb{R}^d\)

where \(w \in C (\mathbb{R}_+, \mathbb{R}^d)\) is a fixed perturbation, the dot denotes differentiation with respect to time and \(b : \mathbb{R}_+ \times \mathbb{R}^d \rightarrow \mathbb{R}^d\) is a time-dependent vector field. Provided eq. (1) is understood as an integral equation and thanks to the additive nature of the perturbation, there are no particular regularity requirements, apart from continuity, which have to be imposed on the function \(w\). A natural question is then for which classes of vector fields \(b\) eq. (1) is well-posed and if, for certain sets of perturbations \(w\), one can obtain well-posedness results in classes which are known to lead to an ill-posed problem when \(w = 0\).

One possible approach to this problem is to consider \(w\) a sample path of a stochastic process \(W\) defined on a probability space \((\Omega, \mathcal{F}, \mathbb{P})\). Indeed, in recent years there has been a lot of activity in understanding the possible role of random perturbations to improve the well-posedness of ordinary (or partial) differential equations (ODE/PDE) (see [21] for a recent review). This approach has, however, certain limitations:

With respect to point a) one can use other assumptions to justify specific choices. Within the class of time-dependent continuous random processes, for example, Brownian motion has suitable features of universality and Markovianity, making it a natural choice. Furthermore, a large set of theoretical tools is available to analyse the effect of Brownian perturbations to deterministic evolutions and this topic has a long and extensive literature [8, 15, 16, 23, 37, 58, 59]. Other classes of random perturbations, like fractional Brownian motion (fBm) have been more recently analysed, or even more exotic variants (e.g. \(\alpha\)-stable and log regular processes) [1, 6, 38, 45, 49, 18]. Let us finally mention the remarkable results from [10] concerning rates of convergence of numerical schemes for (1).

As for the technical limitations in point b), a possible solution is to modify the probabilistic setting in order to derive path-wise statements:

In this work we take a conceptually different approach and consider the regularisation by noise problem from the point of view of generic perturbations, in particular without reference to any (specific) probabilistic setting.

We will say that a property holds for almost every path \(w\) if it holds for a prevalent set of paths. Prevalence [46] is a notion of “Lebesgue measure zero sets” in infinite dimensional complete metric vector spaces. Such sets cannot be naively defined due to the fact that there cannot exist \(\sigma\)-additive, translation invariant measures in infinite dimensional spaces. It was first introduced by Christensen in [13] in the context of abelian Polish groups and later rediscovered independently by Hunt, Sauer and Yorke in [35] for complete metric vector spaces.

Prevalence has been used in different contexts in order to study the properties of generic functions belonging to spaces of suitable regularity. For instance, it was proved in [34] that almost every continuous function is nowhere differentiable, while in [26, 27] the multi-fractal nature of generic Sobolev functions was shown. Recently, prevalence has also attracted a lot of attention in the study of dimension of graphs and images of continuous functions, see among others [25, 7].

A key advantage of prevalence, with respect to other notions of genericity, is that it allows the use of probabilistic methods in the proof. However the statements are fully non-probabilistic and the kind of problems one encounters in formulating prevalence results are quite distinct from those of a purely probabilistic setting, extensively investigated in the probabilistic literature.

Armed with this “noiseless” notion of “almost every path”, we can already state informally one of the results of the paper as follows:

In order to proceed and precise the above claims we will need a suitable notion of solution to (1) which makes sense for distributional fields \(b\). The key observation in this direction comes from the work [12], which started the study of analytic properties of paths which affects the regularisation of ODEs.

In particular, the work [12] introduces the averaging operator \(T\) as a tool to study the regularisation properties of a path \(w\). It is the operator acting on time-dependent vector fields \(b\) and paths \(w\) as as

It is a linear operator in \(b\), so that one can fix \(w\) and consider the operator \(T^w : b \mapsto T^w b\) as above; in this case we say that \(T^w\) is the averaging operator associated to \(w\). Alternatively, one can fix \(b\) and vary \(w\), \(w \mapsto T^w b\); to stress the latter case, we say that \(T^w b\) is an averaged field.

Averaging is connected to an alternative formulation of the ODE via the theory of non-linear Young integration. Assume for the moment \(b : [0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d\) smooth and consider the ODE (1) in integral form

with \(w \in C ([0, T] ; \mathbb{R}^d)\). Then, this equation admits a unique solution of the form \(x = w + C^1\), in the sense that the difference \(x - w\) is a \(C^1\) path, regardless the regularity of \(w\). Applying the change of variables \(\theta := x - w\) we get the new integral equation

Since both \(b\) and \(\theta\) are continuous, the last integral can be approximated via Riemann–Stieltjes type sums as follows

where the limit is taken over all possible partitions \(\Pi = \{ t_0, \ldots, t_n \}\) with \(0 = t_0 < t_1 < \cdots < t_n = t\) with mesh \(| \Pi | = \sup_i | t_{i + 1} - t_i |\) converging to \(0\) and where for a function \(A = A (t, x)\) we adopt the compact notation \(A_{s, t} (x) := A (t, x) - A (s, x)\). The r.h.s. of equation (4) depends now on the averaged field \(T^w b\). The key observation of [12] is that, under suitable space-time regularity conditions on \(T^w b\), it is possible to show convergence of the above Riemann–Stieltjes type sums to a unique limit even when \(b\) is not continuous anymore, thus allowing to define the integral on the l.h.s. of (4) as their limit. This limit is called in [12] a non-linear Young integral and denoted as

Eq. (3) takes then the form of an integral equation involving non-linear Young integrals:

The analysis of such equations (existence, uniqueness, regularity of the flow) for irregular \(b\) depends essentially on the regularity properties of the averaged field \(T^w b\) and a substantial part of the present paper will be dedicated to analyse them in detail. For example we will prove that:

A quantitative version of the statements above, which collects some of the main results of this paper, is the following one.

Remark 2. In this theorem we could allow time dependent \(b \in L^{\infty}_t B^{\alpha}_{\infty, \infty}\) provided \(\delta < 1 / 2\). This is due to some technical limitations in the proof technique.

Let us point out that this results is the first general statement which supports the heuristics “the rougher the noise, the better the regularisation” observed in the probabilistic literature since e.g. [12] but so far never discussed abstracting from a particular probabilistic model of the perturbation.

We conclude this introduction by discussing possible extensions are relations with related work. The averaging operator \(T^w\) is, in many respect, a key tool introduced in [12] to study analytically the regularisation properties of perturbations in dynamical problems. In this paper we refrain to investigate more thoroughly this operator from the point of view of prevalence since this will be the main objective of the companion paper [29]. There we continue the study of the prevalent properties of path which are associated to the regularisation by noise phenomenon by concentrating on the notion of \(\rho\)-irregularity of a path, as introduced in [12], and the related notion of occupation measure, obtaining as a by-product information on the prevalent properties of \(T^w\).

The setting we propose in this paper opens up a completely new research subject with many natural problems, one prominent among them is to investigate the zero noise limit, that is the limit as \(\varepsilon \rightarrow 0\) for solutions to the equation \(\dot{x}_{\varepsilon} = b (x_{\varepsilon}) + \varepsilon w\). Already in the probabilistic setting this limit is not well understood, especially from the path-wise perspective and the dependence of the limit on the law assumed for \(w\) is not clear.

On a more technical level several improvement of our results could be possible. For example it would be interesting to obtain estimates for the averaging in \(L^p\)-based spaces with \(p \in [1, 2)\), see Remark 22 below and the related discussion in Appendix A.3. In particular let us note that the natural Conjecture 1.2 from [12] is still partially open; after the first draft of this work appeared, Nicolas Perkowski presented us a proof that answers negatively the conjecture in the case \(H < 1 / d\) for general \(d \geqslant 3\) and \(H \leqslant 1 / 2\) for \(d = 2\).

While we were finalizing the present paper, two related preprints appeared. Harang and Perkowski [32] study the flow of the ODE (1) perturbed with a Gaussian process very similar to that considered in [1] but from the pathwise point of view of [12]. Along the way they give proofs of some results on the flow of Young differential equations alternative to those we give below. In [2] Amine, Mansouri and Proske study with techniques very different from ours, the path-by-path uniqueness for transport equations driven by fBm with Hurst index \(H < 1 / 2\) and with bounded vector-fields. It is to be noted that while both works obtain interesting results, they still consider very specific probabilistic models. Therefore they are both far from the novel point of view we propose here and in the companion paper [29] and from the specific results it generates.

Let us finally mention the very recent work [31] in which Gerencser provides instances of regularisation by noise for \(w \in C^{\delta}\) with \(\delta > 1\).

Structure of the paper. We start by introducing the concept prevalence and its basic properties. Section 3 is devoted to the study of prevalence statements for averaged fields. Fractional Brownian motion (fBm) enters into the picture as a suitable transverse measure for prevalence. Thanks to a functional Ito–Tanaka type formula, we deduce regularity estimates for distributions averaged by fBm, which are strong enough to lead to prevalence statements. Section 4 is devoted to the application of the results from the previous section to perturbed ODEs via the theory of nonlinear Young integrals. After recalling and expanding the results from [12], we provide conditions (in terms of the regularity of \(T^w b\)) under the ODE admits a flow with prescribed regularity. Combined with Section 3, this allows to prove Theorem 1. Finally, we consider the case of perturbed transport type PDEs, for which it is again possible to establish well-posedness under suitable regularity conditions on \(T^w b\). We choose to put in the Appendix reminders of standard facts and certain technical results.

Acknowledgments. We thank Mark Veraar and Simone Floreani for a very useful discussion on integration in UMD Banach spaces.

Notation. We will use the notation \(a \lesssim b\) to mean that there exists a positive constant \(c\) such that \(a \leqslant c b\); we use the index \(a \lesssim_x b\) to highlight the dependence \(c = c (x)\). \(a \sim b\) if and only if \(a \lesssim b\) and \(b \lesssim a\), similarly for \(a \sim_x b\).

We will always work on a finite time interval \([0, T]\) unless stated otherwise. Whenever useful we adopt the convention that \(f_t\) stands for \(f (t)\) for a function \(f\) indexed on \(t \in [0, T]\), but depending on the context we will use both notations; similarly for the increments of \(f_{s, t} = f_t - f_s\).

For \(x \in \mathbb{R}^d\), \(| x |\) denotes the Euclidean norm, \(x \cdot y\) the scalar product. For any \(R > 0\), \(B_R\) stands for \(B (0, R) = \{ x \in \mathbb{R}^d : | x | \leqslant R \}\).

We denote by \(\mathcal{S} (\mathbb{R}^d ; \mathbb{R}^m)\) and \(\mathcal{S}' (\mathbb{R}^d ; \mathbb{R}^m)\) respectively the spaces of vector-valued Schwarz functions and tempered distributions on \(\mathbb{R}^d\); similarly \(C^{\infty}_c (\mathbb{R}^d ; \mathbb{R}^m)\) is the set of vector-valued smooth compactly supported functions.

Given a separable Banach space \(E\), we denote by \(L^q (0, T ; E) = L^q_t E\) the Bochner–Lebesgue space of \(E\)–valued measurable functions \(f : [0, T] \rightarrow E\) such that

with the essential supremum in the limit case \(q = \infty\). \(C^{\alpha} ([0, T] ; E) = C^{\alpha}_t E\) is the space of \(E\)–valued \(\alpha\)-Ho^¨lder continuous functions, for \(\alpha \in (0, 1)\), i.e. \(f : [0, T] \rightarrow E\) such that

A similar definition holds for \(\operatorname{Lip} ([0, T] ; E) =\operatorname{Lip}_t E\). More generally, for a given modulus of continuity \(\omega\) (possibly defined only in a neighbourhood of \(0\)), we denote by \(C^{\omega} ([0, T] ; E) = C^{\omega} E\) the set of all \(E\)-valued continuous functions with modulus of continuity \(\omega\), \(\| f \|_{C^{\omega} E}\) and \(\mathopen{\unicode{x27E6}} f \mathclose{\unicode{x27E7}}_{C^{\omega} E}\) defined as above.

Whenever \(E =\mathbb{R}^d\), we will refer to \(w \in C^{\alpha}_t = C^{\alpha} ([0, T] ; \mathbb{R}^d)\) as a path and in this case we allow \(\alpha \in [0, \infty)\) with the convention that \(w \in C_t^{\alpha}\) it is has continuous derivatives up to order \(\lfloor \alpha \rfloor\) and \(D^{\lfloor \alpha \rfloor} \varphi\) is \(\{ \alpha \}\)–Ho^¨lder continuous, where \(\lfloor \alpha \rfloor\) and \(\{ \alpha \}\) denote respectively integer and fractional part.

\(B^s_{p, q} (\mathbb{R}^d ; \mathbb{R}^m)\), \(L^{s, p} (\mathbb{R}^d ; \mathbb{R}^m)\) and \(F^s_{p, q} (\mathbb{R}^d ; \mathbb{R}^m)\) will denote respectively vector-valued Besov, Bessel potential and Triebel–Lizorkin spaces (see Appendix A.2), \(L^p (\mathbb{R}^d ; \mathbb{R}^m)\) standard Lebesgue spaces. Whenever it doesn't create confusion, we will just write \(B^s_{p, q}\), \(L^{s, p}\), \(F^s_{p, q}\) and \(L^p\) for short. For \(\alpha \in \mathbb{R} \setminus \mathbb{N}_0\), \(C^{\alpha} (\mathbb{R}^d ; \mathbb{R}^m) = C^{\alpha}_x = B^{\alpha}_{\infty, \infty}\); instead for \(\alpha \in \mathbb{N}_0\), \(C^n (\mathbb{R}^d ; \mathbb{R}^m) = C^n_x\) denotes the Banach space of all continuous functions with continuous derivatives up to order \(n\), endowed with the norm

Let us stress in particular that by saying that \(f \in C^n_x\), we are implying that we have a uniform bound on the whole \(\mathbb{R}^d\) for its derivatives. If instead we want to say that \(f\) has continuous derivatives up to order \(n\), we will write \(f \in C^n_{\operatorname{loc}}\). We will adopt short-hand notations of the form \(L^q_t L^p_x = L^q (0, T ; L^p (\mathbb{R}^d ; \mathbb{R}^m))\), \(C^{\alpha}_t C^{\beta}_x = C^{\alpha} ([0, T] ; C^{\beta} (\mathbb{R}^d ; \mathbb{R}^m))\).

Whenever a stochastic process \(X = (X_t)_{t \geqslant 0}\) is considered, even when it is not specified we imply the existence of an abstract underlying filtered probability space \((\Omega, \mathcal{F}, \{ \mathcal{F}_t \}_{t \geqslant 0}, \mathbb{P})\) such that \(\mathcal{F}\) and \(\mathcal{F}_t\) satisfy the usual assumptions and \(X_t\) is adapted to \(\mathcal{F}_t\). If \(\mathcal{F}_t\) is said to be the natural filtration generated by \(X\), then it is tacitly implied that it is actually its right continuous, normal augmentation. We denote by \(\mathbb{E}\) integration (equiv. expectation) w.r.t. the probability \(\mathbb{P}\).

2Prevalence

Here we follow the setting and the terminology given in [35] even if, for our purposes, we will be interested only in the case of a Banach space \(E\).

Sometimes it is said more informally that the measure \(\mu\) “witnesses” the prevalence of \(A^c\).

It follows immediately from part i. of the definition that, if needed, one can assume \(\mu\) to be a compactly supported probability measure on \(E\). If \(E\) is separable, then any probability measure on \(E\) is tight and therefore i. is automatically satisfied.

The following properties hold for prevalence (all proofs can be found in [35]):

1. If \(E\) is finite dimensional, then a set \(A\) is shy if and only if it has zero Lebesgue measure.

2. If \(A\) is shy, then so is \(v + A\) for any \(v \in E\).

3. Prevalent sets are dense.

4. If \(\dim (E) = + \infty\), then compact subsets of \(E\) are shy.

5. Countable union of shy sets is shy; conversely, countable intersection of prevalent sets is prevalent.

From now, whenever we say that a statement holds for a.e. \(v \in E\), we mean that the set of elements of \(E\) for which the statement holds is a prevalent set. Property 1. states that this convention is consistent with the finite dimensional case.

In the context of a function space \(E\), it is natural to consider as probability measure the law induced by an \(E\)-valued stochastic process. Namely, given a stochastic process \(W\) defined on a probability space \((\Omega, \mathcal{F}, \mathbb{P})\), taking values in a separable Banach space \(E\), in order to show that a property \(\mathcal{P}\) holds for a.e. \(f \in E\), it suffices to show that

Clearly, we are assuming that the set \(A = \{ w \in E : \text{$w$ satisfies property $\mathcal{P}$} \}\) is Borel measurable and if \(E\) is not separable, then we need to require in addition that the law of \(W\) is tight, so as to satisfy point i. of Definition 3.

As a consequence of properties 4. and 5., the set of all possible realisations of a probability measure on a separable Banach space is a shy set, as it is contained in a countable union of compact sets (this is true more in general for any tight measure on a Banach space). This highlights the difference between a statement of the form

and, for instance,

where this last statement corresponds to \(\mu (\text{Property $\mathcal{P}$ holds}) = 1\), \(\mu\) being the Wiener measure on \(C ([0, 1])\). Indeed, the second statement doesn't provide any information regarding whether the property might be prevalent or not. Intuitively, the elements satisfying a prevalence statement are “many more” than just the realisations of the Wiener measure.

3Averaging operators

We introduce in detail the averaging operator \((w, b) \mapsto T^w b\) and analyse its prevalent properties in various functional spaces. Fractional Brownian motion is used as a convenient tranverse measure to detect prevalent regularisation properties of paths.

3.1Definition of averaging operator and basic properties

In this section we provide the definition of the averaging operator \(T^w\) for measurable \(w : [0, T] \rightarrow \mathbb{R}^d\), together with some basic properties which will be fundamental for later sections and our first main prevalence result. Our definition is rather abstract and works for a general class of Banach spaces \(E\), but keep in mind that for our purposes \(E\) will always be either a Bessel space \(L^{s, p}\) or a Besov space \(B^s_{p, q}\) with \(p \in [2, \infty)\). Also, we consider for simplicity the scalar-valued case, i.e. \(E \subseteq \mathcal{S}' (\mathbb{R}^d)\). Everything generalises immediately to the vector-valued case \(\mathcal{S}' (\mathbb{R}^d ; \mathbb{R}^m)\) reasoning component by component.

Let us assume that \(E\) is a separable Banach space that continuously embeds into \(\mathcal{S}' (\mathbb{R}^d)\) (so that there is also a dual embedding \(\mathcal{S} (\mathbb{R}^d) \hookrightarrow E^{\ast}\)) such that translation \(\tau^v : f \mapsto \tau^v f = f \left( \cdot \, + v \right)\) act continuously on it and leave the norm invariant: \(\| \tau^v f \|_E = \| f \|_E\) for all \(v \in \mathbb{R}^d\) and \(f \in E\). Assume moreover that the map \(v \mapsto \tau^v\) is continuous in the sense that if \(v_n \rightarrow v\), then \(\tau^{v_n} f \rightarrow \tau^v f\) for all \(f \in E\).

The definition is meaningful, since by the continuity properties of \(v \mapsto \tau^v\), the map \(s \mapsto \tau^{w_s} b (s)\) is still measurable and by the invariance of \(\| \cdot \|_E\) under translations \(\| b \|_{L^1 E} = \| \tau^{w_{\cdot}} b \|_{L^1 E}\). Continuity of \(T^w b\) and the bound \(\| T^w b \|_{C^0 E} \leqslant \| b \|_{L^1 E}\) follow from standard properties of Bochner integral, as well as the linearity of the map \(b \mapsto T^w b\). Similarly, it is easy to see that, in the case \(b\) enjoys higher integrability, \(T^w\) can also be defines as a linear bounded operator from \(L^q_t E\) to \(C_t^{1 - 1 / q} E\). Furthermore, if \(w\) and \(\tilde{w}\) are such that \(w_t = \tilde{w}_t\) for Lebesgue-a.e. \(t \in [0, T]\), then \(T^w b\) and \(T^{\tilde{w}} b\) coincide for all \(b\), so that \({T^w} \) can be defined for \(w\) in an equivalence class.

Proof. We can assume in addition that \(w^n_t \rightarrow w_t\) for Lebesgue-a.e. \(t\), the general case following from applying the reasoning to any possible subsequence that can be extracted from \(\{ T^{w_n} b \}_n\). Since \(\tau^{w^n_t} b_t \rightarrow \tau^{w_t} b_t\) for Lebesgue-a.e. \(t\) and \(\| \tau^{w^n_t} b_t - \tau^{w_t} b_t \|^q \lesssim \| b_t \|^q \in L^1\), it follows from dominated convergence that

which gives the conclusion. \(\Box\)

The advantage of the above definition of \(T^w\) is that it is intrinsic and does not depend on any approximation procedure by mollifiers. However, a possibly more intuitive description of \(T^w b\) can be given by duality. Recall that in the sense of distributions \((\tau^v)^{\ast} = \tau^{- v}\), so that for any \(\varphi \in \mathcal{S} (\mathbb{R}^d) \hookrightarrow E^{\ast}\) it holds

where the pairing is integrable since \(\left| \langle b_s, \varphi \left( \cdot \, - w_s \right) \rangle \right| \lesssim_{\varphi} \| b_s \|_E\). The above relation holds for all \(\varphi \in \mathcal{S} (\mathbb{R}^d)\) and therefore uniquely identifies \(T^w b (t)\) as an element of \(\mathcal{S}' (\mathbb{R}^d)\), for all \(t \in [0, T]\). The advantage now is that the map \((t, x) \mapsto \varphi (x - w_t)\) can be regarded as an element of \(L^{\infty} (0, T ; \mathcal{S} (\mathbb{R}^d))\), to which standard operations on \(\mathcal{S} (\mathbb{R}^d)\) such as differentiation and convolution can be applied.

Proof. Both statements follow easily from the duality formulation. For any \(\varphi \in \mathcal{S} (\mathbb{R}^d)\) and \(t \in [0, T]\) it holds

If \(K \in C^{\infty}_c (\mathbb{R}^d)\), then denoting by \(\tilde{K}\) its reflection, by duality it holds

A similar computation shows the other part of the identity.\(\Box\)

Remark 7. Let us point out that if \(w \in L^{\infty}\), then the averaging operator has finite speed of propagation and so behaves well under localisation. Indeed, if \(b \in L^1_t E\) is such that \(\operatorname{supp}b_t \subset B_R\) for all \(t \in [0, T]\), then \(\operatorname{supp}T^w b (t) \subset B_{R + \| w \|_{\infty}}\) for all \(t \in [0, T]\) and similarly if \(b\) and \(\tilde{b}\) are such that their restrictions to \(B_R\) coincide for all \(t\), then \(T^w b\) and \(T^w \tilde{b}\) will still coincide on \(B_{R - \| w \|_{\infty}}\).

In view of the applications in Section 4, our main goal is to establish conditions under which \(T^w b \in C^{\gamma}_t F\), where \(\gamma > 1 / 2\) and \(F\) is another Banach space which enjoys better regularity properties than the original space \(E\): typically \(F = C^{\beta}_x\) for suitable values of \(\beta\). For this reason, we are going to assume from now on that \(b \in L^q_t E\) for some \(q > 2\). The idea behind this restriction is that sometimes averaging allows to trade off time regularity for space regularity (think of the analogy with parabolic regularity theory) and therefore in order to have \(T^w b \in C^{\gamma}_t F\), knowing a priori only that \(T^w b \in C^{1 - 1 / q}_t E\), we need to require at least

Remark 8. Despite our use of the terminology “regularisation by averaging”, what we mean is really that we fix a drift \(b\) and we want to establish that for a.e. path \(w\) the averaged function \(T^w b\) has nice regularity properties. This is different from trying to establish that the averaging operator \(T^w\) as a linear operator from \(L^q_t E\) to \(C^{\gamma}_t F\) is bounded, which is clearly false due to the time dependence of the drifts we consider. Indeed, given any \(b \in E\), defining \(\tilde{b}_t = \tau^{- w_t} b\), by definition of averaging we obtain \(T_{s, t}^w \tilde{b} = (t - s) b\), which shows that for such choice of \(\tilde{b}\), \(T^w \tilde{b}\) cannot have better spatial regularity than \(\tilde{b}\). The situation is more interesting if one defines \(T^w\) for time independent drifts only. Prevalence statements for that case will be analysed in the companion paper [29].

In order to show prevalence of regularisation by averaging, we first need to show that such a property indeed defines Borel sets in suitable spaces of paths. To this end, we require \(F\) to be another Banach spaces which embeds into \(\mathcal{S}' (\mathbb{R}^d)\) which enjoys the following Fatou type property: if \(\{ x_n \}_n\) is a bounded sequence in \(F\) such that \(x_n\) converge to \(x\) in the sense of distributions, then \(x \in F\) and \(\| x \|_F \leqslant \liminf \| x_n \|_F\).

In the next lemma we allow any \(\bar{\gamma} \in (0, 1)\), but our primary focus will be \(\bar{\gamma} = 1 / 2\).

Remark 10. Any weakly-\(\ast\) compact Banach space \(F\) which embeds in \(\mathcal{S}' (\mathbb{R}^d)\) satisfies the Fatou property. In the following we will always work with \(L^p_x\)-based function spaces with \(p \in [2, \infty]\), so the property holds automatically. Let us also point out that the proof actually works more generally for conditions of the form \(T^w b \in C^{\omega}_t F\), where \(\omega\) is a prescribed modulus of continuity.

We are now ready to provide a first prevalence statement.

Proof of Theorem 11. By Lemma 9, the set

is Borel in \(C^{\delta}_t\). For simplicity we will adopt the notation \(b \in L^{\alpha}_t E\), as the reasoning is the same for \(E = L^{s, p}_x\) or \(E = B^s_{p, q}\). In order to prove the statement, it remains to find a suitable tight probability distribution \(\mu\) on \(C^{\delta}_t\) such that for any \(\varphi \in C^{\delta}_t\) it holds

Thanks to the translation invariance of \(\| \cdot \|_E\), we can reduce the above problem to an easier one. Indeed, setting \(\tilde{b}_t := \tau^{\varphi_t} b_t\) for all \(t \in [0, T]\), \(\tilde{b} \in L^{\alpha}_t E\) and it holds \(T^{\varphi + w} b = T^w \tilde{b}\). In particular in order to show that (6) holds for fixed \(b \in L^{\alpha}_t E\) and for all \(\varphi \in C^{\delta}_t\), it actually suffices to find \(\mu\) such that

Considering equation (7) for the choice \(E = L^{s, p}\) (resp. \(E = B^s_{p, q}\)), it suffices to show that for all \(\beta\) satisfying (5) there exists a tight measure \(\mu_{\beta, \delta}\) on \(C^{\delta}_t\) such that

The rest of the section will be devoted to the identification of such a measure. In particular, using Theorem 15 (resp. Theorem 19) combined with Remark 24 below, we can choose \(\mu_{\beta, \delta} = \mu^H\) to be the law of a fractional Brownian motion of parameter \(H \in (0, 1)\) such that \(H > \delta\) and

We conclude this section with a lemma on approximation by mollifications which will be very useful in Section 4.

Proof. It follows immediately from the property \((T^w b)^{\varepsilon} = T^w b^{\varepsilon}\) that

and moreover that \(T^w b^{\varepsilon} (t) \rightarrow T^w b (t)\) in \(\mathcal{S}' (\mathbb{R}^d)\) as \(\varepsilon \rightarrow 0\). For any \(R > 0\) and \(\delta > 0\), thanks to the above uniform bound, we can extract by Ascoli-Arzelà a (not relabelled) subsequence such that \(T^w b^{\varepsilon} \left|_{[0, T] \times B_{_R}} \right.\) converges in \(C^{\gamma - \delta} ([0, T] ; C^{\beta - \delta} (B_R))\) to a suitable limit; by the above convergence in probability, the limit must necessarily coincide with \(T^w b \left|_{[0, T] \times B_{_R}} \right.\) and since the reasoning holds for any subsequence we can extract, the whole sequence must converge to \(T^w b \left|_{[0, T] \times B_{_R}} \right.\).\(\Box\)

3.2Fractional Brownian motion and Ito^{^}–Tanaka formula

In view of concluding the proof of Theorem 11 we give here the essential details on the fractional Brownian motion (fBm), whose law will be used as a transverse measure for prevalence.

In the literature, it is more common the use of probes, that is finite dimensional transverse measures in order to establish prevalence properties. The only other work we are aware of using general stochastic processes in this context is [7]. However see also [47] and the references therein for the study of properties of fractional Brownian motion with deterministic drift.

The material on fractional Brownian motion presented here is classical and taken from [44] and [48]. A one dimensional fBm \((W^H_t)_{t \geqslant 0}\) of Hurst parameter \(H \in (0, 1)\) is a mean zero continuous Gaussian process with covariance

When \(H = 1 / 2\), it coincides with standard Brownian motion and for \(H \neq 1 / 2\) it is not a semi-martingale nor a Markov process. However it shares many properties of Brownian motion, such as stationarity, reflexivity and self-similarity. The trajectories of fBm are \(\mathbb{P}\)-a.s. \(\delta\)-Hölder continuous for any \(\delta < H\) and nowhere \(\delta\)-Hölder continuous for any \(\delta \geqslant H\); it follows from Ascoli–Arzelà that its law \(\mu^H\) is tight on \(C^{\delta}_t\) for any \(\delta < H\).

A \(d\)-dimensional fBm \(W^H\) of Hurst parameter \(H \in (0, 1)\) is an \(\mathbb{R}^d\)-valued Gaussian process with components given by independent one dimensional fBms; we state for simplicity in the rest of the section all the results for \(d = 1\) but they generalise immediately to higher dimension reasoning component by component.

A very useful property of fBm is that it admits representations in terms of stochastic integrals. Given a two-sided Brownian motion \(\{ B_t \}_{t \in \mathbb{R}}\), a fBm of parameter \(H \neq 1 / 2\) can be constructed by

where \(c_H = \Gamma (H + 1 / 2)^{- 1}\) is a suitable renormalising constant. Such a representation is usually called non canonical as the filtration \(\mathcal{F}_t = \sigma (B_s : s \leqslant t)\) is strictly larger than the one generated by \(W^H\); it is useful as it immediately shows that, for any pair \(0 \leqslant s < t\), the variable \(W^H_t\) decomposes into the sum of two mean zero Gaussian variables, \(W^H_t = W^{1, H}_{s, t} + W^{2, H}_{s, t}\), where

with \(W^{2, H}_{s, t}\) being \(\mathcal{F}_s\)-measurable and \(W^{1, H}_{s, t}\) being independent of \(\mathcal{F}_s\) and with variance

where \(\tilde{c}_H = c_H^2 / (2 H)\). In particular this implies that

which is a local nondeterminism property. Loosely speaking, it means that for any \(s < t\), the increment \(W_t^H - W^H_s\) contains a part which is independent of the the history of the path \(W^H_{\cdot}\) up to time \(s\) and therefore makes the path \(W^H_{\cdot}\) “intrinsically chaotic”. The local nondeterminism property was first formulated by Berman in [9] in a different context; it plays a major role in the proofs of this section and indeed the prevalence statement can be alternatively proved by using the laws of other locally nondeterministic Gaussian processes, see Remark 23.

We are going to prove an Ito^{^}–Tanaka type formula for averaged functionals, in the same spirit of the one considered in [14]. We first need to recall the Clark–Ocone formula, see [44]. Given a two-sided standard Brownian motion \(B\) on a space \((\Omega, \mathcal{F}, \mathbb{P})\), \(\mathcal{F}_t = \sigma (B_s, s \leqslant t)\), and given a Malliavin differentiable random variable \(A\) with Malliavin derivative \(D_{\cdot} A\), the Clark–Ocone formula states that

From (11) it follows immediately that, for any \(s \in \mathbb{R}\), we have the more general identity

We do not provide here the general definition of Malliavin derivative of a Brownian variable, which can be found in [44]; we only provide it in the following specific case, which is the one of our interest: given a smooth function \(f\) and a variable \(X = \int_{- \infty}^{+ \infty} K_s \, \mathrm{d} B_s\), the Malliavin derivative of \(A : = f (X)\) is given by

In the next statement, \(P_t\) denotes the heat kernel, i.e. \(P_t f = p_t \ast f\) where

Proof. For \(H = 1 / 2\) the above formula is well known and coincides with a standard application of the Ito^{^}–Tanaka trick together with a representation formula for solution of the heat equation, see for instance the discussion in [14]; so we can assume \(H \neq 1 / 2\). Let us fix \(x \in \mathbb{R}^d\). Since \(b\) is smooth, for fixed \(r\) we can apply Clark–Ocone formula to \(b (r, x + W^H_r)\) to obtain

where we used both the representation of \(W^H\) in terms of a stochastic integral and the decomposition \(W^H_r = W^{1, H}_{u, r} + W^{2, H}_{u, r}\) with \(W^{1, H}_{u, r}\) independent of \(\mathcal{F}_u\). Integrating over \([s, t]\) and applying stochastic Fubini's theorem (which is allowed since we are assuming \(b\) smooth and compactly supported) we obtain

which gives the conclusion.\(\Box\)

The previous result can be strengthened by considering for instance \(b \in C_b^1\) instead of smooth, or showing that we can find a set of probability \(1\) on which the identity holds for all \(0 \leqslant s \leqslant t \leqslant T\); we don't do it here since it is not needed for our purposes. Instead, we need to strengthen the result to the following functional equality.

We postpone the proof of this result to Appendix A.3, as it is quite technical and requires some knowledge of stochastic integration in UMD spaces. Up to technical details, it is mostly a rewriting of the statement already contained in Lemma 13 without further insights.

3.3Regularity estimates in Bessel and Besov spaces

We provide here the regularity estimates for \(T^w b\) when \(w\) is sampled as a fBm of parameter \(H\), in view of establishing (8).

The main ingredients of the proof are the use of the functional Ito^{^}–Tanaka formula (14) together with Burkholder's inequality (Theorem 87 below), heat kernel and interpolation estimates from Lemmata 84 and 83. We refer the reader to Appendices A.2 and A.3 for more information on these tools. Let us point out that the strategy of proof is fairly general and in principle could work also in other classes of spaces, up to the requirement that the above tools are still available. However, in order to apply Burkholder's inequality, we need to restrict to scales of \(L^p\)-based spaces with \(p \geqslant 2\). See Appendix A.3 for a deeper discussion of this point.

Although our main aim is to establish prevalence results, our results are also new in the probabilistic setting and therefore we will try to give their sharpest versions. In particular we will always achieve exponential integrability whenever it is possible.

Proof. Without loss of generality, we can assume \(s = 0\). Indeed, if \(b \in L^q_t L^{s, p}_x\), then \(b = G^s \, \tilde{b}\), where \(\tilde{b} \in L^q_t L^p_x\) and \(\| b \|_{L^q L^{s, p}} = \| \tilde{b} \|_{L^q L^p}\); once the statement is shown for \(\tilde{b}\), we can use the fact the commutating property of averaging operators \(T^{W^H} b = T^{W^H} \left( G^s \, \tilde{b} \right) = G^s \, (T^{W^H} \tilde{b})\) to obtain the analogue statement for \(b\) as well.

Let us first assume \(b\) to be a smooth function. By the Ito–Tanaka formula,

From now on for simplicity we will drop the constants \(c_H\), \(\tilde{c}_H\) as they don't play any significant role in the following calculations. For the first term, we can apply the deterministic estimate:

where we used the heat kernel estimates for Bessel spaces, see Lemma 84, and the fact that the \(L^{\rho, p}\)-norm of \(b_r\) is not affected by a translation of \(W^{2, H}_{r, s}\). Observe that \(\rho H \, q' < 1\) is granted by condition (15). Moreover, (15) implies that \(1 - 1 / q - \rho H > 1 / 2\) and therefore we deduce that there exists \(\gamma > 1 / 2\) such that, uniformly in \(\omega \in \Omega\),

For the second term, applying Burkholder's inequality (88) (which is allowed since \(L^{\rho, p}\) with \(p \geqslant 2\) is a martingale type 2 space), we obtain

We can then estimate the inner integral by deterministic estimates similar to the ones above:

where again we used the fact that \((H \rho + 1 / 2) q' < 1\), thanks to (15). Set \(\varepsilon := 1 - 2 / q - 2 H \rho\); inserting the estimate inside (18) we obtain that, for a suitable \(C' > 0\), it holds

But then we have

as soon as \(\lambda < (C' e)^{- 1}\). It follows from Lemma 75 that, for any \(\varepsilon' < \varepsilon\), \(I^{(2)} \in C_t^{1 / 2 + \varepsilon'} L_x^{\alpha, p}\) and that there exists another \(\lambda > 0\) (not relabelled for simplicity) such that

for a constant \(K\) independent of \(b\), which together with (17) proves the claim for smooth \(b\).

Now let \(b\) be a generic element of\(L^q_t L^p_x\); let us consider the case \(q < \infty\) first. We can then find a sequence \(b_n\) of smooth functions such that \(\| b - b_n \|_{L^q L^p} \rightarrow 0\) as \(n \rightarrow \infty\); we know that in this case \(\| T^{W^H} (b_n - b) \|_{C^0 L^p} \rightarrow 0\), uniformly on \(\omega \in \Omega\). On the other hand, it follows from (16), applied to \(b_n - b_m\), that for any \(k\) it holds

which implies that the sequence \(T^{W^H} b_n\) is Cauchy in \(L^{2 k} (\Omega, \mathbb{P}; C_t^{\gamma} L_x^{\rho, p})\), hence it admits a limit. But then the limit must coincide with \(T^{W^H} b \in L^{2 k} (\Omega, \mathbb{P}; C_t^{\gamma} L_x^{\rho, p})\). Applying Fatou lemma we deduce

which gives the conclusion. In the case \(q = \infty\), since \(b \in L^{q'}_t L^p_x\) for every \(q' < \infty\), for any fixed \(\rho\) we can find \(q'\) big enough such that (15) still holds and apply the result for such \(q'\).\(\Box\)

Remark 16. Theorem 15 immediately implies that, under assumption (15), the random averaging operator \(T^{W^H} : b \mapsto T^w b\) is a linear bounded map from \(L^q_t L^{s, p}_x\) into \(L^k (\Omega, \mathbb{P}; C^{\gamma}_t L^{s + \rho, p}_x)\), for any \(k \in \mathbb{N}\). Observe the difference with Remark 8.

We can actually even improve the regularity result of Theorem 15.

Proof. As before, we can assume without loss of generality \(s = 0\). If \(\rho\) satisfies (15), then there exists \(\varepsilon > 0\) such that also \(\rho + \varepsilon\) satisfies (15); it then follows from Lemma 83 that

where \(\theta = \varepsilon / (s + \varepsilon)\) and we used the fact that \(q > 2\) due to condition (15). It follows that

where \(1 / \theta = (s + \varepsilon) / \varepsilon =: \beta\). Applying Theorem 15 to \(\rho + \varepsilon\), we obtain that there exist \(\bar{\lambda}\), \(\bar{K}\) independent of \(b\) such that

Since \(\beta > 1\), the conclusion follows with the constant \(K (\lambda)\) given by the optimal deterministic constant such that \(\exp (\lambda x^2) \leqslant K (\lambda) \exp (\bar{\lambda} x^{2 \beta}) / \bar{K}\) for all \(x \geqslant 0\).\(\Box\)

In the limiting case in which (15) becomes an equality, slightly more careful estimates still allow to obtain a regularity result in space at the cost of lower time regularity.

Proof. As before, we can assume \(s = 0\), \(b\) smooth; again we decompose \(T^{W^H} b = I^{(1)} + I^{(2)}\). Going through the same calculations for \(I^{(1)}\), we obtain

where the estimate is uniform in \(\omega \in \Omega\); it follows immediately that

and therefore we only need to focus on \(I^{(2)}\). By Burkholder's inequality, we have

and as before we want to estimate the integral inside in a deterministic manner. Going through similar calculations we obtain

and now due to the assumption on the coefficients, we can apply the Hardy-Littlewood-Sobolev inequality to obtain

which implies

The conclusion then follows by expanding the exponential and choosing \(\lambda\) sufficiently small as before.\(\Box\)

Going through the exact same calculations as above, an analogue result can be obtained in the case of Besov spaces \(B^s_{p, q}\) with \(p, q \in [2, \infty)\). In order to avoid unnecessary repetitions, we omit the proof.

We end this section with several remarks discussing various technical point and extensions, and which can be skipped on a first reading.

Remark 20. Heuristically, condition (21) can be seen as a time-space weighted regularity condition, where time counts as \(1 / H\) times space (which is in agreement with parabolic regularity in the case \(H = 1 / 2\) of Brownian motion). Indeed, we know that the averaging operator \(T^w\) maps \(L^{\alpha} B^s_{p, q}\) into \(W^{1, \alpha} B^s_{p, q}\); if we assume that regularity can be distributed between time and space, it should also map \(L^{\alpha} B^s_{p, q}\) into \(W^{\theta, \alpha} B^{s + (1 - \theta) / H}_{p, q}\) for any \(\theta \in (0, 1)\). In order to achieve \(1 / 2 + \varepsilon\) regularity in time it is then required \(\theta - 1 / \alpha > 1 / 2\), which implies that the regularity gain in space is at most

which matches exactly condition (21) for \(\rho\).

Remark 21. The restriction to work with \(B^s_{p, q}\) with \(q \in [2, \infty)\), is not particularly relevant since by Besov embedding if \(b \in L^{\alpha}_t B^s_{p, q}\), then it also belongs to \(L^{\alpha}_t B^s_{p, q'}\) for any \(q' > q\) and to \(L^{\alpha}_t B^{s - \varepsilon}_{p, q'}\) for any \(q' < q\) and \(\varepsilon > 0\), so that we can first embed it for a choice \(q' \in [2, \infty)\) and then apply the estimate there. Also the restriction \(p \neq \infty\) can be overcome, for instance by first localising it as \(b\)^~ in a ball \(B_R\) and then embedding it into some \(p < \infty\); by the properties of averaging, we know that \(T^{W^H} b = T^{W^H} \tilde{b}\) in \(B_{R - \| W^H \|_{\infty}}\) and we can choose \(R\) big enough such that \(\mathbb{P} (R - \| W^H \|_{\infty} < R / 2)\) is very small, to deduce local estimates for \(T^{W^H} b\) which hold with high probability. Alternatively, estimates for averaging in Besov-Ho^¨lder spaces have been given by a different technique in [12], Section 4.1. However for simplicity, when dealing with \(b \in L^{\alpha}_t B^s_{\infty, \infty}\), we will always assume that \(b\) has compact support in space, uniformly in time, so that we can embed it in \(L^{\alpha}_t B^s_{p, p}\) for any \(p < \infty\) and then apply estimates there.

Remark 22. The restriction to work with \(L^p\)-based spaces with \(p \geqslant 2\) is more restrictive and it would be of fundamental importance to weaken it, especially reaching the case \(p = 1\); this was already pointed out in Conjecture 1.2 from [12]. The reason is that, by the properties of averaging, we know that for any \(K \in C^{\infty}_c\) and time independent \(b\) it holds \(K \ast T^w b = T^w (K \ast b) = (T^w K) \ast b\); if we were able to show that \(T^w K \in C^{\gamma}_t W^{\rho, 1}_x\) with an estimate that only depends on the \(L^1\)-norm of \(K\), then we could automatically deduce regularity estimates of the form \(K \ast T^w b \in C^{\gamma}_t L^{\rho, p}\) with \(b \in L^p\) for any \(p \in [1, \infty]\). We could then consider a family of mollifiers obtained by rescaling \(K\) (which all have the same \(L^1\)-norm, so the same estimate in \(C^{\gamma}_t W^{\rho, 1}_x\)) to get estimates for the map \(b \mapsto T^w b\) in any \(L^p\) based space with \(p \in [1, \infty]\) (as above, only time independent \(b\) considered).

Remark 23. A closer look at the proofs shows that both the Ito^{^}–Tanaka formula from Theorem 14 and the regularity estimates from Theorems 15 and 19 can be generalised to Gaussian processes \(X\) different from fBm and of the form

for some deterministic matrix-valued function \(K\), such that for some \(H \in (0, 1)\) it holds

where \(\mathcal{F}_t = \sigma (B_s : s \leqslant t)\). Condition (23) is a type of strong local nondeterminism (SLND) and these type of processes satisfy many interesting properties, which are studied in detail in [29].

Remark 24. It follows immediately from the above results and from Bessel (resp. Besov) embeddings (see Appendix A.2) that if \(b \in L^{\alpha}_t L^{s, p}_x\) (resp. \(b \in L^{\alpha}_t B^s_{p, q}\)) for some \(\alpha > 2\), \(p, q \in [2, \infty)\), then for any \(\beta\) such that

there exists \(\gamma > 1 / 2\) such that \(T^{W^H} b \in C^{\gamma}_t C^{\beta}_x\) with full probability. For instance in the case \(s = 0\), i.e. \(b \in L^{\alpha}_t L^p_x\), in order to require \(T^{W^H} b \in C^{\gamma}_t C^0_x\) it is enough

while in order to require \(T^{W^H} b \in C^{\gamma}_t C^1_x\) it suffices

If \(b \in L^{\alpha}_t B^s_{\infty, \infty}\) with spatially compact support, uniform in time, then \(T^{W^H} b \in C^{\gamma}_t C^n_x\) if

Remark 25. Finally, let us compare our results for \(T^{W^H} b\) with existing literature; with the exception of the case \(H = 1 / 2\), in which classical stochastic calculus provides more refined information, the only references we are aware of are the aforementioned [12, 38]. The technique applied in [12] allows to deal only with time independent \(b\); however, introducing suitable weighted spaces, it does not require \(b\) to belong to \(B^s_{p, p}\) for some \(p < \infty\). The results from Section 7 of [38], where \(b \in L^q_t B^s_{\infty, \infty}\) is considered, are in line with those from Remark 24; still, the techniques used therein, based on moment estimates and Garsia-Rodemich-Rumsay lemma, do not provide global regularity estimates for \(T^{W^H} b\) (only local ones) nor the exponential integrability (16). Both such features will be fundamental in the solution theory presented the next section: global estimates avoid finite time blow-up of solutions, exponential integrability allows the use of Girsanov's theorem. Finally, let us point out that both references only provide estimates for \(T_{s, t}^{W^H} b\) in \(B^s_{\infty, \infty}\), not covering other scales \(B^s_{p, q}\) with \(p, q < \infty\).

4Application to perturbed ODEs

Now we are going to transfer the prevalence results for the averaged vector-field to prevalence of well-posedness to perturbed ODEs including regularity of the flow. The key technical tool to achieve this connection is a simple theory of nonlinear Young equations which we recall and adapt to our specific setting.

4.1Perturbed ODEs as nonlinear Young differential equations

In this section we provide a summary of the results contained in [12] on nonlinear Young differential equations (YDEs). Sometimes we will provide slightly different statements which fit better our context and in order to facilitate the understanding we will provide self-contained proofs whenever possible.

Let us fix some notation first. Given \(A \in C^{\gamma}_t C^{\nu}_x = C^{\gamma} ([0, T] ; C^{\nu} (\mathbb{R}^d ; \mathbb{R}^d))\) for \(\gamma, \nu \in (0, 1)\), we denote by the norm \(\| A \|_{C^{\gamma} C^{\nu}}\) and the semi-norm \(\mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma} C^{\nu}}\) respectively the quantities

and

One of the main results of [12] is the rigorous construction of the nonlinear Young integral.

The statement is a (less general) version of Theorem 2.4 from [12]; we omit the proof, but let us mention that an elementary proof based on the Sewing Lemma has been also given in [33]. The statement above can be localised, i.e. it is enough to require \(A \in C^{\gamma}_t C^{\nu}_{\operatorname{loc}}\) and in this case all the estimates depend on the \(C^{\gamma}_t C^{\nu}_x\)-norm (resp. semi-norm) of \(A\) restricted to \([0, T] \times B_{\| \theta \|_{\infty}}\).

With this tool at hand, we can provide an alternative definition of solutions to the perturbed ODE which is meaningful even when \(b\) is distributional in space. Since we want to apply the results from Section 3, from now on when we say that \(b\) is distributional we are always going to implicitly assume that there exists \(q > 2\) such that \(b \in L^q_t E\), where \(E\) is a suitable space of distributions as the ones described in Section 3.1.

Observe that the condition \(\gamma (1 + \nu) > 1\) immediately implies \(\gamma > 1 / 2\), in line with standard Young differential equations; in the case of continuous \(b\), it follows from the discussion in the introduction that the condition \(x \in w + C^{\gamma}\) is trivially satisfied and so the two formulations (25) and (26) are equivalent, (25) being interpreted as the classical integral equation.

Remark 28. From now on we will mostly focus on solving (26) with \(T^w b = A\) being regarded as an abstract element in a class \(C^{\gamma}_t C^{\nu}_x\); however, whenever \(b\) is spatially bounded, the ODE formulation for \(\theta\) is still useful, as it provides additional regularity estimates for \(\theta\) compared to the ones given by the Young integral formulation: for instance if \(b \in L^{\infty}_{t, x}\), then any solution \(\theta\) of the integral equation is automatically Lipschitz with \(\mathopen{\unicode{x27E6}} \theta \mathclose{\unicode{x27E7}}_{\operatorname{Lip}} \leqslant \| b \|_{L^{\infty}}\), while Point 3. of Theorem 26 only provides estimate for \(\| \theta \|_{C^{\gamma}}\), where \(\gamma < 1\) (usually we will take \(\gamma\) as small as possible, namely \(\gamma \sim 1 / 2\)).

Proof. The existence of solutions is granted under milder conditions on \(T^w b\) by Theorem 2.9 from [12], so here we only show the a-priori estimates. Let \(\theta \in C^{\gamma}\) be a solution and for any \(\Delta > 0\) define the semi-norm

Let \(\Delta\) be a parameter to be fixed later; for any \(s < t\) such that \(| s - t | \leqslant \Delta\) it holds

where in the last passage to used the trivial inequality \(a^{\nu} \leqslant 1 + a\) for all \(a \geqslant 0\) and \(\nu \in (0, 1]\). Dividing both sides by \(| t - s |^{\gamma}\) and taking the supremum \(s\), \(t\) such that \(| s - t | \leqslant \Delta\) we get

Choosing \(\Delta\) small enough such that \(C \Delta^{\gamma \nu} \| T^w b \|_{C^{\gamma} C^{\nu}} \leqslant 1 / 2\), we obtain

If we can take \(\Delta = T\), this provides an estimate for \(\mathopen{\unicode{x27E6}} \theta \mathclose{\unicode{x27E7}}_{C^{\gamma}}\), which together with \(\| \theta \|_{C^0} \leqslant | \theta_0 | + T^{\gamma} \mathopen{\unicode{x27E6}} \theta \mathclose{\unicode{x27E7}}_{C^{\gamma}}\) gives the conclusion. If this is not the case, we can choose \(\Delta\) as above such that in addition \(C \Delta^{\gamma \nu} \| T^w b \| \geqslant 1 / 4\) and then by the simple inequality (see for instance Exercise 4.24 from [28])

It follows that

where in the last line we used the fact that \(\gamma (1 + \nu) > 1\) implies \((1 - \gamma) / (\gamma \nu) < 1\). The conclusion again follows by the standard inequality \(\| \theta \|_{C^{\gamma}} \lesssim | \theta_0 | + T^{\gamma} \mathopen{\unicode{x27E6}} \theta \mathclose{\unicode{x27E7}}_{C^{\gamma}}\).\(\Box\)

Given that in general we consider \(\gamma\) to be very close to \(1 / 2\), in order to have existence in general we need \(\nu\) to be arbitrarily close to \(1\), thus we will usually require directly \(T^w b \in C^{\gamma}_t \operatorname{Lip}_x\) (with the quantities \(\| T^w b \|_{C^{\gamma} \operatorname{Lip}}\) and \(\mathopen{\unicode{x27E6}} T^w b \mathclose{\unicode{x27E7}}_{C^{\gamma} \operatorname{Lip}}\) defined as above).

To establish uniqueness of solutions, we need the following lemma of independent interest.

Proof. Suppose first that in addition \(\partial_t A \in C^0_t C^2_x\), then by Taylor expansion

where all manipulations in this case are allowed by the properties of Young integral and the fact that we are assuming \(A\) regular; in particular by hypothesis \(\nabla A \in C^{\gamma}_t C^{\nu}_x\) and \(\theta^i \in C^{\rho}_t\) with \(\gamma + \nu \rho > 1\), so the interpretation of the integrals as nonlinear Young integrals is legit. Observe that the map \(A \mapsto V (A)\) is linear by construction and we have the estimate

which gives the conclusion in this case. The general case follows by approximation, considering a sequence of regular \(A^n \rightarrow A\) locally in \(C^{\gamma - \delta}_t C^{1 + \nu - \delta}_x\), on a ball of radius \(R > \| \theta^i \|_{\infty}\), for \(\delta\) small enough such that \(\gamma - \delta + (\nu - \delta) \rho > 1\).\(\Box\)

With the above lemma at hand, we can provide a comparison principle, which estimates the difference between solutions. It comes in two versions, which apply to different scenarios.

Proof. We show in detail the derivation of (29) and briefly sketch the one of (30) as the structure of the proof is the same. By the assumptions and Lemma 30 applied to \(A = T^w b^1\), which is allowed for the choice \(\nu = 1\), \(\rho = \gamma\), the difference \(v = \theta^1 - \theta^2\) satisfies

This is a linear Young differential equation, for which standard estimates are available; Theorem 29, Lemma 30 and properties of nonlinear Young integral provide

where the constants appearing all depend on \(\gamma, T, R\). Combining this estimates with Lemma 76 from Appendix A.1 yields the conclusion.

The proof in the second case is analogue, but we have the additional estimate \(\mathopen{\unicode{x27E6}} \theta^i \mathclose{\unicode{x27E7}}_{\operatorname{Lip}} \leqslant \| b^i \|_{L^{\infty}}\) coming from the ODE integral interpretation of the YDE and so we can apply as above Lemma 30 to \(T^w b^1\) this time for the choice \(\nu = 1 / 2\), \(\rho = 1\).\(\Box\)

Remark 32. It follows immediately from the above result that if \(T^w b \in C^{\gamma}_t C_x^2\) or \(b \in L^{\infty}_{t, x}\) and \(T^w b \in C^{\gamma}_t C^{3 / 2}_x\), then for any \(\theta_0 \in \mathbb{R}^d\) there exists a unique solution to the YDE (26) and moreover the solution map \(\theta_0 \mapsto \theta_{\cdot}\) is Lipschitz continuous w.r.t \(\theta_0\); the solution constructed this way is also stable under approximation of \(T^w b\) by other drifts \(T^w \tilde{b}\), which can be combined with Lemma 12, as we can take \(\tilde{b} = b^{\varepsilon} = \rho^{\varepsilon} \ast b\) for some spatial mollifier \(\rho^{\varepsilon}\).

The above version of the Comparison Principle is of straightforward application, as it only requires good regularity estimates on \(T^w b\). The next version is instead slightly more subtle and can be regarded as a conditional Comparison Principle, as it allows to deduce estimates under less regularity on \(T^w b\) imposing the existence of a solution with suitable properties; however, the existence of such solutions is not granted a priori by the deterministic theory and in order to construct them probabilistic tools will be needed, specifically Girsanov transform.

The proof requires the following technical lemma.

Proof. If \(b\) is jointly continuous in \((t, x)\) then the result is straightforward by the equivalence between the Young integral formulation and the standard integral formulation. Next, if \(b\) satisfies the hypothesis and in addition \(b \in L^1_t C^{\alpha}_x\) for some \(\alpha > 0\), then for any \(t > 0\) and for any sequence of partitions \(\Pi_n\) of \([0, t]\) such that \(| \Pi_n | \rightarrow 0\) it holds

which proves the statement in this case. For a general \(b\), consider \(b^{\varepsilon} = \rho^{\varepsilon} \ast b\), where \(\rho^{\varepsilon}\) is a sequence of spatial mollifiers; for \(b^{\varepsilon}\) by the previous step identity (32) is true and by Lemma 12 \(T^w b^{\varepsilon} \rightarrow T^w b\) locally in \(C^{\gamma - \delta}_t C^{1 - \delta}_x\), similarly for \(T^{w + \theta} b^{\varepsilon} \rightarrow T^{w + \theta} b\). Choosing \(\delta\) small such that \(\gamma - \delta + (1 - \delta) \gamma > 1\) and using the continuity of Young integral we obtain the conclusion in the general case.\(\Box\)

Proof. (of Theorem 33) The idea of the proof is the same as that of Theorem 31, and it is based on finding a Young differential equation for \(v = \theta^2 - \theta^1\), only we now need to exploit the additional information on \(T^{w + \theta^1} b^1\). By the assumptions combined with Lemma 34, \(v\) satisfies

where \(\psi_t\) is defined in the usual way and \(A (t, x) = T^{w + \theta^1} (t, x) - T^{w + \theta^1} (t, 0)\), so that \(A \in C^{\gamma}_t \operatorname{Lip}_x\) with \(\mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma} \operatorname{Lip}} = \mathopen{\unicode{x27E6}} T^{w + \theta^1} b \mathclose{\unicode{x27E7}}_{C^{\gamma} \operatorname{Lip}}\) and \(A (t, 0) = 0\) for all \(t \in [0, T]\). We can then apply the estimates from Lemma 77 from Appendix A.1 to deduce

for some constant \(C = C (\gamma, T)\) which together with the estimate

yields the conclusion.\(\Box\)

Remark 35. It follows immediately from Theorem 33 that, if there exists a solution \(\theta\) to the YDE associated to \(T^w b\) with initial data \(\theta_0\) such that \(T^{w + \theta} \in C^{\gamma}_t \operatorname{Lip}_x\), then this is necessarily the unique solution with initial data \(\theta_0\) and it is stable under perturbation. This provides a nice “duality principle”: existence of solutions is granted if \(T^w b \in C^{\gamma}_t \operatorname{Lip}_x\), uniqueness instead if there exists a solution with similar averaging properties. In the case \(b\) is continuous, so that by Peano Theorem existence of a solution \(x = w + \theta \in w +\operatorname{Lip}\) is automatic, the statement can be rephrased as the fact that uniqueness for the Cauchy problem associated to \(x_0\) holds under the condition \(T^x b \in C^{\gamma}_t \operatorname{Lip}_x\) for some \(\gamma > 1 / 2\).

Remark 36. For the sake of simplicity we considered from the start \(T^w b \in C^{\gamma}_t C^{\beta}_x\) in order to develop a global theory in space, but many results from Section 4 can be localised, thanks to Remark 7, in a similar fashion to what is done in Section 2.3 of [12]. For instance local existence holds for \(T^w b \in C^{\gamma}_t \operatorname{Lip}_{\operatorname{loc}}\), while local existence and uniqueness holds for \(T^w b \in C^{\gamma}_t C^2_{\operatorname{loc}}\); in the second version of the Comparison Principle, if there exists a solution \(x\) defined on \([0, T^{\ast})\) such that \(T^x b \in C^{\gamma}_t \operatorname{Lip}_{\operatorname{loc}}\), then it is the unique solution on \([0, T^{\ast})\). Analogue considerations hold for the results from Section 4.3 on the regularity of the flow.

4.2Prevalence for the Cauchy problem

In this section we focus on establishing conditions under which, for a given drift \(b\) and a given initial datum \(x_0 \in \mathbb{R}^d\), for almost every \(\varphi \in C^{\delta}_t\) the Cauchy problem (from now on referred to as \((\operatorname{CP}_{x_0})\))

is well-posed, for suitable values of \(\delta\). Here by well-posedness for \((\operatorname{CP}_{x_0})\) we mean the following: \(\varphi\) is such that \(T^{\varphi} b \in C^{\gamma}_t \operatorname{Lip}_x\) for some \(\gamma > 1 / 2\), so that it makes sense to talk about solutions to (33) in the sense of Definition 27, and there exists a unique such solution in the class \(x \in \varphi + C^{\gamma}\). The main results we are going to prove are the following.

In the second statement we have used the terminology “path-by-path uniqueness” as it appears frequently in regularisation by noise results, see [21], but in the framework introduced above it just amounts to stating that there exists \(\gamma > 1 / 2\) such that

The section is organised as follows: we first prove Theorem 37 in Section 4.2.1 relying on the validity of Theorem 38; then we pass to the proof of the latter, which is based on an application of Theorem 33 in combination with Girsanov transform for fBm, which is introduced in Section 4.2.2. The proof of Theorem 38 is completed in Section 4.2.3, along with several other results of the same nature. We leave the details to the following subsections, but let us point out already here that we will exploit crucially the general principle

Such a principle is not new and was crucially exploited in [17] and [12]. However, we believe it is the first time it is properly formalised as in Lemma 46 and its general structure allows to apply it in other situations.

4.2.1Proof of Theorem 37

We need a few preparations first. Recall that in order to establish prevalence of well-posedness for \((\operatorname{CP}_{x_0})\) in \(C_t^{\delta}\), we need to find a set \(\mathcal{A} \subset C_t^{\delta}\) and a tight probability \(\mu\) on \(C_t^{\delta}\) such that: i) \(\mathcal{A}\) is Borel w.r.t. the topology of \(C_t^{\delta}\); ii) for all \(w \in \mathcal{A}\), \((\operatorname{CP}_{x_0})\) is well-posed; iii) for all \(\varphi \in C_t^{\delta}\), \(\mu (\varphi + \mathcal{A}) = 1\).

A good candidate for the set \(\mathcal{A}\) is given by Theorem 33 as follows: for \(\gamma > 1 / 2\), define

For such an \(\mathcal{A}_{\gamma}\), it is now rather clear by the statement of Theorem 38 that we plan to use as a measure \(\mu^H\) for suitable choice of \(H\). But we first need to check that condition i) holds, which is the aim of the following lemma.

Proof. The idea of the proof is the usual one: we write the set \(\mathcal{A}_{\gamma}\) as the countable union

In order to conclude it is then sufficient to show that, for each \(N\), the set \(\mathcal{A}_N\) is closed under the topology of \(C^{\delta}\). We can restrict ourselves to the case \(C^0\), since any other convergence we consider is stronger than this one.

Let \(w^n\) be a sequence of elements of \(\mathcal{A}_N\) such that \(w^n \rightarrow w\), then by Lemma 9 we know that \(T^w b \in C^{\gamma}_t \operatorname{Lip}_x\) with the bound \(\| T^w b \|_{C^{\gamma}_t \operatorname{Lip}_x} \leqslant N\). For each \(n\), denote by \(x^n = \theta^n + w^n\) the associated solution of \((\operatorname{CP}_{x_0})\) such that \(\| T^{x^n} b \|_{C^{\gamma}_t \operatorname{Lip}_x} \leqslant N\); by the a priori estimates from Theorem 29, together with \(\| T^{w^n} b \|_{C^{\gamma}_t \operatorname{Lip}_x} \leqslant N\), we deduce that \(\| \theta^n \|_{C^{\gamma}}\) are uniformly bounded. We can therefore (up to subsequence) consider \(\theta^n \rightarrow \theta\) in \(C^{\gamma - \varepsilon}\) for suitable \(\varepsilon > 0\). Since \(w^n + \theta^n \rightarrow w + \theta\) in \(C^0\), again it must hold \(\| T^{w + \theta} b \|_{C^{\gamma}_t \operatorname{Lip}_x} \leqslant N\).

In order to conclude it remains to show that \(x = \theta + w\) is a solution of the \((\operatorname{CP}_{x_0})\) associated to \(T^w b\). Since the sequence \(T^{w^n} b \rightarrow T^w b\) in the sense of distributions and it is uniformly bounded in \(C^{\gamma}_t \operatorname{Lip}_x\), reasoning as in the proof of Lemma 12 we deduce that also local convergence in \(C^{\gamma - \varepsilon}_t C^{1 - \varepsilon}_x\) holds, for any \(\varepsilon > 0\). Choosing \(\varepsilon > 0\) small enough such that \(\gamma - \varepsilon + (1 - \varepsilon) / (\gamma - \varepsilon) > 1\), by continuity of nonlinear Young integral it holds \(\int_0^{\cdot} T^{w^n} b (\mathrm{d} s, \theta^n_s) \rightarrow \int_0^{\cdot} T^w b (\mathrm{d} s, \theta_s)\) in \(C^{\gamma - \varepsilon}\). Taking the limit as \(n \rightarrow \infty\) of

we deduce that \(x\) is a solution w.r.t. \(T^w b\) of \((\operatorname{CP}_{x_0})\), which concludes the proof.\(\Box\)

Proof of Theorem 37. In order to conclude it suffices to show that we can find \(\gamma > 1 / 2\) and \(H > \delta\) such that \(\mu^H (\varphi + \mathcal{A}_{\gamma}) = 1\) for all \(\varphi \in C_t^{\delta}\). Let us choose \(\varepsilon > 0\) small enough such that

We need to find \(\gamma > 1 / 2\) such that for any fixed \(\varphi \in C_t^{\delta}\),

By definition of the averaging operator we have \(T^{w + \varphi} b = T^w \tilde{b}\), where \(\tilde{b} (t, \cdot) = b \left( t, \cdot \, + \varphi_t \right)\); moreover, \(x \in (w + \varphi) + C_t^{\gamma}\) solves \((\operatorname{CP}_{x_0})\) if and only if \(\tilde{x} := x - \varphi \in w + C_t^{\gamma}\) is again a solution to another Cauchy problem of the same type. Indeed, by definition of solution, \(\theta = x - (w + \varphi) = \tilde{x} - w\) must solve

where \(\tilde{x}_0 = x_0 - \varphi_0\), so that \(\tilde{x}\) is a solution to the Cauchy problem associated to \(\tilde{x}_0\), \(\tilde{b}\) and \(w\). Moreover by properties of averaging operators it holds \(T^{\tilde{x}} \tilde{b} = T^x b\).

By the translation invariance of the \(C^{\alpha}_x\)-norm, it holds \(\tilde{b} \in C^{\alpha}_x\), \(\| \tilde{b} \|_{C^{\alpha}} = \| b \|_{C^{\alpha}}\); moreover \(\tilde{b}\) has still compact support in space, uniformly in time. Since condition (35) implies \(\alpha > 1 - (2 H)^{- 1}\), we can apply Theorem 38 for the choice \(\tilde{x}_0\), \(T^w \tilde{b}\) to find \(\gamma > 1 / 2\) (independent of \(\varphi\)) such that

which gives the conclusion.\(\Box\)

Remark 40. For simplicity we have preferred to give the statement of Theorem 37 as above, but it will be clear from the contents of Section 4.2.3 that similar prevalence statements can be formulated under other hypothesis on \(b\) and \(\delta\) simply by going through the same proof and applying in the end either Theorem 48 or Corollary 49.

4.2.2Girsanov transform

Before introducing Girsanov Theorem, we need to recall another representation formula for fBm, different from the one given in Section 3.2, which can be found in [44], [48]. The representation is based on fractional calculus, which we also quickly introduce and for which we refer the interested reader to [50].

Given \(f \in L^1 (0, T)\) and \(\alpha > 0\), the fractional integral of order \(\alpha\) of \(f\) is defined as

where \(\Gamma\) denotes the Gamma function. For \(\alpha \in (0, 1)\) and \(p > 1\), the map \(I^{\alpha}\) is an injective bounded operator on \(L^p\) and we denote by \(I^{\alpha} (L^p)\) the image of \(L^p\) under the \(I^{\alpha}\), which is a Banach space endowed with the norm \(\| f \|_{I^{\alpha} (L^p)} := \| g \|_{L^p}\) if \(f = I^{\alpha} g\). On this domain, \(I^{\alpha}\) admits an inverse, which is the fractional derivative of order \(\alpha\), given by

With this notation in mind, a fBm of Hurst parameter \(H \in (0, 1)\) can be constructed starting from a standard Brownian motion \(B\) on the interval \([0, T]\) by setting \(W^H = K_H (\mathrm{d} B)\), where the operator \(K_H\) is defined as

where the notation \(s^{\beta}\) denotes the multiplication operator with the function \(s \mapsto s^{\beta}\). It can be shown that this definition of \(W^H\) is meaningful and that the operator \(K_H\) corresponds to a Volterra kernel \(K_H (t, s)\), so that the above representation is equivalent to

The explicit expression for \(K_H\) in the case \(H > 1 / 2\) is given by

in the case \(H < 1 / 2\) it is more complicated and we omit it as we will not need it. It can be shown that the operator \(K_H\) can be inverted, which implies that the processes \(B\) and \(W^H\) generate the same filtration, which makes it a canonical representation; moreover this implies that given any fBm \(W^H\) on a probability space, it is possible to construct the associated \(B\) by setting \(B_{\cdot} = \int_0^{\cdot} (K^{- 1}_H W^H)_s \mathrm{d} s\). The inverse operator \(K_H^{- 1}\) is given by

We will use the following terminology: given a filtered space \((\Omega, \mathcal{F}, \{ \mathcal{F}_t \}_{t \geqslant 0}, \mathbb{P})\), we say that a process \(W^H\) is an \(\mathcal{F}_t\)-fBm if it is a fBm under \(\mathbb{P}\) and the associated \(B\) is an \(\mathcal{F}_t\)-Bm in the usual sense.

The result is taken from [45], Theorem 2, with the exception of the final part which is just classical Novikov condition; in the original statement from [45], the process \(h\) is taken of the form \(h_{\cdot} = \int_0^{\cdot} u_s \mathrm{d} s\), but this doesn't play any role in the proof, which indeed holds also in the case \(h\) is not of bounded variation.

In order to apply Theorem 41 in cases of interest, we first need to establish conditions under which (43) holds, which requires a good control of \(\| K^{- 1}_H h \|_{L^2}\) in terms of \(h\).

Since \(K^{- 1}_H\) is defined in terms of fractional derivatives, the following fact will be quite useful: if \(f \in C^{\beta}\) and \(f_0 = 0\), then \(D^{\alpha} f\) is well defined for any \(\alpha < \beta\) and moreover \(D^{\alpha} f \in C^{\gamma}\) for any \(\gamma < \beta - \alpha\) together with the estimate

For a self-contained proof of this fact see Theorem 2.8 from [48] (on a finite interval \([0, T]\), the space \(\mathbb{H}^{\beta, 0}\) considered therein corresponds to the functions \(f \in C^{\beta}_t\) such that \(f_0 = 0\)).

Proof. We have

Since \(h \in C^{\beta}\), it clearly also belongs to \(L^2\), so we only need to control the term

Since \(\alpha \in (0, 1 / 2)\), \(t^{- \alpha} \in L^2_t\) and so we deduce that the overall expression belongs to \(L^2_t\), as well as estimate (45). Regarding the second statement, the case \(H = 1 / 2\) is straightforward since \(K^{- 1}_H h = h'\). In the case \(H > 1 / 2\), by the formula for \(K^{- 1}_H\) combined with estimates (44) and (45) for the choice \(\alpha = H - 1 / 2\), choosing \(\varepsilon > 0\) sufficiently small we have

the case \(H < 1 / 2\) is analogous.\(\Box\)

Remark 43. We have given an explicit proof of Lemma 42, but a similar (stronger) type of result can be achieved by a more abstract argument. Indeed it follows from the proof of Theorem 5.4 from [48] that \(\| s^{\alpha} D^{\alpha} (s^{- \alpha} h) \|_{L^2} \sim \| D^{\alpha} h \|_{L^2}\) and similarly \(\| K^{- 1}_H h \|_{L^2} \sim \| D^{H + 1 / 2} h \|_{L^2}\); we have already seen that if \(h \in C^{\beta}\) with \(\beta > \alpha\) and \(h_0 = 0\), then \(D^{\alpha} h\) is a continuous function, so its \(L^2\)-norm is trivially finite. The inclusion \(C^{\beta} \subset I^{\alpha} (L^2)\) is strict and therefore the hypothesis of Lemma 42 are non optimal, but they are rather useful when dealing with functions \(h\) not of bounded variation.

We can now state a general result on the applicability of Girsanov transform together with a good control on the density defining \(\mathbb{Q}\).

Proof. By hypothesis (47) and Lemma 42 it follows immediately that

Therefore Novikov criterion is satisfied and Girsanov transform is applicable. The proof of second part of the statement follows from classical arguments, but we include it for the sake of completeness. Let us prove integrability of the moments: for any \(\alpha \geqslant 1\) it holds

where in the second line we used the fact that the integrand in the first term is again a probability density by Novikov's criterion, this time applied to the process \(\tilde{h} = 2 \alpha h\). Now in order to show that the measures \(\mathbb{Q}\) and \(\mathbb{P}\) are equivalent, we need to show that the inverse density \(\mathrm{d} \mathbb{P}/ \mathrm{d} \mathbb{Q}\) is integrable w.r.t. \(\mathbb{Q}\). Again by Girsanov, since we have \(W^H = \tilde{W}^H - h\), the inverse density is given by

where \(\tilde{B}\) now denotes the standard Bm associated to \(\tilde{W}^H\), i.e. such that \(\tilde{W}^H_t = \int^t_0 K_H (t, s) \mathrm{d} \tilde{B}_s\). Since we have

we can conclude, again by applying Novikov, that \(\mathrm{d} \mathbb{P}/ \mathrm{d} \mathbb{Q}\) is integrable w.r.t. \(\mathbb{Q}\). Reasoning as before it can be shown that \(\mathrm{d} \mathbb{P}/ \mathrm{d} \mathbb{Q}\) admits moments of any order w.r.t. \(\mathbb{Q}\), which gives the conclusion.\(\Box\)

4.2.3Path-by-path uniqueness for SDEs driven by additive fBm

Girsanov's Theorem allows to construct a probabilistically weak solution of \((\operatorname{CP}_{x_0})\), which we define in the following way.

We have given a non classical notion of weak solution, which is well suited when dealing with a distributional \(b\); depending on the context, this is not the only possible definition, see for instance [3] and [24] for different choices.

We are now ready to provide a general principle to establish path-by-path uniqueness.

Proof. Consider \(\gamma > 1 / 2\) as in the assumption and the set \(\mathcal{A}_{\gamma}\) defined as in (34); by Theorem 33, in order to conclude it is enough to show that \(\mu^H (\mathcal{A}_{\gamma}) = 1\). By hypothesis, the first half of the statement defining \(\mathcal{A}_{\gamma}\) is already satisfied on a set of full probability, so we only need to concentrate on the second half. By the definition of \(h\), the process \(X = x_0 + W^H\) satisfies

by hypothesis Girsanov theorem is applicable, so we can construct a new probability measure \(\mathbb{Q}\) which is absolutely continuous w.r.t. to \(\mathbb{P}\) such that \(\tilde{W}^H\) is an \(\mathcal{F}_t\)-fBm under \(\mathbb{Q}\). Observe that \(\mathbb{P}\)-a.s. \(T^X b = \tau^{x_0} T^{W^H} b \in C^{\gamma}_t \operatorname{Lip}_x\) and so \(\mathbb{P}\)-a.s. the difference \(X_{\cdot} - \tilde{W}^H_{\cdot} = x_0 + T^X \left( \cdot \,, 0 \right) \in C^{\gamma}\) (if \(T^X b \in C^{\gamma}_t \operatorname{Lip}_x\), then it also belongs to \(C^0_x C^{\gamma}_t\)); then by Lemma 34, on a set of full measure \(\mathbb{P}\) equation (49) is equivalent to

and so \(X\) is \(\mathbb{P}\)-a.s. a solution to \((\operatorname{CP}_{x_0})\) in the sense of Definition 27. Since \(\mathbb{Q} \ll \mathbb{P}\), all the above statements also hold on a set of \(\mathbb{Q}\)-full measure. But then since \(\tilde{W}^H\) has law \(\mu^H\) under \(\mathbb{Q}\), we obtain

which gives the conclusion.\(\Box\)

Remark 47. We cannot apply directly the Yamada-Watanabe theorem to deduce existence of a strong solution under the assumptions of Lemma 46, because our path-by-path uniqueness statement holds only in the class \(w + C^{\gamma}\) and not in the class of all possible continuous paths (although in the case of continuous \(b\) the two classes coincide). There is however a more direct way to show that the path-by-path unique solution \(X\) is adapted to the filtration generated by \(W^H\). Consider a sequence \(\varepsilon_n \, \downarrow 0\) and \(b^n := \rho^{\varepsilon_n} \ast b\), where as usual \(\{ \rho^{\varepsilon} \}_{\varepsilon > 0}\) is a sequence of spatial mollifiers, and consider \(X^n\) solution to

by classical theory \(X^n\) is unique and adapted to the filtration generated by \(W^H\). Then by Theorem 33 (possibly combined with Lemma 12), \(\mathbb{P}\)-a.s. \(X^n_{\cdot} \rightarrow X_{\cdot}\) in \(C^{\gamma}\), which implies that \(X\) is adapted as well and thus a strong solution.

All the results obtained so far are of abstract nature. Now we are going to show how to apply them to establish path-by-path uniqueness for \((\operatorname{CP}_{x_0})\) in our context. In particular, Theorem 38 is a direct consequence of the following more general result.

Proof. In both cases, in order to conclude, we need to show that we can apply Lemma 46 to the process \(h_{\cdot} = \int_0^{\cdot} b (s, x_0 + W^H_s) \mathrm{d} s\); in order to do so, we will check that the conditions of Theorem 44 are satisfied. Up to shifting \(b\), we can assume without loss of generality \(x_0 = 0\).

Let \(H > 1 / 2\), then by the hypothesis \(b \in C^0_t C^{\alpha}_x\) and Theorem 19 we know that \(T^{W^H} \in C^{\gamma}_t C^{1 + \varepsilon}_x\) (at least locally) for some \(\gamma > 1 / 2\) and \(\varepsilon > 0\); the process \(h\) belongs to \(C_t^{H + 1 / 2 + \varepsilon}\) if and only if the map \(t \mapsto b (t, W^H_t) \in C_t^{H - 1 / 2 + \varepsilon}\). Recall that for any \(\gamma < H\), \(W^H \in C_t^{\gamma}\); then by the hypothesis it holds

and so we can find \(\varepsilon > 0\) small enough such that \(\gamma = H - \varepsilon\) and

As the exponent \(\alpha < 1\), by Fernique Theorem we deduce that

Consider now the case \(H < 1 / 2\). By Theorem 19 (as the support of \(b\) is compact uniformly in time, we have the embedding \(C^{\alpha} \hookrightarrow B^{\alpha}_{p, p}\) for any \(p < \infty\)) we know that

for some \(\gamma > 0\) and \(\varepsilon > 0\) sufficiently small, therefore the process \(h_t = T^{W^H} b (t, 0)\) is a well defined element of \(C^{\gamma}_t\). We now want to show that it actually belongs to \(C_t^{H + 1 / 2 + \varepsilon}\);we can do so by interpolation, using the fact that \(T^{W^H}\) has higher spatial regularity. Indeed by properties of the averaging operator \(T^{W^H} b \in \operatorname{Lip}_t C^{\alpha}_x\) and so for any \(\theta \in (0, 1)\) it holds

where \(\beta = (1 - \theta) \alpha + \theta (\alpha + (2 H)^{- 1} - \varepsilon)\) and thanks to the hypothesis we can choose \(\theta \in (0, 1)\) s.t.

For this choice of \(\theta\) therefore we obtain

and since the exponent \(2 \theta < 2\), and we have exponential integrability for the term on the r.h.s. by Theorem 19, we get the conclusion.\(\Box\)

In the regime \(H > 1 / 2\), the hypothesis required on \(b\) is the same as in [45], although therein path-wise uniqueness is shown only in the case \(d = 1\), while here we obtain path-by-path uniqueness in any dimension. In the case \(H = 1 / 2\), we can allow \(b \in L^{\infty}_t C^{\alpha}_x\) for any \(\alpha > 0\); this result is comparable to the one from [17], in which sharper estimates allow to reach \(b \in L^{\infty}_{t, x}\), see also [52, 53] for further extensions. Observe that in the regime \(H < 1 / 2\) we can allow \(b\) to be only distributional; in this case, we recover the results from [12]. Unfortunately, the original proof from [12] is wrong, due to an incorrect version of the formula defining \(K_H^{- 1}\) (see the formula for \(H^n\) just before Lemma 4.8 therein), which is why we have decided to give an alternative proof rather than directly invoking the results from [12].

The driving principle given by Lemma 46 is fairly general and can be applied under different hypothesis on \(b\), especially when we combine it with Theorems 15 and 19.

Proof. It follows from hypothesis (50), combined with Theorem 19 and the Besov embeddings \(B^{\alpha + s}_{p, p} \hookrightarrow C^{\alpha + s - d / p}\), that we can choose \(s\) satisfying (21) such that \(T^{W^H} b \in C^{\gamma}_t C^1_x\) for some \(\gamma > 1 / 2\). As before, we can now assume \(x_0 = 0\) and it remains to show that the process \(h_t = T^{W^H} b (t, 0) \in C^{\beta}_t\) for some \(\beta > H + 1 / 2\) and satisfies integrability conditions like those of Theorem 44. By the properties of the averaging operator, on a set of full probability it holds

for any \(s\) such that \(H s + 1 / q < 1 / q\). Therefore by interpolation, for any \(\theta \in (0, 1)\), it holds

In order to deduce that \(T^{W^H} b \left( \cdot \,, 0 \right) \in C^{\beta}_t\) with \(\beta > H + 1 / 2\), we need to find parameters \(s > 0\) and \(\theta \in (0, 1)\) such that

A few algebraic manipulations show that the above system is equivalent to condition (50); from interpolation we then obtain, for \(\beta = (1 - 1 / q) (1 - \theta) + \theta / 2\) as above,

and since the parameter \(\theta \in (0, 1)\), we deduce that \(\| h \|_{C^{\beta}}\) satisfies (47).

In the case \(b \in L^q_t L^p_x\), using the embedding \(L^p_x \hookrightarrow B^{- \varepsilon}_{p, p}\) for any \(\varepsilon > 0\) (see Appendix A.2) and applying the previous result for \(\varepsilon\) sufficiently small we get the conclusion.\(\Box\)

In the case \(b \in L^q_t L^p_x\), it was already shown in [38] that pathwise uniqueness holds. Here we have strengthened the result to path-by-path uniqueness. The case \(b \in L^q_t B^{\alpha}_{p, p}\) with \(\alpha < 0\) to the best of our knowledge has not been considered in the literature so far. Condition (50) actually holds also in the regime \(\alpha > 0\), but this is not particularly interesting as one can use fractional Sobolev embeddings (see [19]) to deduce \(L^q_t B^{\alpha}_{p, p} \hookrightarrow L^q_t L^{p^{\ast}}_x\) with

and then reduce it to the case (51).

Remark 50. The guiding principle of Lemma 46 is fairly general, but there are situations in which it is possible to establish path-by-path uniqueness even if Girsanov theorem is not applicable (or at least we are currently not able to find suitable estimates in order to apply it). Consider for instance the case of \(H > 1 / 2\) and \(b \in L^{\infty}_t C^{\alpha}_x\) for \(\alpha \in (0, 1)\) such that

observe that the condition is non trivial for every \(H \in (1 / 2)\). Then by Theorem 19 (possibly after a localisation procedure) \(T^{W^H} b \in C^{\gamma}_t \operatorname{Lip}_x\) (at least locally) and so by Theorem 31 path-by-path uniqueness holds for the whole ODE. However, lack of continuity in time of \(b\) prevents us from applying Girsanov.

4.3Regularity of the flow

4.3.1Variational formula for flow of diffeomorphisms

It follows from Theorem 31 that, if \(b\) and \(T^w b\) satisfy the regularity assumptions, the solution map \((\theta_0, t) \mapsto \theta_t\) is Lipschitz in space, uniformly in time (more precisely, it follows from (29) and (30) that it is \(C^{\gamma}_t \operatorname{Lip}_{\operatorname{loc}}\)). However we cannot yet talk about a flow, as we haven't shown the invertibility of the solution map, nor the flow property; this is accomplished by the following two lemmas.

Proof. Let \(\Pi\) be a partition of \([0, t]\) given by \(0 = t_0 < t_1 < \cdots < t_n = t\) and define \(\tilde{t}_i = T - t_i\), which defines a partition of \([T - t, T]\) (up to the fact that it is decreasing w.r.t. \(i\)); it holds

where the remainder term \(J\) satisfies

By basic properties of the averaging operator we have \(T^w b_{T - t, T - s} (x) = T^{\tilde{w}} \tilde{b}_{s, t} (x)\) and so overall we obtain

Taking a sequence of partitions \(\Pi_N\) such that \(\| \Pi_N \| \rightarrow 0\) and taking the limits on both sides we obtain the first statement. Regarding the second statement, if \(\theta\) is a solution of the YDE, then by (52) for any \(t \in [0, T]\) it holds

which implies the conclusion.\(\Box\)

Similar arguments also provide the following lemma, whose proof is therefore omitted.

We are now ready to provide sufficient conditions for the existence of a Lipschitz flow.

Proof. The proof is a straightforward application of Theorem 31 and Lemmata 51 and 52. In both cases of time reversal and translation we have \(\| T^{\tilde{w}} \tilde{b} \|_{C^{\gamma} C^2} \leqslant \| T^w b \|_{C^{\gamma} C^2}\) (same for \(\| \cdot \|_{C^{\gamma} C^{3 / 2}}\) and \(\| \cdot \|_{L^{\infty}_{t, x}}\)) so that uniqueness holds also for the reversed/translated YDE, with the same continuity estimates; this provides respectively invertibility of the solution map and flow property.\(\Box\)

Further estimates for \(\Phi\) are available, since Theorem 31 actually implies that

uniformly in \(s \in [0, T]\), \(x, y \in \mathbb{R}^d\).

Let us denote by \(\Phi_t\) the map \(\Phi_t (x) = \Phi (0, t, x)\); from now on we are only going to consider the map \(\Phi_{\cdot} = \Phi_t (x)\), which by an abuse of notation and language, will be just denoted by \(\Phi\) and referred to as the flow of the YDE. This is just to keep the notation simple and indeed all the proofs below can be easily adapted to the whole flow \(\Phi (s, t, x)\).

We will keep using the incremental notation \(\Phi_{s, t} (x) = \Phi_t (x) - \Phi_s (x)\); it follows from the above estimates that \(\Phi \in C^{\gamma}_t \operatorname{Lip}_{\operatorname{loc}}\), since

Similarly, we define \(\psi_t (x) = \psi (0, t, x)\), so that \(\psi_t = \Phi_t^{- 1}\) as a map from \(\mathbb{R}^d\) to itself.

We now state a specialised version of Theorem 31 which is quite useful for practical purposes, as it clearly identifies a way to approximate the flow associated to \(T^w b\), which by the YDE formulation is well defined when \(b\) is only a distribution, by means of more regular flows, associated to drifts \(b^{\varepsilon}\) for which also the ODE interpretation is meaningful.

Proof. We only prove the statement in the case \(T^w b \in C^{\gamma}_t C^2_x\), the other one being almost identical. By the properties of mollifiers it holds \(T^w b^{\varepsilon} = (T^w b)^{\varepsilon}\), so that \(\| T^w b^{\varepsilon} \|_{C^{\gamma}_t C^2_x} \leqslant\)\(\| T^w b \|_{C^{\gamma}_t C^2_x}\) for all \(\varepsilon > 0\), thus the hypothesis of Theorem 31 are satisfied uniformly in \(\varepsilon > 0\). Once we fix \(R > 0\), by the a priori estimates from Theorem 29 we have a uniform bound of the form

in particular we can localise \(T^w b\) and \(T^w b^{\varepsilon}\) in such a way that they all have support contained in a sufficiently big ball (say for instance \(B_{2 R}\)) in such a way that for \(x \in B_R\), \(\Phi \left( \cdot \,, x \right)\) and \(\Phi^{\varepsilon} \left( \cdot \,, x \right)\) are not affected by it. Now take any \(\tilde{\gamma} \in (1 / 2, \gamma)\), then by (29) in order to conclude it is enough to show that \(T^w b^{\varepsilon} \rightarrow T^w b\) locally in \(C^{\tilde{\gamma}}_t \operatorname{Lip}_x\); but this is an immediate consequence of Lemma 12.\(\Box\)

From now on we will adopt the following notation: whenever all the Young integrals involved are well defined, we write

so that in particular, whenever \(\varphi\) is regular enough for \(T^w \varphi\) to make sense both as a Young integral and a Lebesgue integral, it holds

We are now ready to further improve the regularity of the flow \(\Phi\) and provide a variational equation for \(D_x \Phi\), as well as an expression for its Jacobian. In the case \(A = T^w b \in C^{\gamma}_t C^2_x\) a similar result was proved in [33], Section 3.3; our derivation is of different nature and based on approximating \(b\) by more regular \(b^{\varepsilon}\), for which standard ODE theory applies. The case \(T^w b \in C^{\gamma}_t C^{3 / 2}_x\) appears to be new.

Proof. As before, to avoid repetitions we give a detailed proof only in the case \(T^w b \in C^{\gamma}_t C^2_x\); we provide in the end the main differences of the proof in the case \(b \in L^{\infty}_{t, x}\), \(T^w b \in C^{\gamma}_t C^{3 / 2}_x\).

We divide the proof in several steps, but the main idea is the following: in the case of spatially smooth \(b\), the result is just a reformulation of the standard ODE results; in the general case we can recover the result by reasoning by approximation with the help of Lemma 54.

Step 1: Proof in the case of regular \(b\). Let us first assume in addition that \(b \in L^q_t C_x^2\) for some \(q > 2\); then in this case we know that the YDE formulation is equivalent to the ODE one, so that the flow \(\Phi\) associated to \(b\) satisfies

moreover by standard ODE theory we have the variational equation

The term in the last line now makes perfectly sense as a Young integral, as the term

is a well defined \(C_t^{\gamma}\) map for \(\gamma > 1 / 2\), since \(b \in L^q_t C^1_x\), proving the first part of the claim.

Step 2: Approximation and characterisation of the limit as \(\varepsilon \rightarrow 0\) of \(D_x \Phi^{\varepsilon}\). Consider a sequence \(T^w b^{\varepsilon}\), \(\Phi^{\varepsilon}\) defined by spatial mollification as in Lemma 54. By Step 1, for any \(\varepsilon > 0\), \(D_x \Phi^{\varepsilon}\) satisfies the variational equation, which for fixed \(x\) is a linear YDE in the unknown \(D_x \Phi^{\varepsilon} (\cdot, x)\) with drift \(\int_0^{\cdot} T^w D_x b^{\varepsilon} (\mathrm{d} r, \Phi^{\varepsilon}_r (x))\); thanks to the a priori bounds given by Theorem 29, which for fixed \(x\) are uniform in \(\varepsilon\), we have the estimate

which implies by Proposition 76 in Appendix A.1 that for fixed \(x\) we have the uniform estimate

As in the proof of Lemma 54, for any \(\delta > 0\) we have \(T^w D b^{\varepsilon} \rightarrow T^w D b\) locally in \(C^{\gamma - \delta}_t C^{2 - \delta}_x\), as well as \(\Phi^{\varepsilon} (\cdot, x) \rightarrow \Phi (\cdot, x)\) in \(C^{\gamma - \delta}_t\), thus choosing \(\delta\) sufficiently small such that \((\gamma - \delta) (2 - \delta) > 1\) by the continuity of nonlinear Young integral it holds

By the a priori estimates on \(D_x \Phi^{\varepsilon} (\cdot, x)\), we can extract a subsequence converging to a limit in \(C^{\beta}\) for any \(1 / 2 < \beta < \gamma\); let us denote this limit by \(g (\cdot, x)\) (the notation will be clear in a second). By Step 1, \(D_x \Phi^{\varepsilon}\) satisfy variational equations with drifts \(\int_0^{\cdot} T^w D_x b^{\varepsilon} (\mathrm{d} r, \Phi^{\varepsilon} (r, x)) \rightarrow \int_0^{\cdot} T^w D_x b (\mathrm{d} r, \Phi (r, x))\), which implies that \(g (\cdot, x)\) must satisfy the linear YDE

But the solution to this linear equation unique, thus so is the limit of any subsequence we can extract, showing that the whole sequence \(\{ D_x \Phi^{\varepsilon} (\cdot, x) \}_{\varepsilon > 0}\) converges to such \(g (\cdot, x)\). The reasoning holds for any \(x \in \mathbb{R}^d\).

Step 3: Continuity of the map \((t, x) \mapsto g (t, x)\). This step is very similar to the previous one, so we only sketch it. Continuity in \(t\) is clear, we only need to prove continuity in \(x\); by the continuity of the flow, for any sequence \(x_n \rightarrow x\) we have \(\Phi (\cdot, x_n) \rightarrow \Phi (\cdot, x)\) in \(C^{\gamma - \delta}\) for any \(\delta > 0\) and since all \(x_n\) lie in a bounded ball, we have uniform estimate both on \(\int_0^{\cdot} T^w D b (\mathrm{d} r, \Phi_r (x_n))\) and \(g (\cdot, x_n)\). Therefore by the usual compactness argument we deduce that \(g (\cdot, x_n)\) converge in \(C^{\gamma - \delta}\) to the unique solution of the YDE associated to \(\int_0^{\cdot} T^w D b (\mathrm{d} r, \Phi_r (x))\), namely \(g (\cdot, x)\).

Step 4: Flow of diffeomorphisms. We know that for any \(R > 0\), the flows \(\Phi^{\varepsilon}\) are spatially Lipschitz in \(B_R\), uniformly in \([0, T]\) and \(\varepsilon > 0\), and that they converge uniformly on compact sets to \(\Phi\), while their spatial derivatives \(D_x \Phi^{\varepsilon}\) converge to the continuous function \(g (t, x)\). Therefore we deduce that \(g (t, x) = D_x \Phi_t (x)\), thus showing that \(\Phi\) is \(C^1\) in space, uniformly in time; moreover by construction \(g\) is the unique solution to the variational equation (54). The reasoning applies to \(\psi = \Phi^{- 1}\) as well, as it can be represented through the flow associated to the time reversed drift \(T^{\tilde{w}} \tilde{b}\), which enjoys the same regularity as \(T^w b\).

Step 5: Jacobian. As before, let us first assume \(b\) spatially smooth, then by standard ODE theory it holds

which gives equation (55) in this case. The general case is accomplished as above by an approximation procedure, using the continuity of Young integrals. Regarding the bound on \(J \Phi\), by Point 4 of Theorem 26 combined with the a priori estimates on \(\Phi\), we obtain

which gives the upper bound; the lower bound follows from \((J \Phi_t (x))^{- 1} = J \psi_t (\Phi_t (x)) \leqslant C\).

Step 6: Differences in the case \(b \in L^{\infty}_{t, x}\) with \(T^w b \in C^{\gamma}_t C^{3 / 2}_x\). The proof in this case goes along the exact same lines, with only slightly different regularity estimates. Indeed in this case we know that \(\Phi (\cdot, x)\) is Lipschitz with \(\mathopen{\unicode{x27E6}} \Phi (\cdot, x) \mathclose{\unicode{x27E7}}_{\operatorname{Lip}} \leqslant \| b \|_{L^{\infty}}\) for all \(x \in \mathbb{R}^d\) and so the drift associated to the variational equation is controlled by

Moreover by Lemma 54, we now have \(\Phi^{\varepsilon} \left( \cdot \,, x \right) \rightarrow \Phi (\cdot, x)\) in \(C^{1 - \delta}\) for all \(\delta > 0\) and so all the reasonings related to compactness and continuity of Young integrals still work. A similar reasoning goes for equation (55) and the two-sided estimates for \(J (t, x)\).\(\Box\)

Remark 56. A closer look at the proof shows that the result can be further generalised to include the case of \(T^w b \in C^{\gamma}_t C^{\nu}_x\) with \(b \in L^p_t L^{\infty}_x\), under the conditions \(\gamma > 1 / 2\) and \(\nu \geqslant 1 + q / 2\), \(q\) being the conjugate of \(p\), i.e. \(1 / p + 1 / q = 1\).

Remark 57. Recall that in the case of spatially smooth \(b\), differentiating the relation \(\psi_t (\Phi_t (x)) = x\) w.r.t. \(t\), one obtains that \(\psi\) satisfies the PDE

Equation (56) still holds if \(b \in C^0_{t, x}\) and \(T^w b \in C^{\gamma}_t C^{3 / 2}_x\), since in this case \(\Phi\) is locally \(C^1_{t, x}\) and the same holds for \(\psi\).

In the general case \(T^w b \in C^{\gamma}_t C^2_x\), reasoning by approximation, if \(\psi \in C^{\gamma}_t C^1_{\operatorname{loc}}\) then the equation is still satisfied in the following generalised sense:

where the r.h.s. is a Young integral in time, for fixed \(x \in \mathbb{R}^d\).

However, the regularity requirement \(\psi \in C^{\gamma}_t C^1_{\operatorname{loc}}\) does not need to hold; in general the only information available is \(\psi \in C^{\gamma}_t C^0_{\operatorname{loc}} \cap C^0_t C^1_{\operatorname{loc}}\). Indeed, by the group property

it holds

where the estimate is uniform in \(x \in \mathbb{R}^d\); establishing \(\psi \in C^{\gamma}_t C^1_{\operatorname{loc}}\) requires an analogue estimate for \(D_x \psi\), where

It's easy to see from the above expression that if \(\psi \in C^0_t C^2_{\operatorname{loc}}\) (which by time reversal is equivalent to \(\Phi \in C^0_t C^2_{\operatorname{loc}}\)), then it belongs to \(C^{\gamma}_t C^1_{\operatorname{loc}}\) as well. As shown in the next section, this condition is met if \(T^w b\) is regular enough.

4.3.2Higher regularity

Similarly to the standard ODE case, we can show that the flow \(\Phi\) inherits the spatial regularity of \(T^w b\), i.e. to a more regular averaged functional \(T^w b\) corresponds a more regular flow of solutions.

Proof. As before, we give a detailed proof in the case \(T^w b \in C^{\gamma}_t C^{n + 1}_x\) and in the end highlight the main differences in the other case. The idea of the proof, similarly to that of Theorem 55, is to reason by approximation and establish first that, for \(b^{\varepsilon} = \rho^{\varepsilon} \ast b\), it holds \(\Phi^{\varepsilon} \in C^{\gamma}_t C_x^{n + 1}\) with an estimate which is uniform in \(\varepsilon > 0\); then the conclusion follows from taking the limit as \(\varepsilon \rightarrow 0\). In order to get uniform estimates, we will show that for any \(k \leqslant n\), \(D^k_x \Phi^{\varepsilon}\) satisfies a variational type equation in which the leading term is a linear Young integral. We split the proof in several steps.

Step 1: \(k\)-th order variation equation. We start by assuming \(b \in L^q_t C^{\infty}_x\) in addition to the assumptions, so that by standard ODE theory the associated flow has \(C^{\infty}\) spatial regularity. We now adopt the following convention: the symbol \(\circ\) denotes a suitably chosen matrix product, which can change from line to line. We claim that, for any \(1 \leqslant k \leqslant n\), \(D^k_x \Phi\) satisfies the variational-type equation

where the first integral makes sense in the Young sense and the \(F_k\) are “polynomial” functions of the form

where the internal sum is taken over all possible \(\beta = (\beta_1, \ldots, \beta_{k - 1})\) with \(\beta_i \in \{ 1, \ldots, k \}\) such that \(\sum_i i \beta_i = k\) and \(a_{\beta}\) are suitable coefficients of combinatorial nature. Observe that, in terms of the variable \(D^k_x \Phi (\cdot, x)\), equation (58) is a linear YDE of the form \(y_t = \int_0^t A_{\mathrm{d} s} y_s + h_t\), as the second term does not have any dependency on \(D^k_x \Phi\).

The proof is by induction on \(k\), the case \(k = 1\) being immediate. In the case \(k = 2\), differentiating both terms in the variational equation associated to the drift \((t, x) \mapsto b (t, x + w_t)\)

we obtain

which is exactly of the form (58). Now assume that the statement is true for \(k\), then differentiating (58) on both sides we obtain

where \(\tilde{F}_{k + 1} (\{ D^i \Phi (\cdot, x) \}_{i \leqslant k + 1}) = D_x F_k (\{ D^i \Phi (\cdot, x) \}_{i \leqslant k})\) and it is easy to check that it is still of “polynomial type”.

Step 2: Inductive estimate on \(D^k_x \Phi\). Fix \(R > 0\); we claim that, for any \(2 \leqslant k \leqslant n\), there exists a constant \(C_k < \infty\) (which depends on \(R\)), which is independent of \(\| b \|_{L^q_t C^{\infty}_x}\), such that

Again the proof is inductive, mainly relying on the fact that \(D^i \Phi\) solves the linear YDE (58) in combination with the a priori bounds given by Proposition 76.

We start by proving the claim in the case \(k = 2\). In this case we already know by Theorems 29 and 55 that \(\sup_{x \in B_R} (\| \Phi (\cdot, x) \| + \| D \Phi (\cdot, x) \|_{C^{\gamma}_t}) \leqslant C\); moreover by properties of Young integral we have

as well as the bound \(\left\| \int_0^{\cdot} D_x T^w b (\mathrm{d} r, \Phi (r, x)) \right\|_{C^{\gamma}} \lesssim \| T^w b \|_{C^{\gamma}_t C^2_x} \| \Phi (\cdot, x) \|_{C^{\gamma}}\). Applying again Proposition 79 yields the conclusion in this case.

Assume now that the claim holds for \(k\), then by the inductive hypothesis all the term appearing in the sum defining \(F_{k + 1}\) can be estimated by

which together with the estimate for \(\left\| \int_0^{\cdot} D T^w b (\mathrm{d} r, \Phi (r, x)) \right\|_{C^{\gamma}}\) and the application of Proposition 76 yields a new constant \(C_{k + 1}\).

Step 3: Approximation procedure. Let \(b^{\varepsilon} = \rho^{\varepsilon} \ast b\) denote by \(\Phi\) and \(\Phi^{\varepsilon}\) the flows associated to \(b\) and \(b^{\varepsilon}\) respectively. Then \(\| T^w b^{\varepsilon} \|_{C^{\gamma}_t C^{n + 1}_x} \leqslant\)\(\| T^w b \|_{C^{\gamma}_t C^{n + 1}_x}\) for all \(\varepsilon > 0\) and so by the previous step we deduce that for any \(R > 0\) there exists a suitable constant \(C\) such that

But \(\Phi^{\varepsilon} \rightarrow \Phi\) uniformly in \([0, T] \times B_R\), which together with the weak-\(\ast\) compactness of balls in \(W^{n, \infty} (B_R)\) implies that \(\Phi \in L^{\infty}_t W^{n, \infty} (B_R)\). A slightly more refined argument, analogue to the one from Theorem 55, allows to show that, for any fixed \(x \in \mathbb{R}^d\), \(D^n \Phi^{\varepsilon} (\cdot, x)\) must converge as \(\varepsilon \rightarrow 0\) to the unique solution of the variational-type equation (58) associated to \(\Phi\); with this information at hand it is then possible to show that the limit varies continuously in \(x\) and must coincide with \(D^n \Phi (\cdot, x)\), thus showing that \(\Phi\) is not only in \(W^{n, \infty}_{\operatorname{loc}}\) but also \(C^n\). We omit the details in order to avoid unnecessary repetitions.

Step 4: The case \(b \in L^{\infty}_{t, x}\) with \(T^w b \in C^{\gamma}_t C^{n + 1 / 2}_x\). In this case Step 1 and Step 3 are identical to the ones above, the only change is in the estimates from Step 2, as we can use the information \(\left\| \Phi \left( \cdot \,, x \right) \right\|_{\operatorname{Lip}_t} < \infty\) uniformly in \(x \in B_R\) to require less regularity for \(T^w b\). For instance in the case \(k = 2\) we have the estimates

and \(\left\| \int_0^{\cdot} D_x T^w b (\mathrm{d} r, \Phi (r, x)) \right\|_{C^{\gamma}} \lesssim \| T^w b \|_{C^{\gamma}_t C^{3 / 2}_x} \| \Phi (\cdot, x) \|_{\operatorname{Lip}}\). The general inductive step similar.\(\Box\)

5Application to transport type PDEs

The aim of this section is to apply the theory of Section 4 in order to solve perturbed first order linear PDEs of the form

where \(\dot{w}\) denotes the time derivative of \(w\); at this stage, the equation is only formal. However, if we assumed everything smooth, then applying the change of variables \(\tilde{u} (t, x) = u (t, x + w_t)\) (similarly for \(\tilde{b}\), \(\tilde{c}\)), (59) would be equivalent to

Equation (60) is now meaningful in the classical sense if for instance \(\tilde{b}, \tilde{c} \in C^0_{t, x}\), which is equivalent to \(b, c \in C^0_{t, x}\); it also makes sense in the weak sense under suitable integrability assumptions on \(b, c\). Moreover the transformation that defines \(\tilde{u}\) in function of \(u\) is well defined whenever \(w\) is a continuous path.

Based on the above reasoning, we will adopt the convention that \(u\) is a solution to (59) if and only if \(\tilde{u}\) defined as above is a solution to (60) and we will study systematically the latter equation. Let us mention that in the case \(w\) is a rough path, it is possible to give meaning to (59) and the passage from (59) to (60) can be rigorously justified, see [11].

Although the above discussion holds for general \(c\), we will focus only on two cases of interest, given by transport and continuity equations, namely for \(c = 0\) and \(c =\operatorname{div}b\) (resp. \(\tilde{c} = 0\) and \(\tilde{c} =\operatorname{div} \tilde{b}\)).

While in Section 4 all the proofs were almost identical for \(b \in C^0_{t, x}\) with \(T^w b \in C^{\gamma}_t C^{3 / 2}_x\) and \(T^w b \in C^{\gamma}_t C^2_x\), here the difference becomes relevant and the first case is much easier to treat compared to the latter; to our surprise, even if the existence of a Lipschitz flow for the associated ODE is already known, the case \(T^w b \in C^{\gamma}_t C^2_x\) requires the application of refined tools like commutators and the Sewing lemma. For this reason, we split the results in two subsections, with the proofs becoming gradually more complex, so that the difficulties arising in the second case become apparent.

5.1The case of continuous bounded \(b\)

Let us mention that in this case the transport equation has been treated with similar techniques in [11], while the continuity equation in Chapter 9 from [41]. More recently, in the case \(b \in L^{\infty}_{t, x}\), the transport equation has been investigated with different techniques in [2].

We start by considering the case \(c \equiv 0\). Recall that \(\tilde{b} (t, x) = b (t, x + w_t)\) and that in this case the YDE associated to \(\theta\) corresponds to the ODE associated to \(\tilde{b}\), for which existence of a locally \(C^1_{t, x}\) flow \(\Phi\) is known. Let us also recall the notation from Section 4.3, namely \(\Phi_t (x) = \Phi (0, t, x)\), \(\psi (s, t, \cdot) = \Phi (s, t, \cdot)^{- 1}\) and \(\psi_t = \Phi_t^{- 1}\). With a slight abuse, from now on we will denote \(\tilde{u}\) with \(u\) instead.

Proof. Recall that by Remark 57 \(\psi \in C^1_{t, x}\) solves equation

Therefore \(u (t, x) := u_0 (\psi (t, x)) \in C^1_{t, x}\) and satisfies

which shows that it is a solution.

Conversely, let \(u\) be a solution and for a given \(x \in \mathbb{R}^d\) define \(z_t = u (t, \Phi_t (x))\). \(\dot{\Phi}_t (x) = \tilde{b} (t, \Phi_t (x))\), therefore \(z\) solves

which implies that \(u (t, \Phi_t (x)) = u_0 (x)\) for all \(x\) and thus \(u (t, x) = u_0 (\psi (t, x))\).\(\Box\)

We now turn to the case \(c =\operatorname{div}b\), i.e. the continuity equation. Since in general \(\operatorname{div}b\) is only defined as a distribution, it makes sense to interpret the equation in a weak sense.

We adopt the following notation: \(\mathcal{M}_x (\mathbb{R}^d) = \mathcal{M}_x\) denotes the Banach space of all finite signed Radon measures on \(\mathbb{R}^d\), endowed with the total variation norm. We say that \(v \in L^{\infty}_t \mathcal{M}_x\) is weakly continuous if the map \(t \mapsto v_t\) is continuous \(\mathcal{M}_x\) endowed with the weak-\(\ast\) topology, equivalently if for any \(\varphi \in C^{\infty}_c (\mathbb{R}^d)\), the map \(t \mapsto \langle v_t, \varphi \rangle\) is continuous.

Remark 62. Whenever \(\operatorname{div}b \in C^0_{t, x}\), equation (64) corresponds to the classical formulation

Under the above assumptions equation (64) is still meaningful as a nonlinear Young integral, since \(\operatorname{div}T^w b = T^w \operatorname{div}b \in C^{\gamma}_t C^{1 / 2}_x\) and the map \(s \mapsto \psi (s, t, x)\) is Lipschitz. However, we will only use formula (65) in the proof, as it is more practical for explicit computations.

Proof. Let \(v\) be defined by (65), then for any \(\varphi \in C^{\infty}_c ([0, T] \times \mathbb{R}^d)\) we have

which shows that \(v\) is a weak solution of (62).

Since equation (63) is linear, it is enough to establish uniqueness in the case \(v_0 \equiv 0\). Let \(v\) be a given weak solution, then by standard density arguments (63) extends to all \(\varphi \in C^1_c ([0, T] \times \mathbb{R}^d)\); take \(\varphi_t (x) = u (\psi_t (x))\) with \(u \in C^{\infty}_c (\mathbb{R}^d)\), so that \(\varphi \in C^1_c ([0, T] \times \mathbb{R}^d)\) and it solves \(\partial_t \varphi + \nabla \varphi \cdot b = 0\). Then we obtain

By usual density arguments, the relation then extends to all continuous bounded \(u\); for fixed \(t\), taking \(u (x) = \tilde{u} (\Phi_t (x))\), we deduce that \(\langle \tilde{u}, v_t \rangle = 0\) for all \(\tilde{u} \in C^0_b\), which implies \(v_t \equiv 0\) for all \(t\).\(\Box\)

5.2The case of distributional \(b\)

We now pass to the case \(T^w b \in C^{\gamma}_t C^2_x\), without assuming any regularity on the distribution \(b\). To the best of our knowledge, this case has never been considered in literature so far; although perturbed linear PDEs have been previously treated in [11, 43], it is always assumede therein at least \(b \in L^{\infty}_{t, x}\) (which can be treated analogously to Section 5.1). However, our approach in the “Young regime”, namely for time regularity \(\gamma > 1 / 2\), is undoubtedly similar (and even simpler) to that in the “rough regime” \(\gamma \in (1 / 3, 1 / 2]\) treated in [5]. The use of a commutator lemma also reflects the work [20] and Chapter 9 from [41]. Abstract transport equations in Ho^¨lder media have been treated also in [33]; however the results there are, in our opinion, not completely clear, see Remark 68 below.

Remark 64. The integral appearing in (67) is meaningful as a Young integral, since by assumptions the map \(t \mapsto \operatorname{div} (T^w b (t, \cdot) \varphi)\) belongs to \(C^{\gamma}_t C^0_c\) while \(t \mapsto u_t \in C^{\gamma}_t C^0_{\operatorname{loc}}\). An equivalent more pratical formulation of (67) is the following one: for any \(\varphi \in C^{\infty}_c (\mathbb{R}^d)\), we have the estimate

which is uniform over \((s, t) \in \Delta_T\) but depends on \(K =\operatorname{supp} \varphi\); choosing \(\varphi = \rho^{\varepsilon} (x - \cdot)\) with \(x \in B_R\) and \(\rho^{\varepsilon}\) standard mollifier, since \(T^w_{s, t} b \cdot \nabla u_s\) is a well defined distribution, we obtain

If in addition \(u \in C^0_t C^1_{\operatorname{loc}}\), we can integrate by parts in (68) back to obtain

if \(u \in C^{\gamma}_t C^1_{\operatorname{loc}}\), then this necessarily implies the pointwise identity

which is meaningul since \(T^w b (\cdot, x), \nabla u (\cdot, x) \in C^{\gamma}_t\). It is therefore clear that for regular \(b\), any classical solution of (61) is also a solution in the sense of Definition 63.

We start by showing that our candidate solution satisfies Definition 63.

Proof. The regularity of \(u \in C^{\gamma}_t C^0_x \cap C^0_t C^1_{\operatorname{loc}}\) follows from Remark 57, since \(\psi\) satisfies

combined with the regularity of \(u_0\). Recall that by (55), for any \(s < t\) it holds

where

By Young chain rule, we have the estimates

which can all be taken uniform over \(x\) belonging to a compact set \(K\); combining them we deduce that

in the sense of the equality holding up to a term of order \(| t - s |^{2 \alpha}\). Therefore

which implies the conclusion.\(\Box\)

Remark 66. If \(T^w b \in C^{\gamma}_t C^3_x\), then by Theorem 58 and Remark 57 \(\psi \in C^0_t C^2_x \cap C^{\gamma}_t C^1_x\). It is then possible to check with standard calculations that for \(u_0 \in C^2_x\), the solution \(u\) constructed as above belongs to \(C^0_t C^2_x \cap C^{\gamma}_t C^1_x\) as well. Therefore in this case, by Remark 64, \(u\) also satisfies the stronger pointwise identity (70).

By the method of characteristics, we are able to obtain the following preliminary uniqueness result. It is however of limited applicability, see Remark 68 below.

Proof. In order to conclude, it is enough to show that the function \(f_t := u (t, \Phi_t (x))\) is constant; in particular, it suffices to prove that \(| f_{s, t} | \lesssim | t - s |^{2 \gamma}\) since \(\gamma > 1 / 2\). By the regularity assumption on \(u\), it satisfies (70) and therefore

Choosing appropriately \(R\) we have

where in the last passage we used the fact that \(\Phi_{s, t} (x) = \int_s^t T^w b (\mathrm{d} r, \Phi_r (x)) \mathrm{d} r\).\(\Box\)

Remark 68. The hypothesis \(u \in C^{\gamma}_t C^2_x\) is required in order to justify the passage

which is not true in general under the sole assumption \(u \in C^{\gamma}_t C^1_x\). However, for general \(T^w b \in C^{\gamma}_t C^2_x\), we only know that \(\psi \in C^{\alpha}_t C^0_{\operatorname{loc}} \cap C^0_t C^1_{\operatorname{loc}}\) and so the solution constructed by \(u_t (x) = u_0 (\psi_t (x))\) is not a priori in the class \(C^{\gamma}_t C^2_x\). For this reason, Lemma 67 is potentially vacuous, as it might only imply the non existence of \(C^{\gamma}_t C^2_x\)-solutions, while leaving open the problem of uniqueness in the class where \(u\) constructed as in Lemma 65 lives. We believe the same issue arises in Theorems 3.6 and 3.7 from [33], which do not settle the problem of uniqueness.

Observe that the above issue is typical of the Young regime and is completely absent in the case \(b \in C^0_{t, x}\), where uniqueness follows immediately from standard arguments.

In order to prove uniqueness of solutions to (66) in the class \(C^{\gamma}_t C^0_{\operatorname{loc}} \cap C^0_t C^1_{\operatorname{loc}}\), we need to use an appropriate commutator lemma, in the style of [20]. The basic idea is as follows: let \(\{ \rho_{\varepsilon} \}_{\varepsilon > 0}\) be a family of standard mollifiers (assume \(\rho _1 = \rho\) to be supported on \(B_1\) for simplicity), denote \(u^{\varepsilon} = \rho_{\varepsilon} \ast u\); by equation (69) we deduce that for any \(R > 0\), adopting the notation \(C^0_R = C^0_{B_R}\), it holds

where the estimate is uniform in \(\varepsilon > 0\) and the commutator \(R^{\varepsilon}\) appearing is the bilinear operator

Now \(u^{\varepsilon} \in C^{\gamma}_t C^2_{\operatorname{loc}}\) and so we can apply the same idea of the proof of Lemma 67, i.e. study the function \(f^{\varepsilon}_t = u^{\varepsilon}_t (\Phi_t (x))\), which we expect to be quasi constant; in the estimates, terms of the form \(R^{\varepsilon} (u_s, T^w_{s, t} b) (\Phi_s (x))\) will then start to appear, and so we need to control them as \(\varepsilon \rightarrow 0\). For this reason we need the following lemma.

Proof. The proof is analogue to the one of Lemma II.1 from [20]. It holds

Thus claim i. follows from \(\| (h\operatorname{div}g)^{\varepsilon} \|_{C_R^0} \leqslant \| h \|_{C_{R + 1}^0} \| g \|_{C_{R + 1}^1}\) and

where the estimate is uniform in \(x \in B_R\). Now fix \(R > 0\); we can assume that \(h\), \(g\), \(D g\) all have modulus of continuity \(\omega\) on \(B_{R + 1}\). By known properties of convolutions, \((h\operatorname{div}g)^{\varepsilon} \rightarrow h\operatorname{div}g\) uniformly on compact sets; moreover for all \(x \in B_R\) it holds

combined with a similar estimate for \(| h (x - \varepsilon z) - h (x) |\), this implies that, uniformly in \(x \in B_R\),

which implies claim ii. . The statements for \(R^{\varepsilon} : C_{\operatorname{loc}}^1 \times C_{\operatorname{loc}}^2 \rightarrow C_{\operatorname{loc}}^1\) follow immediately once we observe that \(\partial_i R^{\varepsilon} (h, g) = R^{\varepsilon} (\partial_i h, g) + R^{\varepsilon} (h, \partial_i g)\) as we apply the previous results.\(\Box\)

We have now all the ingredients to show uniqueness in the class \(C^{\gamma}_t C^0_x \cap C^0_t C^1_x\).

Proof. As before, it is enough to show that for any \(x \in \mathbb{R}^d\), the function \(f_t : = u_t (\Phi_t (x))\) satisfies \(| f_{s, t} | \lesssim | t - s |^{2 \gamma}\), as it implies that \(f\) is constant. Recall that \(\Phi\) satisfies the estimate \(| x - \Phi_t (x) | \lesssim | t |^{\gamma}\) uniformly in \(x\), therefore we can fix \(B_R\) such that \(\Phi_t (x) \in B_{2 R}\) for all \(t \in [0, T]\) and all \(x \in B_R\); from now on all the norms appearing will be localised on \(B_{2 R}\) without writing it explicitly.

Since \(u\) is a solution of (66), it satisfies (69) and therefore \(u^{\varepsilon}\) is such that

Define \(f^{\varepsilon}_t = u^{\varepsilon}_t (\Phi_t (x))\); using the above property and going through similar calculations as in the proof of Lemma 67, we deduce that

The estimate above a priori depends on \(\varepsilon\), as it involves \(\| u^{\varepsilon} \|_{C^{\gamma} C^2}\), but we are now going to show that under the assumptions on \(T^w b\) and \(u\) it is actually uniform in \(\varepsilon > 0\). This is accomplished with the help of the Sewing lemma, see Lemma 78 from Appendix A.1. Define

so that relation (72) can be rephrased as \(| f^{\varepsilon}_{s, t} - \Gamma^{\varepsilon}_{s, t} | \lesssim | t - s |^{2 \gamma}\). We can estimate \(\| \delta \Gamma^{\varepsilon} \|_{2 \gamma}\) as follows:

where we used the fact that \(\mathopen{\unicode{x27E6}} \Phi_{\cdot} (x) \mathclose{\unicode{x27E7}}_{\gamma} \lesssim 1\) by Theorem 53 and the estimate is uniform in \(\varepsilon\), since \(\| R^{\varepsilon} \|_{\mathcal{L}^2 (C^i \times C^{i + 1} ; C^i)} \leqslant C_1\) for \(i = 1, 2\) by Lemma 69. It follows that \(\| \delta \Gamma^{\varepsilon} \|_{2 \gamma} \leqslant C_2\) for some constant independent of \(\varepsilon\) and therefore by Lemma 78 (specifically estimate (83)) there exists \(C_3\) such that

Since \(u^{\varepsilon}_t (\Phi_t (x)) \rightarrow u_t (\Phi_t (x))\) and by part ii. of Lemma 69 \(R^{\varepsilon} (u_s, T^w_{s, t} b) (\Phi_s (x)) \rightarrow 0\), taking the limit as \(\varepsilon \rightarrow 0\) in (73) we deduce that \(| u_t (\Phi_t (x)) - u_s (\Phi_s (x)) | \leqslant C_3 | t - s |^{2 \gamma}\), which gives the conclusion.\(\Box\)

We now pass to study weak solutions of the continuity equation associated to \(T^w b\). Given a distribution \(v\), we say that \(v \in (C^1_x)^{\ast}\) if there exists a constant \(C\) such that \(| \langle v, \varphi \rangle | \leqslant C \| \varphi \|_{C^1}\) for all smooth \(\varphi\). We denote by \(\| v \|_{(C^1)^{\ast}}\) the optimal constant \(C\)\(\). Note that, when \(v\) is a measure, \(\| v_{s, t} \|_{(C^1)^{\ast}}\) is the \(1\)-Wasserstein distance between \(v_t\) and \(v_s\).

Remark 72. As before, it can be shown that for smooth \(b\), any classical solution of

is also a solution in the sense of the definition above. Equations (74) and (75) can be rephrased as \(v\) satisfying the functional Young integral equation

where the integral inside the divergence is a well defined element of \((C^1)^{\ast}\) since the product between \(C^1\) and \((C^1)^{\ast}\) is still an element of \((C^1)^{\ast}\). Formulation (75) is however more useful for our purposes.

Proof. Let us first show that \(v\) defined as above belongs to \(C^{\gamma}_t (C^1_x)^{\ast}\). It holds

where we used estimate (27); it follows that \(\| v \|_{C^{\gamma}_t (C^1_x)^{\ast}} \lesssim \| v_0 \|_{\mathcal{M}}\). We now check that \(v\) is a solution in the sense of Definition 71. It holds

where as before we used (27) and the estimate is uniform in \(x\); similarly

Combining the two estimates we obtain

which yields

and thus the conclusion.\(\Box\)

Proof. As before, by linearity it is enough to show that there exists a unique solution for the initial condition \(v_0 \equiv 0\). The basic strategy is the usual one: given any \(u_0 \in C^{\infty}_c\), setting \(u_t (x) = u_0 (\psi_t (x))\), it is enough to show that the function \(f_t := \langle v_t, u_t \rangle\) is constant, as it implies

and thus reasoning as in the proof of Proposition 61 that \(v_t \equiv 0\). Observe that the function \(u\) has compact space-time support, so we don't need to introduce localisations here.

Now we reason following the same lines as in Theorem 70, namely we spatially mollify \(u\) so that now \(u^{\varepsilon}\) solves

and all the terms are in \(C^1_x\) due to the mollification. Define \(f^{\varepsilon}_t = \langle v_t, u^{\varepsilon}_t \rangle\), then

The last term trivially satisfies \(| \langle v_{s, t}, u^{\varepsilon}_{s, t} \rangle | \lesssim_{\varepsilon} | t - s |^{2 \gamma}\). Combining the estimates

which come respectively from \(v\) being a solution of (75) and (76) above, we overall obtain

As before, the estimate a priori depends on \(\varepsilon\), but we can apply the Sewing lemma for the choice \(\Gamma_{s, t} = \langle v_s, R^{\varepsilon} (u_s, T^w_{s, t} b) \rangle\) for which, by analogue computations to the ones of Theorem 70, it holds

uniformly in \(\varepsilon > 0\). Therefore there exists a constant \(C\) independent of \(\varepsilon\) such that

By the properties of \(R^{\varepsilon}\), taking \(\varepsilon \rightarrow 0\) we deduce \(| \langle v_t, u_t \rangle - \langle v_s, u_s \rangle | \lesssim | t - s |^{2 \gamma}\) which implies the conclusion.\(\Box\)

Appendix ASome tools

This appendix collect some technical estimates and some reminder of various standard results, from certain functional spaces to stochastic integration in Banach setting.

A.1Some useful lemmas

The following chaining lemma is a slight variation on the one from [12], Lemma 3.1.

Proof. Without loss of generality we can assume \(T = 1\). Also, we will only show that proof in the case \(\alpha = 1 / 2\), the other cases being entirely analogue. Let us define the random variable

Then it follows from the assumption that \(\mathbb{E} [R (\lambda)] \leqslant C\). We can then apply Lemma 3.1 from [12] to deduce that there exist deterministic positive constants \(K, \beta\) such that

which implies by taking the logarithm and dividing by \(- \log | t - s |\) that

which yields the conclusion. Alternatively, it follows from the assumption that

which implies that we can apply Garsia-Rodemich-Rumsey Theorem (see [30]) for the choice \(\psi (x) = e^{\lambda x^2}\), \(p (x) = x^{\alpha}\), which gives

and from which we can again deduce that

and the exponential integrability bound. The final claim follows immediately.\(\Box\)

We also provide here a simple lemma on a priori bounds on solutions to linear Young differential equations, in the style of Section 6.2 from [39].

Proof. Since \(A \in C^{\gamma}_t C^{\infty}_x\), uniqueness of solutions is well known (see for instance [39]), so we are only interested in proving the bounds (79) and (80). Up to renaming \(x_0\), we can assume \(h_0 = 0\); we can also assume up to rescaling everything that \(T = 1\).

We adopt the following notation: for \(\Delta \leqslant 1\), we consider

Let \(\Delta > 0\) to be chosen later, \(s < t\) such that \(| t - s | \leqslant \Delta\), by (78) it holds

and so dividing both sides by \(| t - s |^{\gamma}\), taking the supremum over \(s, t\) and choosing \(\Delta\) such that \(C \Delta^{\gamma} \mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma}} \leqslant 1 / 2\) we obtain

We now distinguish two cases. If \(\mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma}}\) is such that \((2 + C) \mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma}} \leqslant 1 / 2\), then it follows from (81) with the choice \(\Delta = 1\) and the trivial estimate \(\| x \|_{C^0} \leqslant | x_0 | + \mathopen{\unicode{x27E6}} x \mathclose{\unicode{x27E7}}_{\gamma}\) that

which immediately implies the conclusion. Suppose instead the opposite and choose \(\Delta\) such that \(1 / 4 \leqslant (2 + C) \Delta^{\gamma} \mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma}} \leqslant 1 / 2\); define \(I_n = [(n - 1) \Delta, n \Delta]\), \(J_n = \sup_{t \in I_n} | x_t |\), then estimates similar to the one done above show that

which implies recursively that for a suitable constant \(C\) it holds \(J_n \leqslant C^n (| x_0 | + \mathopen{\unicode{x27E6}} h \mathclose{\unicode{x27E7}}_{C^{\gamma}})\). Since \(n \sim \Delta^{- 1} \sim \mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma}}^{1 / \gamma}\) we deduce that

which gives (80); this combined with \(\Delta^{- \gamma} \sim \mathopen{\unicode{x27E6}} A \mathclose{\unicode{x27E7}}_{C^{\gamma}}\), estimate (81) and the basic inequality

yields estimate (79).\(\Box\)

Similarly to the above lemma, we also have the following result.

Proof. Since \(x\) is a solution to the nonlinear YDE, for any \(s < t\) it holds

where in the second line we used the fact that \(A (s, 0) = 0\) by hypothesis. The rest of the proof from here on is identical to the one of Lemma 76 and we omit it. The inequality (82) is a combination of inequalities (79) and (80).\(\Box\)

We conclude this section by recalling the Sewing lemma, which is a fundamental tool in the theory of rough paths. Consider an interval \([0, T]\) and a Banach space \(E\); let \(\Delta_n\) denote the \(n\)-simplex on \([0, T]\), so that \(\Delta_n = \{ (t_1, \ldots, t_n) : 0 \leqslant t_1 \leqslant \ldots \leqslant t_n \leqslant T \}\). Given a map \(\Gamma : \Delta_2 \rightarrow E\), we define \(\delta \Gamma : \Delta_3 \rightarrow E\) by

We say that \(\Gamma \in C^{\alpha, \beta}_2 ([0, T] ; E)\) if \(\Gamma_{t, t} = 0\) for all \(t \in [0, T]\) and \(\| \Gamma \|_{\alpha, \beta} < \infty\), where

Let us remark that for a map \(f : [0, T] \rightarrow E\), we still denote by \(f_{s, t}\) the increment \(f_t - f_s\).

For a proof, see Lemma 4.2 from [28]. Let us point out that estimate (83) is extremely useful even in cases even when \(\mathcal{I} \, \Gamma\) is already known, as it asserts that in order to control \(\left\| \left( \mathcal{I} \, \Gamma \right)_{s, t} - \Gamma_{s, t} \right\|\) it is enough to have an estimate for \(\| \delta \Gamma \|_{\beta}\).

A.2Function spaces

We recall here the definition and basic properties of the function spaces we consider, which are Bessel potential spaces \(L^{s, p} (\mathbb{R}^d)\) and Besov spaces \(B^s_{p, q} (\mathbb{R}^d)\). In particular, in view of application to regularity estimates from Section 3.3, we need interpolation estimates and heat kernel estimates for such spaces. Bessel potential spaces are a subclass of Triebel–Lizorkin spaces, which will be also introduced. Most of the material is classical and covered in the monographs [4] and [55].

It follows immediately from the definition and the semigroup property that \(G^t\) provides an isomorphism of \(L^{s, p}\) and \(L^{s + t, p}\) in the sense that \(\| G^t f \|_{L^{s + t, p}} = \| f \|_{_{L^{s, p}}}\). This allows also to define \(L^{s, p}\) for negative values of \(s\), as the set of distributions \(f\) such that \(G^s f \in L^p\). Whenever \(s = m\) integer, the space \(L^{s, p}\) coincides with the classical Sobolev space \(W^{m, p}\), with equivalent norm. Similarly to Sobolev spaces, Bessel embeddings are available; in particular if \(s p > d\), we have the continuous embedding \(L^{s, p} \hookrightarrow C^{\gamma}\) with \(\gamma = s - d / p\), whenever \(\gamma\) is not an integer.

The spaces \(B^s_{2, 2}\) coincide with the (fractional) Sobolev spaces \(H^s\) which also coincide with \(L^{s, 2}\); however, for \(p \neq 2\) Bessel and Besov spaces do not coincide. \(B^s_{\infty, \infty}\) coincide with \(C^s\) whenever \(s\) is not an integer. Also in the case of Besov spaces, embedding theorems are available; in particular, \(B^s_{p, q} \hookrightarrow B^{s - d / p}_{\infty, \infty}\), which coincides with \(C^{\gamma}\) whenever \(\gamma = s - d / p\) is not an integer. Let us also point out that the exponent \(q\) is most of the time not particularly relevant, as for any \(\tilde{q} < q\) and any \(\varepsilon > 0\) we have the embeddings \(B^s_{p, \tilde{q}} \hookrightarrow B^s_{p, q} \hookrightarrow B^{s - \varepsilon}_{p, \tilde{q}}\).

Both definitions of Besov and Triebel–Lizorkin spaces are independent of the dyadic pair \((\chi, \varphi)\), in the sense that different pairs yields the same space of distributions with equivalent norms. Bessel spaces \(L^{s, p}\) correspond to \(F^s_{p, 2}\); the spaces \(F^s_{p, q}\) and \(B^s_{p, q}\) coincide if and only if \(p = q\), in which case \(F^s_{p, p} = B^s_{p, p} = W^{s, p}\) are sometimes referred to as fractional Sobolev spaces, see [19]. In the case \(p \neq q\), suitable embeddings between \(F^s_{p, q}\) and \(B^s_{p, q}\) follow immediately from Minkowski's inequality, since their norms can be regarded respectively as \(L^p (\mathbb{R}^d, \lambda ; \ell^q (\mathbb{N}, \mu))\)- and \(\ell^q (\mathbb{N}, \mu ; L^p (\mathbb{R}^d, \lambda))\)-norms, where \(\lambda\) is the Lebesgue measure in \(\mathbb{R}^d\) and \(\mu\) is the counting measure on \(\mathbb{N}\). In particular, for \(p > q\) it holds \(B^s_{p, q} \hookrightarrow F^s_{p, q}\) while for \(p < q\) we have the reversed embedding.

We now state a simple interpolation-like inequality for Bessel and Besov spaces. Since we don't have a direct reference for this result, we also provide a quick proof.

Proof. We use here the equivalent norm for \(L^{s, p}\) given by \(\| \cdot \|_{F^s_{p, 2}}\) as defined above. For any \(N \in \mathbb{N}\) it holds

Choosing suitable \(N\) such that \(2^{s N} \| f \|_{L^p} \sim 2^{- \varepsilon N} \| f \|_{F^{s + \varepsilon}_{p, 2}}\) we obtain the conclusion for \(L^{s, p}\).

An similar proof can be carried out for \(B^s_{p, q}\); alternatively in this case one can use Hölder inequality as follows:

which gives the conclusion for the choice \(\theta = s / (s + \varepsilon)\).\(\Box\)

We also need to recall the action of the heat flow \(P_t\) on such spaces; with a slight abuse of notation we will denote by \(P_t\) both the convolution operator and the Gaussian density itself.

Both statements are classical, the first one following immediately from the fact that, due to the scaling \(P_t = t^{- d / 2} P_1 (t^{- 1 / 2} \cdot)\) which implies \(\| P_t \|_{L^{\rho, 1}} = t^{- \rho / 2} \| P_1 \|_{L^{\rho, 1}}\); see Proposition 5 at page 2414 of [42], for a proof in a more general context of the second statement.

A.3A primer on stochastic integration in UMD Banach spaces

In this appendix we recall several results on abstract stochastic integration which are needed in order to complete the proof of Theorem 14; we believe they are also of independent interest and therefore provide a general presentation. In view of application to Section 3.3 we only need results for martingale type 2 spaces, which however yield the restriction to work on \(L^p\)-based spaces with \(p \geqslant 2\); weakening this condition to the case \(p = 1\) would highly enhance the results, as discussed in Remark 22, which is why in this appendix we also discuss UMD Banach spaces. Even with this more general theory we are currently not able to overcome the obstacle, we believe it might be of help for future developments and improvements.

All the material presented here is taken from [56], [57]. Also, we restrict for simplicity to the case \(W\) is a real valued Brownian motion (the extension to the vector valued case \(W \in \mathbb{R}^d\) being straightforward) but the theory is far more general as it considers the case of \(H\)-cylindrical Brownian motion, \(H\) being an abstract Hilbert space. This gives rise to \(\gamma\)-Radonifying norms \(\gamma (H, E)\); in our simple setting, \(H =\mathbb{R}\), for any Banach space \(E\) it holds \(\| \cdot \|_{\gamma (H, E)} = \| \cdot \|_E\).

Examples of martingale type spaces are the following:

• Every Banach space has martingale type 1.

• Every Hilbert space has martingale type 2.

• A closed subspace of a Banach space of martingale type \(p\) has still martingale type \(p\).

• If \(E\) has martingale type \(p\) and \((S, \mathcal{A}, \mu)\) is a measure space, then \(L^r (S ; E)\) with \(r \in [1, \infty)\) has martingale type \(p \wedge r\); in particular Lebesgue spaces \(L^p (\mathbb{R}^d)\) have martingale type \(p \wedge 2\).

• Let \((E_0, E_1)\) be an interpolation couple such that \(E_i\) has martingale type \(p_i \in [1, 2]\), let \(\theta \in (0, 1)\) and consider \(p \in [1, 2]\) such that \(1 / p = (1 - \theta) / p_0 + \theta / p_1\). Then both the complex and real interpolation spaces \(E_{\theta}\) and \(\tilde{E}_{\theta}\) have martingale type \(p\).

For the last two examples see Propositions 7.1.3 and 7.1.4 from [36]. It follows from the previous list of examples that Sobolev spaces \(W^{k, p} (\mathbb{R}^d)\) with \(p \in [2, \infty)\) have martingale type as they can be identified with closed subspaces of \(L^p (\mathbb{R}^d)^{\otimes n}\) for suitable \(n\); Bessel potential spaces \(L^{s, p} (\mathbb{R}^d)\) with general \(s\) are isomorphic to \(L^p (\mathbb{R}^d)\), with isomorphism given by \(G^s = (1 - \Delta)^{s / 2}\), therefore for \(p \in [2, \infty)\) they have martingale type 2. In the case of Besov spaces \(B^s_{p, q}\) with \(p, q \in [2, \infty)\), again it can be shown that they have martingale type 2, either by constructing them as interpolation spaces (see for instance Section 17.3 from [40]) or reasoning as follows: by definition, any \(\varphi \in B^s_{p, q}\) can be identified with a sequence \(\{ \Delta_j \varphi \}_j \subset L^p (\mathbb{R}^d)\) with suitable summability, namely such that it belongs to \(\ell^q (\mathbb{N}, \mu ; L^p (\mathbb{R}^d))\), where \(\mu (\{ j \}) = 2^{- s q j}\); in the case \(p, q \in [2, \infty)\) by the previous examples it has martingale type 2.

Now let \(W\) be a real valued \(\mathcal{F}_t\)-Brownian motion on a filtered probability space \((\Omega, \mathcal{F}, \{ \mathcal{F}_t \}_{t \geqslant 0}, \mathbb{P})\), \(\{ \mathcal{F}_t \}_{t \geqslant 0}\) being a filtration satisfying the usual conditions. For martingale type 2 spaces it is possible to define stochastic integrals analogously to the standard case: for an adapted elementary process \(\phi : \mathbb{R}_+ \times \Omega \rightarrow E\), namely of the form

where \(0 \leqslant t_1 < t_2 < \cdots < t_n\), \(x_i \in E\), \(F_i \in \mathcal{F}_{t_i}\), we set

Using the martingale type 2 property it is then possible to show that the \(L^2\) norm of the process defined in this way is controlled by \(\| \phi \|_{L^2 (\mathbb{R}_+ \times \Omega, E)}\), see Theorem 4.6 from [57]. By standard approximation procedures, together with Doob's maximal inequality, the following analogue of standard Itô integration can then be proven.

Let us also remark that it follows immediately from the definition for simple processes and the usual approximation procedure that, for any \(\phi\) as above and any deterministic \(\varphi^{\ast} \in E^{\ast}\), the following identity holds

where the integral on the r.h.s. is a standard real valued stochastic integral.

We are now ready to complete the proof of Theorem 14.

Proof. (of Theorem 14) Let us first show the following general fact: given a separable Banach space \(E\) and two \(E\)-valued random variables \(X\) and \(Y\) such that for any \(\varphi^{\ast}\) in a linearly dense subspace of \(E^{\ast}\) it holds

then necessarily \(X = Y\) \(\mathbb{P}\)-a.s. Indeed, it follows from the linear density assumption that relation (87) holds for any \(\varphi^{\ast} \in E^{\ast}\); by separability of \(E\) and Hahn-Banach Theorem, it is possible to find a countable collection \(\{ \varphi_n^{\ast} \}_n \subset E^{\ast}\) such that \(\| \varphi^{\ast}_n \|_{E^{\ast}} = 1\) for all \(n\) and

By (87) and the fact that the supremum is over a countable set, we can find a set \(\Gamma\) of full probability such that

which proves the claim. Now let \(b \in C^{\infty}_c ([0, T] \times \mathbb{R}^d ; \mathbb{R})\), so that it can be identified with an element of \(L^2 (0, T ; H^{\alpha} (\mathbb{R}^d))\) for any \(\alpha > 0\); choose \(\alpha\) big enough so that \(H^{\alpha}\) embeds into continuous functions vanishing at infinity. Then thanks to relation (86), equation (13) can be written as: for a given \(x \in \mathbb{R}^d\), \(\mathbb{P}\)-a.s. it holds

where the first two integrals are interpreted as (random) Bochner integrals while the last one as a stochastic integral in \(H^{\alpha}\) (with the inner integral being a random Bochner integral); integrability and predictability are straightforward due to the regularity of \(b\) and the properties of \(W^{2, H}_{u, r}\). Finally, as the collection \(\{ \delta_x \}_{x \in \mathbb{R}^d}\) is linearly dense in \(H^{- \alpha}\) and \(H^{\alpha}\) is separable, we can apply the general fact above to deduce that, for \(s < t\) fixed, the random variables above coincide on a set of full probability, without the need of testing against \(\delta_x\). This is exactly formula (14).\(\Box\)

In the setting of martingale type 2 spaces a one-sided Burkholder's inequality is available; we state it with the optimal asymptotic behaviour of the constants, which is needed in the estimates in Section 3.3. It was first shown by Seidler in [51].

This concludes the exposition of results needed in the proofs of this work. In the rest of this appendix, we present a brief account on stochastic integration in UMD Banach spaces.

Some of the major drawbacks of martingale type 2 spaces are the fact that they do not include \(L^p\) spaces with \(p < 2\), Burkholder's inequality is in general only one-sided and it is not sharp, which is troublesome in applications to maximal regularity of mild solutions of SPDEs. This motivates the introduction of a larger class of spaces. As before, we only consider the case of a real valued \(W\), but the theory extends to \(W\) being a cylindrical \(H\)-Brownian motion for an Hilbert space \(H\).

It can be shown that if \(E\) is UMD for some \(p \in (1, \infty)\), then it is actually UMD for all \(p \in (1, \infty)\). Examples are the following (here \(p' \in (1, \infty)\) denotes the conjugate of \(p\)):

• Every Hilbert space \(H\) is UMD with \(\beta_{p, H} = \max \{ p, p' \}\).

• If \(E\) is a UMD Banach space and \((S, \mathcal{A}, \mu)\) is measure space, then \(L^p (\mu ; E)\) is a UMD space with \(\beta_{p, L^p (\mu ; E)} = \beta_{p, E}\).

• \(E\) is UMD if and only if \(E^{\ast}\) is UMD and it holds \(\beta_{p, E} = \beta_{p', E^{\ast}}\).

In the case of UMD spaces, it is possible again to construct stochastic integrals in a suitable class of predictable processes and to obtain two-sided Burkholder inequalities.

In the above statement, \(\gamma^p (L^2 (\mathbb{R}_+), X)\) stands for the \(p\)-th \(\gamma\)-Radonifying norm; we omit the precise definition, which can be found in [56], [57]. There are special cases in which the \(\gamma\)-Radonifying norm is equivalent to other norms with a simpler expression, in particular when \(E = L^q (\mu)\), in which case there is an isomorphism of Banach spaces

and so the previous inequality can be reformulated as

In the case \(q \geqslant 2\), an application of Minkowski's inequality then yields

which is consistent with the aforementioned results for martingale type 2 spaces. In the general case instead, assuming we want to estimate the \(L^q (\mathbb{R}^d)\) norm of an averaged operator by means of the Ito^{^}–Tanaka formula (14), we would then need to estimate a term of the form (we omit the constants for simplicity)

which we are currently not able to do. The techniques employed in Section 3.3 rely quite crucially on the simplifications given by a formula of the form (88).

Bibliography