1Introduction

Consider the following stochastic hyper-viscous Navier–Stokes equation on \(\mathbb{R}_+ \times \mathbb{T}^2\)

where \(\mathbb{T}^2\) is the two dimensional torus, \(A = - \Delta\) on \(\mathbb{T}^2\), \(\nabla^{\perp} := (\partial_2, - \partial_1)\), \(\theta > 1\), and \(\xi\) denotes a space-time white noise. The initial condition for \(u\) will be taken distributed according to the white noise on \(\mathbb{T}^2\) or an absolute continuous perturbation thereof with density in \(L^2\). The white noise on \(\mathbb{T}^2\) is formally invariant for the dynamics described by (1.1) and the existence theory for the corresponding stationary process has been addressed by Gubinelli and Jara in [14] using the concept of energy solutions for any \(\theta > 1\). Uniqueness was left open in the aforementioned paper, and the main aim of the present work, which can be thought of as a continuation of [17], is to introduce a martingale problem formulation (1.1) for which we can prove uniqueness.

In order to properly formulate the martingale problem, we need to investigate the infinitesimal generator for eq. (1.1) and uniqueness will result from suitable solutions of the associated Kolmogorov backward equation.

The variable \(u\) appearing in eq. (1.1) represents physically the velocity of a fluid. Rewriting the equation for the vorticity \(\omega := \nabla^{\perp} \cdot u\) yields

We also have the relation \(u = K \ast \omega\), for the Biot-Savart kernel \(K\) on \(\mathbb{T}^2\) given by

where \(k^{\perp} = (k_2, - k_1)\) and \(\mathbb{Z}_0^2 =\mathbb{Z}^2 \backslash \{ 0 \}\). It is more convenient to work with the scalar quantity \(\omega\) and with eq. (1.2).

The standard stochastic Navier-Stokes equation corresponds to the case \(\theta = 1\). However, this regime is quite singular for the white noise initial condition and no results are known, not even existence of a stationary solution, e.g. from limit of Galerkin approximations. While a bit unphysical, we will stick here to the hyper-viscous regime, namely \(\theta > 1\). Note that the noise has to be coloured accordingly in order to preserve the white noise as invariant measure. Moreover, we call energy measure the law under which the velocity field is a (vector-valued, incompressible) white noise. In terms of vorticity \(\omega\), the kinetic energy of the fluid configuration \(u\) is

where \(\hat{f} : \mathbb{Z}^2 \rightarrow \mathbb{C}\) denotes the Fourier transform of \(f : \mathbb{T}^2 \rightarrow \mathbb{R}\) defined as to have \(f (x) = \sum_{k \in \mathbb{Z}^2} e^{2 \pi \iota k \cdot x} \hat{f} (k)\). The energy measure is thus formally given by

where \(\mathrm{d} \omega\) denotes the “Lebesgue measure” on functions on \(\mathbb{T}^2\). Rigorously, this of course means the product Gaussian measure

with the restriction that \(\hat{\omega} (- k) = \overline{\hat{\omega} (k)}\). For \(f, g \in C^{\infty} (\mathbb{T}^2)\), we have

We can use the right-hand side as the definition of the covariance of \((\omega (f))_{f \in C^{\infty} (\mathbb{T}^2)}\), which determines the law of \(\omega\) as a centred Gaussian process indexed by \(H^1 (\mathbb{T}^2)\). If \(\eta\) is a white noise on \(L^2 (\mathbb{T}^2)\), then \(\mu\) has the same distribution as \(A^{1 / 2} \eta\) and it is only supported on \(H^{- 2 -} (\mathbb{T}^2)\).

A different situation occurs if we consider initial conditions distributed according to the enstrophy measure, namely the Gaussian measure for which the initial vorticity is a white noise. This measure is more regular than the energy measure and more results are known, both for the Euler dynamics (i.e., without dissipation and noise) and for the stochastic Navier-Stokes dynamics, see e.g. [1, 2, 3].

As we already remarked, we use here the technique introduced in [17] and strongly rooted in the notion of energy solution of Gonçalves and Jara [12], extended in [14]. With respect to [17] we give a slightly different formulation which simplifies certain technical estimates. The core of the argument however remains the same. The main point is to consider the well-posedness problem for (1.1) as a problem of singular diffusion, i.e. diffusions with distributional drift. The papers [10, 11, 8, 5] all follow a similar strategy in order to identify a domain for the formal infinitesimal generator \(\mathcal{L}= \tfrac{1}{2} \Delta + b \cdot \nabla\) of a finite dimensional diffusion. Then they show existence and uniqueness of solutions for the corresponding martingale problem. The key difficulty is that for distributional \(b\) the domain does not contain any smooth functions and instead one has to identify a class of non-smooth test functions with a special structure, adapted to \(b\). Roughly speaking they must be local perturbations of a linear functional constructed from \(b\). Recently other results of regularisation by noise for SPDEs [6, 7] have been obtained. An important difference is that our drift is unbounded and not even a function. The connection between energy solutions and regularisation by noise was first observed in [14].

Plan of the paper

Notation

identifying \(f\) and \(g\) whenever \(\| f - g \|_{H^s (\mathbb{T}^2)} = 0\). From now on, we write \(C (X, Y)\) to indicate the space of continuous functions from \(X\) to \(Y\). We also write \(a \lesssim b\) or \(b \gtrsim a\) if there exists a constant \(C > 0\), independent of the variables under consideration, such that \(a \leqslant C \cdot b\), and \(a \simeq b\) if both \(a \lesssim b\) and \(b \lesssim a\). If the aforementioned constant \(C\) depends on a variable, say \(C = C (x)\), then we use the notation \(a \lesssim_x b\), and similarly for \(\gtrsim\). For the sake of brevity, we will also use the notation \(k_{1 : n} = (k_1, \ldots, k_n)\).

2Galerkin approximations

In order to rigorously study the eq. (1.2), consider the solution \((\omega^m_t)_{t \geqslant 0}\) to its Galerkin approximation:

where

and \(\Pi_m\) denotes the projection onto Fourier modes of size less than \(m\), namely \(\Pi_m f (x) = \sum_{| k | \leqslant m} e^{2 \pi \iota k \cdot x} \hat{f} (k)\).

Proof. We can rewrite \(\omega^m\) in Fourier variables as \(\omega^m = w_{\textrm{\operatorname{fin}}}^m + w_{\textrm{\operatorname{lin}}}^m := \Pi_m \omega^m + (1 - \Pi_m) \omega^m\), in such a way that \(w_{\textrm{\operatorname{fin}}}^m\) and \(w_{\textrm{\operatorname{lin}}}^m\) solve a finite-dimensional SDE with locally Lipschitz continuous coefficients and an infinite-dimensional linear SDE, respectively. Global existence and invariance of \(\mu\) follow by Section 7 in [14]. Now, \(w_{\textrm{\operatorname{fin}}}^m\) has compact spectral support and therefore \(w_{\textrm{\operatorname{fin}}}^m \in C (\mathbb{R}_+, C^{\infty} (\mathbb{T}))\), while it can be proved that \(w_{\textrm{\operatorname{lin}}}^m\) has trajectories in \(C (\mathbb{R}_+, H^{- 2 -} (\mathbb{T}^2))\). Thus, \(\omega^m\) has trajectories in \(C (\mathbb{R}_+, H^{- 2 -} (\mathbb{T}^2))\).\(\Box\)

We define the semigroup of \(\omega^m\) for all bounded and measurable functions \(\varphi\) as \(T^m_t \varphi (\omega_0) := \mathbb{E}_{\omega_0} [\varphi (\omega_t^m)]\), where, under \(\mathbb{P}_{\omega_0}\), the process \(\omega^m\) solves (2.1) with initial condition \(\omega_0 \in H^{- 2 -} (\mathbb{T}^2)\).

On such cylinder functions the generator of the semigroup \(T^m\) has an explicit representation: Itô's formula gives, for \(\varphi \in \operatorname{Cyl}_{\mathbb{T}^2}\) as in Definition 2.3,

where \(\mathcal{L}^m := \mathcal{L}_{\theta} +\mathcal{G}^m\) with

and

Here, \((M_t (f_i))_{t \geqslant 0}\) is a continuous martingale with quadratic variation

and therefore \(\int_0^{\cdot} \sum_{i = 1}^n \partial_i \Phi (\omega_t^m (f_1), \ldots, \omega_t^m (f_n)) \mathrm{d} M_t (f_i)\) is a martingale. Consequently, we have

To extend this to more general functions \(\varphi\), we work via Fock space techniques. The Hilbert space \(L^2 (\mu)\) can be identified with the Fock space \(\mathcal{H}= \Gamma H^1_0 (\mathbb{T}^2) := \bigoplus_{n = 0}^{\infty} (H^1_0 (\mathbb{T}^2))^{\otimes n}\) with \(H^1_0 (\mathbb{T}^2) := \{ \psi \in H^1 (\mathbb{T}^2) : \hat{\psi} (0) = 0 \}\) and norm

by noting that any \(\varphi \in L^2 (\mu)\) can be written in chaos expansion \(\varphi = \sum_{n \geq 0} W_n (\varphi_n)\), where \(W_n\) is the \(n\)-th order Wiener-Itô integral and \(\varphi_n \in H^1_0 (\mathbb{T}^2)^{\otimes n}\) for every \(n \in \mathbb{N}\), see e.g. [18, 20] for details. We will use the convention that \(\varphi_n\) is symmetric in its \(n\) arguments, that is, we identify it with its symmetrisation.Note that cylinder functions are dense in \(\mathcal{H}\). We denote by \(\mathcal{N}\) the number operator, i.e. the self–adjoint operator on \(\mathcal{H}\) such that \((\mathcal{N} \varphi)_n := n \varphi_n\). It is well known that the semigroup generated by the number operator satisfies an hypercontractivity estimate, see Theorem 1.4.1 in [20]. We record it in the next lemma.

With these preparations we are ready to give expressions for the operators \(\mathcal{L}_{\theta}\) and \(\mathcal{G}^m\) in terms of the Fock space representation of \(\mathcal{H}\).

Proof. The computations are analogous to those of Lemma 3.7 of [16] for \(\mathcal{L}_{\theta}\) and of Lemma 2.4 and Lemma 2.7 in [17].\(\Box\)

Remark 2.6. \(\mathcal{G}_+^m\) and \(\mathcal{G}_-^m\) are (unbounded) operators which increase and decrease, respectively, the “number of particles” by one. Moreover, we know from (2.6) that they are formally the adjoint of the other (modulo a sign change).

A key result is given by the following bounds for \(\mathcal{G}_{\pm}^m\) acting on weighted subspaces of \(\mathcal{H}\).

These bounds will be proven later on in Section 6. In view of eq. (2.7), it is natural to identify a dense domain \(\mathcal{D} (\mathcal{L}^m)\) for \(\mathcal{L}^m\) as

Note that \(\langle \psi, (\mathcal{L}_{\theta} +\mathcal{G}^m) \varphi \rangle = \langle (\mathcal{L}_{\theta} -\mathcal{G}^m) \psi, \varphi \rangle\) for \(\psi, \varphi \in \mathcal{D} (\mathcal{L}^m)\) and in particular that \(\mathcal{L}_{\theta}\) is dissipative since for all \(\varphi \in \mathcal{D} (\mathcal{L}^m)\) we have

A priori \(\mathcal{L}^m\) is only the restriction to \(\mathcal{D} (\mathcal{L}^m)\) of the generator \(\hat{\mathcal{L}}^m\) of the semigroup \((T^m_t)_t\). However, we will also prove in Lemma 5.2 below that the operator \(\mathcal{L}^m\) is closable and that its closure is indeed the generator \(\hat{\mathcal{L}}^m\).

In order to exploit these pieces of information, we have to work with solutions of Galerkin approximations having “near-stationary” fixed-time marginal.

For an incompressible process \((\omega_t)_{t \geqslant 0}\) it makes sense, using a density argument involving cylinder functions, to define \(s \mapsto \varphi (\omega_s)\) for all \(\varphi \in \mathcal{H}\) as a stochastic process continuous in \(L^1\).

Proof. We get

Incompressibility easily follows from the fact that \(\mu\) is an invariant measure for the Galerkin approximations independently of \(m\).\(\Box\)

We will use the notation \(D_x\) to indicate the Malliavin derivative, see e.g. [17], which acts on cylinder functions \(\varphi\) as in Definition 2.3 as follows,

Proof. If \(\varphi\) is a cylinder function, then we have eq. (2.2) and in that case Doob's inequality and Lemma 2.9 yield, for all \(T > 0\),

The norm appearing on the right-hand side can be estimated as follows:

where we used a symmetrisation in the arguments of \(\hat{\varphi}_n\) in the 5th line. Using the bounds (2.7) and (2.11), one can extend formula (2.11) to all functions in \(\mathcal{D} (\mathcal{L}^m)\) by a density argument.\(\)

As far as (2.12) is concerned, let us remark that, provided the process \(\omega^m\) is started from its stationary measure \(\mu\), then the reversed process \((\tilde{\omega}_t = \omega_{T - t})_{t \geqslant 0}\) is also stationary and with (martingale) generator \(\tilde{\mathcal{L}}^m =\mathcal{L}_{\theta} -\mathcal{G}^m\). The forward–backward Itô trick used in [14] allows us to represent additive functionals of the form \(\int_0^t \mathcal{L}_{\theta} \psi (\omega_s^m) \mathrm{d} s\) as a sum of forward and backward martingales whose quadratic variations satisfy (2.10). Therefore,

Let \(\psi = (1 -\mathcal{L}_{\theta})^{- 1} \varphi\) and exploit (2.13) to compute

which is uniform in \(m\).\(\Box\)

3The cylinder martingale problem

We want now to take limits of Galerkin approximations and have a characterisation of the limiting dynamics. The main problem is that the formal limiting (martingale) generator \(\mathcal{L}\) does not send cylinder functions to \(\mathcal{H}\), therefore we cannot properly formulate a martingale problem for incompressible solutions. However, estimate (2.12) suggests that it is reasonable to ask that any limit process \((\omega_t)_{t \geqslant 0}\) satisfies

for all \(p \geqslant 1\) and all cylinder functions \(\varphi \in \mathcal{C}\). The proof of the next lemma is almost immediate.

Proof. Take \((\varphi_n)_n \subseteq \mathcal{C}\) such that \(\sum_n \| (1 -\mathcal{L}_{\theta})^{1 / 2} \varphi_n - \varphi \| < \infty\), then it is easy to see that \((I (\varphi_n))_n\) is a Cauchy sequence in \(C ([0, T] ; \mathbb{R})\) a.s. with limit \(I (\varphi) \in C ([0, T] ; \mathbb{R})\). It satisfies (3.1) by Fatou's lemma and, therefore, depends only on \(\varphi\) and not on the particular approximating sequence.\(\Box\)

From this we deduce that for such processes we have

in probability and in \(L^p\) for cylinder functions \(\varphi \in \mathcal{C}\). Here, on the right-hand side the quantity \(\mathcal{L} \varphi\) is defined as \(\mathcal{L} \varphi =\mathcal{L}_{\theta} \varphi + \lim_{m \rightarrow \infty} \mathcal{G}^m \varphi\), that is an element of the space of distributions \((1 -\mathcal{L}_{\theta})^{1 / 2} \mathcal{H}\). The limit exists and is unique thanks to the uniform estimates on \(\mathcal{G}^m\) in Lemma 2.7. As a consequence, we have also a notion of martingale problem w.r.t. the operator \(\mathcal{L}\) involving only cylinder functions.

Proof. The proof follows the one for Theorem 4.6 in [17].

Step 1. Consider \(p \geqslant 2\) and \(\varphi \in \mathcal{C}\). We want to derive an estimate for \(\mathbb{E} [| \varphi (\omega^m_t) - \varphi (\omega_s^m) |^p]\). We write then \(\varphi (\omega^m_t) - \varphi (\omega_s^m) = \int_s^t \mathcal{L}^m \varphi (\omega^m_r) \mathrm{d} r + M_t^{m, \varphi} - M_s^{m, \varphi}\), and get from Lemma 2.9 and eq. (2.12) the following bound

The martingale term can be bounded by means of the Burkholder-Davis-Gundy inequality and (2.11) as follows:

Therefore,

The law of the initial condition \(\varphi (\omega_0^m)\) is independent of \(m\), and by Kolmogorov's continuity criterion the sequence of real-valued processes \((\varphi (\omega^m))_m\) is tight in \(C (\mathbb{R}_+ ; \mathbb{R})\) whenever \(p \geqslant 4\) and \(\varphi \in \mathcal{C}\) is such that \(\| c_{4 p}^{\mathcal{N}} (1 -\mathcal{L}_{\theta})^{- 1 / 2} \varphi \| < \infty\). Note that this space contains in particular all the functions of the form \(\varphi (\omega) = \omega (f)\) with \(f \in C^{\infty} (\mathbb{T}^2)\). Hence, we can apply Mitoma's criterion [19] to get the tightness of the sequence \((\omega^m)_m\) in \(C (\mathbb{R}_+ ; \mathcal{S}')\).

Step 2. Since \(\omega_0^m \sim \eta \mathrm{d} \mu\), any weak limit has initial distribution \(\eta \mathrm{d} \mu\). Incompressibility is also clear since, for any \(\varphi \in \mathcal{H}\), we have

Using cylinder functions, we can pass to the limit in eq. (2.12) and prove that any accumulation point \((\omega_t)_t\) satisfies eq. (3.1). It remains to check the martingale characterisation (3.2). Fix \(\varphi \in \mathcal{C}\) and let \((\psi_n)_n \subseteq \mathcal{C}\) be such that \(\psi_n \rightarrow \mathcal{L} \varphi\) in \((1 +\mathcal{L}_{\theta})^{1 / 2} \mathcal{H}\). By convergence in law, incompressibility, eq. (2.12) and eq. (3.1), we have that

where the exchange of limits in the last line is justified by the uniformity in \(m\) of the bound in eq. (2.12). By dominated convergence in the estimates leading to Lemma 2.11 one has

as \(m \rightarrow \infty\). This is enough to conclude (again using eq. (2.12)) that

since \((\omega^m_t)_t\) solves indeed the martingale problem for \(\mathcal{L}^m\). This establishes that any accumulation point \((\omega_t)_t\) is a solution to the cylinder martingale problem for \(\mathcal{L}\). Similarly, one can pass to the limit on the martingales \((M_t^{m, \varphi})_t\) to show that the limiting quadratic variation is as claimed.\(\Box\)

4Uniqueness of solutions

Uniqueness of solutions to the cylinder martingale problem depends on the control of the associated Kolmogorov equation.

The following standard fact on generators of semigroups that will be useful in our further considerations. For the sake of the reader we provide also a proof to illustrate the relation between the Kolmogorov equation for a concrete operator and abstract semigroup theory.

Proof. Since \(\mathcal{A}\) is dissipative, the solution to the Kolmogorov equation is unique and \(\| \varphi (t) \| \leqslant \| \varphi_0 \|\). Then, if we let \(T_t \varphi_0 = \varphi (t)\) for \(\varphi_0 \in \mathcal{U}_{\mathcal{A}}\) we can extend \(T_t\) by continuity to the whole space \(\mathcal{H}\) as a contraction. By uniqueness, we have then \(T_{t + s} \varphi_0 = T_t T_s \varphi_0\), since \(t \mapsto T_{t + s} \varphi_0\) solves the equation with initial condition \(T_s \varphi_0\). Moreover, for \(\varphi_0 \in \mathcal{U}_{\mathcal{A}}\), we have that

which implies that \(t \mapsto T_t \varphi_0\) is strongly continuous. Again by density, we deduce that \((T_t)_{t \geqslant 0}\) is a strongly continuous semigroup. Let now \(\mathcal{B}\) be its Hille–Yosida generator. Then (4.2) implies that \(\mathcal{B} \varphi_0 = \partial_t T_t \varphi_0 |_{t = 0} =\mathcal{A} \varphi_0\) for all \(\varphi_0 \in \mathcal{U}_{\mathcal{A}}\), and therefore for all \(\varphi_0 \in \mathcal{D} (\mathcal{A})\) since \(\mathcal{B}\) is closed. So \(\mathcal{B}\) is an extension of \(\mathcal{A}\) and therefore \(\mathcal{A}\) is closable. Assume now that there exists another extension \(\tilde{\mathcal{B}}\) which is the generator of another strongly continuous semigroup \((S_t)_{t \geqslant 0}\) of contractions. Now, for all \(\varphi_0 \in \mathcal{U}_{\mathcal{A}} \subseteq \mathcal{D} (\mathcal{A}) \subseteq \mathcal{D} (\tilde{\mathcal{B}})\) we have \(\partial_t S_t \varphi_0 = \tilde{\mathcal{B}} S_t \varphi_0\), but also \(\partial_t T_t \varphi_0 =\mathcal{A}T_t \varphi_0 = \tilde{\mathcal{B}} T_t \varphi_0\). Since \(\tilde{\mathcal{B}}\) is dissipative (due to the fact that its semigroup is contractive), the associated Kolmogorov equation must have a unique solution and, as a consequence, \(T_t \varphi_0 = S_t \varphi_0\), which by density implies that \(T = S\) and that \(\mathcal{B}= \tilde{\mathcal{B}}\). Now observe that, if \(\varphi_0 \in \mathcal{U}_{\mathcal{A}}\), then \(T_t \varphi_0 \in \mathcal{D} (\mathcal{A})\) and by standard results on contraction semigroups (see e.g. Proposition 1.1.5 in [9]) we have \(\mathcal{A}T_t \varphi_0 =\mathcal{B}T_t \varphi_0 = T_t \mathcal{B} \varphi_0 = T_t \mathcal{A} \varphi_0\).\(\Box\)

Theorem 5.10 below tells us that we can find a dense domain \(\mathcal{D} (\mathcal{L}) \subseteq \mathcal{H}\) for \(\mathcal{L}\) such that the Kolmogorov equation

has a unique solution in \(C (\mathbb{R}_+ ; \mathcal{D} (\mathcal{L})) \cap C^1 (\mathbb{R}_+ ; \mathcal{H})\) for any initial condition in a dense set \(\mathcal{U} \subseteq \mathcal{H}\). As a first consequence, Lemma 4.1 tells us that \(\mathcal{L}\) is closable and its closure \({\mathcal{L}^{\natural}} \) is the generator of a strongly continuous semigroup \((T_t)_{t \geqslant 0}\) and \(\varphi (t) = T_t \varphi\) for all \(\varphi \in \mathcal{U}\).

Proof. By an approximation argument, it is easy to see that for any \(\varphi \in \mathcal{D} (\mathcal{L})\) the process

is a martingale, where the integral on the right-hand side is now understood as a standard Lebesgue integral of the continuous process \(s \mapsto (\mathcal{L} \varphi) (\omega_s)\) (which is well defined a.s.). The proof of the extension to time-dependent functions follows the same lines as that of Lemma A.3 in [17].\(\Box\)

For an incompressible process we have that, for all \(t \geqslant 0\),

for all \(\varphi \in \mathcal{D} \left( {\mathcal{L}^{\natural}} \right)\), and therefore also that \((T_{t - s} \varphi (\omega_s))_{s \in [0, t]}\) is a martingale for any solution of the cylinder martingale problem for \(\mathcal{L}\). This easily implies the main result of the paper.

Proof. Let us first prove that \((\omega_t)_{t \geqslant 0}\) is Markov. Let \(0 \leqslant t < s\), let \(X\) be an \(\mathcal{F}_t\)-measurable bounded random variable, where \(\mathcal{F}_t = \sigma (\omega_r : r \in [0, t])\), and let \(\varphi \in \mathcal{D} \left( {\mathcal{L}^{\natural}} \right)\), then \((T_{s - t} \varphi (\omega_t))_{t \in [0, s]}\) is a martingale and

i.e., \(\mathbb{E} [\varphi_0 (\omega_s) | \mathcal{F}_t ] = T_{s - t} \varphi (\omega_t) =\mathbb{E} [\varphi_0 (\omega_s) | \omega_t ]\), and the Markov property is a consequence of another density argument. Moreover, its transition kernel is given by the semigroup \((T_t)_{t \geqslant 0}\). By an induction argument, it is clear that any finite-dimensional marginal is determined by \(T\) and by the law of \(\omega_0 \sim \eta \mathrm{d} \mu\). As a consequence, the law of the process is unique. If \(\omega_0 \sim \mu\), then the process is stationary.\(\Box\)

Remark 4.4. As a by-product note that the formula \((T_t \varphi) (\omega_0) =\mathbb{E} [\varphi (\omega_t) | \omega_0]\) allows to extend the semigroup \(T\) to a bounded semigroup in \(L^p\) for all \(p \in [1, \infty]\) since

for all \(\psi, \varphi \in L^{\infty} (\mu)\) and all \(p, q \in [1, \infty]\) with \(1 / p + 1 / q = 1\). Therefore \(\| T_t \varphi \|_{L^q} \leqslant \| \varphi \|_{L^q}\). Moreover for all \(\varphi \in \mathcal{C}\) such that \(\| c_{4 p}^{\mathcal{N}} (1 -\mathcal{L}_{\theta})^{- 1 / 2} \varphi \| < \infty\) we have

as \(t \rightarrow 0\) by eq. (3.3). An approximation argument gives that \((T_t)_{t \geqslant 0}\) is strongly continuous in \(L^p\) for all \(1 \leqslant p < \infty\).

5The Kolmogorov equation

It remains to determine a suitable domain for \(\mathcal{L}\) and solve the Kolmogorov backward equation

for a sufficiently large class of initial data. In order to do so, we consider the backward equation for the Galerkin approximation with generator \(\mathcal{L}^m\) and derive uniform estimates. By compactness, this yields the existence of strong solutions to the backward equation after removing the cutoff. Uniqueness follows by the dissipativity of \(\mathcal{L}\).

5.1A priori estimates

Proof. Take \(h > 0\) and let \(\mathcal{G}^{m, h} = J_h \mathcal{G}^m J_h\), where \(J_h = e^{- h (\mathcal{N}-\mathcal{L}_{\theta})}\). The operator \(\mathcal{G}^{m, h}\) is bounded on \(\mathcal{H}\) by the estimates in Lemma 2.7. Consider \(\varphi_0^m \in \mathcal{D} (\mathcal{L}^m)\). Using the fact that \(\mathcal{L}_{\theta}\) is the generator of a contraction semigroup, we take \((\varphi^m (t))_{t \geqslant 0}\) to be the solution to the integral equation

in \(C (\mathbb{R}_+ ; (1 -\mathcal{L}_{\theta}) \mathcal{H})\) and deduce easily that \(\varphi^{m, h}\) solves the equation \(\partial_t \varphi^{m, h} (t) = (\mathcal{L}_{\theta} +\mathcal{G}^{m, h}) \varphi^{m, h} (t)\). Moreover

for any finite \(t \geqslant 0\) and \(p > 0\) but not uniformly in \(h\) and \(m\). Now

Using \(| (1 +\mathcal{N})^{2 p} -\mathcal{N}^{2 p} | \lesssim (1 +\mathcal{N})^{2 p - 1}\) and the uniform estimates in Lemma 2.7 we have that, for some \(\sigma \in (0, 1 / 2)\),

Therefore,

and by interpolation we can bound this by

for some small \(\delta > 0\). Therefore, we have

uniformly in \(m\) and \(h\). Integrating this inequality gives

for all \(p \geqslant 1\) where the constants are uniform in \(m\) and \(h\). Inserting this a priori bound in the mild formulation in eq. (5.1) we obtain

where we also used that

by the presence of the Galerkin projectors and our (non-uniform) bounds. Indeed, note that

We conclude that

uniformly in \(h\). We can then pass to the limit (by subsequence) as \(h \rightarrow 0\) and obtain a function \(\varphi^m \in C (\mathbb{R}_+, (1 +\mathcal{N})^{- p} (1 -\mathcal{L}_{\theta})^{- 1} \mathcal{H})\) satisfying the estimates

and

for all \(t \geqslant 0\) and \(p \geqslant 1\). As a consequence, \(\varphi^m \in C (\mathbb{R}_+, \mathcal{D} (\mathcal{L}^m))\) for all \(t \geqslant 0\) as soon as \(\| (1 +\mathcal{N})^2 (1 -\mathcal{L}_{\theta}) \varphi_0 \| < \infty\). By passing to the limit in the equation, \(\varphi^m\) also satisfies

Recall that we write \(T^m\) to indicate the semigroup generated by the Galerkin approximation \(\omega^m\). Moreover, if we denote by \(\hat{\mathcal{L}}^m\) its Hille–Yosida generator, we have the following result.

Proof. Let \((\omega^m_t)_{t \geqslant 0}\) be a solution to the Galerkin approximation (2.1) with initial condition \(\omega_0\). If \(\varphi \in \mathcal{C}\) is a cylinder function, then we have

By approximation (using a Bochner integral in \(\mathcal{H}\) on the right-hand side), we can extend this point-wise formula to all \(\varphi \in \mathcal{D} (\mathcal{L}^m)\) obtaining for them that \(T^m_t \varphi - \varphi = \int_0^t T^m_s \mathcal{L}^m \varphi \mathrm{d} s\) in \(\mathcal{H}\). For every \(\varphi \in \mathcal{D} (\mathcal{L}^m)\), Lemma 2.2 implies that the map \(s \mapsto T^m_s \mathcal{L}^m \varphi \in \mathcal{H}\) is continuous, and therefore

with convergence in \(\mathcal{H}\). As a consequence, \(\varphi \in \mathcal{D} (\hat{\mathcal{L}}^m)\) and we conclude that \(\hat{\mathcal{L}}^m\) is an extension of \((\mathcal{L}^m, \mathcal{D} (\mathcal{L}^m))\). By Lemma 4.1, we have that the closure of \(\mathcal{L}^m\) is \(\hat{\mathcal{L}}^m\) and that \(\mathcal{L}^m T^m_t \varphi = T^m_t \mathcal{L}^m \varphi\) for all \(\varphi \in \mathcal{V}\).\(\Box\)

Using the commutation \(\mathcal{L}^m T^m_t \varphi = T^m_t \mathcal{L}^m \varphi\), we are able to get better estimates, uniform in \(m\).

Proof. Recall \(T^m_t \varphi^m_0 = \varphi^m (t)\). We already know

which yields

and

which is what claimed.\(\Box\)

5.2Controlled structures

The a priori bounds (5.2) and (5.3) bring us in position to control \(\| \varphi^m (t) \|\), \(\| \partial_t \varphi^m (t) \|\), and \(\| \mathcal{L}^m \varphi^m (t) \|\) uniformly in \(m\) and locally uniformly in \(t\), but in order to study the limiting Kolmogorov backward equation we have first to deal with the limiting operator \(\mathcal{L}\) and to define a domain \(\mathcal{D} (\mathcal{L})\).

To take care of the term \(\mathcal{G}\) in the limiting operator \(\mathcal{L}\), we decompose it by means of a cut-off function \(\mathcal{M}= M (\mathcal{N})\) as follows

We then set

so that

Proof. We start with the estimate on \(\mathcal{G}_+^{m, \succ}\). We have, for \(\varepsilon \in] 0, 1 / 2 - 1 / (2 \theta) [\), using Lemma 2.7,

The bound on \(\mathcal{G}_-^{m, \succ}\) can be obtained using again Lemma 2.7:

In conclusion, for \(\varepsilon \in] 0, (\theta - 1) / (2 \theta) [\), choosing \(M (n) = L (n + 1)^{3 \theta / (\theta - 1 - 2 \theta \varepsilon)}\), for \(L \geqslant 1\),

Now let \(\varphi^{\sharp} \in w (\mathcal{N})^{- 1} (1 -\mathcal{L}_{\theta})^{- 1 / 2} \mathcal{H}\), the map

satisfies, for some positive constant \(C\),

Namely, \(\Psi^m\) is well-defined and, choosing \(L\) large enough, it is a contraction leaving the ball of radius \(2 \| w (\mathcal{N}) (1 -\mathcal{L}_{\theta})^{1 / 2} \varphi^{\sharp} \|\) invariant. Therefore, it has a unique fixed point \(\mathcal{K}^m \varphi^{\sharp}\) satisfying the claimed inequalities.\(\Box\)

Remark 5.5. In the previous lemma, the cut-off \(M (n)\) depends via \(| w |\) on the weight \(w\). In the following we will only use polynomial weights of the form \(w (n) = (1 + n)^{\alpha}\) with \(| \alpha | \leqslant K\) for a fixed \(K\). In this case \(| w |\) is uniformly bounded and it is possible to select a cut-off which is adapted to all those weights. This will be fixed once and for all and not discussed further.

Proof. By eq. (5.5) we need only to estimate \(\mathcal{G}^{m, \prec} \varphi^m\). We first deal with \(\mathcal{G}_+^{m, \prec}\): we have by (2.8), for \(\delta < 1 / 2 - 1 / (2 \theta)\),

For \(\mathcal{G}_-^{m, \prec}\), it follows in a similar way from estimate (2.9) that, for every \(\delta \in] 0, 1 / (2 \theta)]\),

These bounds and the definition of \(M (n)\) give the claimed bound on \(\mathcal{G}^{m, \prec}\).\(\Box\)

5.3Limiting generator and its domain

Proof. Note that \(w (\mathcal{N})^{- 1} (1 -\mathcal{L}_{\theta})^{- 1} \mathcal{H} \cap w (\mathcal{N})^{- 1} (\mathcal{N}+ 1)^{- \alpha (0)} (1 -\mathcal{L}_{\theta})^{- 1 / 2} \mathcal{H}\) is dense in \(w (\mathcal{N})^{- 1} \mathcal{H}\), therefore, in order to prove Lemma 5.7, it suffices to show that, for any \(\psi \in w (\mathcal{N})^{- 1} (1 -\mathcal{L}_{\theta})^{- 1} \mathcal{H} \cap w (\mathcal{N})^{- 1} (\mathcal{N}+ 1)^{- \alpha (0)} (1 -\mathcal{L}_{\theta})^{- 1 / 2} \mathcal{H}\) and for all \(\nu \geqslant 1\), there exists \(\varphi^{\nu} \in \mathcal{D}_w (\mathcal{L})\) such that

for some \(\delta > 0\). By Lemma 5.4, there exists \(\varphi^{\nu} \in w (\mathcal{N})^{- 1} \mathcal{H}\) such that

and satisfying estimates (5.9)–(5.10). We are left to show that \(\varphi^{\nu} \in \mathcal{D}_w (\mathcal{L})\) and (5.11). Note that

where

In particular, we have \(\mathcal{L} \varphi^{\nu} = \varphi^{\nu} +\mathcal{G}^{\prec} \varphi^{\nu} - (1 -\mathcal{L}_{\theta}) \varphi^{\nu, \sharp}\), and, by Proposition 5.6, it suffices to estimate \(\varphi^{\nu, \sharp}\) in \(w (\mathcal{N})^{- 1} (1 -\mathcal{L}_{\theta})^{- 1} \mathcal{H} \cap w (\mathcal{N})^{- 1} (\mathcal{N}+ 1)^{- \alpha (0)} (1 -\mathcal{L}_{\theta})^{- 1 / 2} \mathcal{H}\). The first contribution, \(\psi\), satisfies the required bounds by assumption, so it is enough to show that the second contribution, which we denote by \(\psi^{\nu}\), satisfies

Notice that \((1 -\mathcal{L}_{\theta}) \psi^{\nu} = - \mathbb{1}_{M (\mathcal{N}) \leqslant | \mathcal{L}_{\theta} | < \nu M (\mathcal{N})} \mathcal{G} \varphi^{\nu}\), hence estimate (5.12) can be obtained from the uniform bounds in Lemma 2.7 as follows (note that those bounds are valid also when \(m = + \infty\)). We have, for \(\mathcal{G}_+\),

For \(\mathcal{G}_-\) we have, instead

which gives estimate (5.12) if we choose \(\bar{\varepsilon}\) small enough. In order to obtain estimate (5.13), note that, for \(\kappa \in] 0, (\theta - 1) / (2 \theta) [\),

Now recall that \(M (n) \simeq (n + 1)^{3 \theta / (\theta - 1 - 2 \theta \bar{\varepsilon})}\) and get by (2.8)–(2.9) the inequality

Applying (5.7) yields the result.\(\Box\)

Proof. Notice that \(\varphi \in \mathcal{D} (\mathcal{L})\) implies \(\mathcal{L}_{\theta} \varphi, \mathcal{G} \varphi \in (1 -\mathcal{L}_{\theta})^{1 / 2} \mathcal{H}\) and \(\varphi \in (1 -\mathcal{L}_{\theta})^{- 1 / 2} (1 +\mathcal{N})^{- 1} \mathcal{H}\). These regularities are enough to proceed by approximation and establish that

where we used the anti-symmetry of the form associated to \(\mathcal{G}\), i.e. \(\langle \varphi, \mathcal{G} \varphi \rangle = 0\).\(\Box\)

5.4Existence and uniqueness for the Kolmogorov equation

Having defined a domain for \(\mathcal{L}\) it remains to study the Kolmogorov equation \(\partial_t \varphi =\mathcal{L} \varphi\). In particular, we consider the equation for \(\varphi^{m, \sharp}\), which was defined in (5.4),

We want to get a suitable bound in terms of \(\varphi^{m, \sharp}_0\) for each term of \(\Phi^{m, \sharp}\). The Schauder estimate in Lemma A.2 will be crucial. We will also need the following result.

Proof. By (5.3) and Lemma 5.4 it follows that

where in the last step we exploited Proposition 5.6.\(\Box\)

For \(\gamma \in] 1 / 2, 1 - 1 / (2 \theta) [\), we have that, by the estimates (2.8) and (2.9),

By interpolation for products, there exists \(q > 0\) such that, for all \(\varepsilon \in] 0, 1 [\),

where the first term on the right-hand side can be controlled via the a priori estimate (5.14), while the second term can be absorbed on the left-hand side. Moreover, we have by (5.8) and by estimate (5.14),

Recalling \(\gamma \in] 1 / 2, 1 - 1 / (2 \theta) [\) and exploiting estimates (2.8)–(2.9), we get

where we used \(3 / 2 + \alpha (\gamma - 1 / 2 + 1 / (2 \theta)) < \alpha (\gamma)\) whenever \(\bar{\varepsilon} < 1 / 3 - 1 / (3 \theta)\). This bound can be controlled via (5.14) as above. As a consequence, we established that, after renaming \(q = q (p, \gamma) > 0\),

and hence, for \(\gamma \in] 1 / 2, 1 - 1 / (2 \theta) [\),

Recall \(\partial_t \varphi^{m, \sharp} = (1 -\mathcal{L}_{\theta}) \varphi^{m, \sharp} + \Phi^{m, \sharp} (t)\), so that

By interpolation, this gives

Introduce now, for \(p > 0\), the sets

and \(\mathcal{U}= \bigcup_{p > \alpha (0)} \mathcal{U}_p\).

Proof. Let \(\varphi_0 \in \mathcal{U}_p\) and set \(\varphi_0^{\sharp} := \mathcal{K}^{- 1} \varphi_0 \in (1 +\mathcal{N})^{- q} (1 -\mathcal{L}_{\theta})^{- 1 - \gamma} \mathcal{H}\) for \(\gamma \in] 1 / 2, 1 - 1 / (2 \theta) [\) and \(p > 0\). For \(m \in \mathbb{N}\), let \(\varphi^m\) be the solution to \(\partial_t \varphi^m =\mathcal{L}^m \varphi^m\) with initial condition \(\varphi^m (0) =\mathcal{K}^m \varphi_0^{\sharp}\). A diagonal argument yields the relative compactness of bounded sets of \((1 +\mathcal{N})^{- p} (1 -\mathcal{L}_{\theta})^{- 1 - \gamma / 2} \mathcal{H}\) in the space \((1 +\mathcal{N})^{- p + \delta} (1 -\mathcal{L}_{\theta})^{- 1} \mathcal{H}\) for \(\delta > 0\), with the consequence that, by Ascoli-Arzelà the sequence \((\varphi^{m, \sharp})_m\) is relatively compact in \(C (\mathbb{R}_+ ; (1 +\mathcal{N})^{- p + \delta} (1 -\mathcal{L}_{\theta})^{- 1} \mathcal{H})\) equipped with the topology of uniform convergence on compact sets. We denote \(\varphi^{\sharp}\) a limit point of such a sequence and let \(\varphi =\mathcal{K} \varphi^{\sharp}\). Then, along the convergent subsequence,

where we exploited our uniform bounds on \(\mathcal{L}_{\theta}\), \(\mathcal{G}^{m, \succ}\), \(\mathcal{K}^m\) and the convergence of \(\varphi^{m, \sharp}\) to \(\varphi^{\sharp}\) as \(m \rightarrow \infty\) to get the 4th equality, while the last step follows from our bounds for \(\mathcal{G}^{\prec}\) and \(\mathcal{K}\), together with the dominated convergence theorem.

If we take \(p > \alpha (0)\), then by definition (cfr. Lemma 5.7) \(\varphi \in \mathcal{D} (\mathcal{L})\). Furthermore, \(\mathcal{L} \varphi \in C (\mathbb{R}_+ ; \mathcal{H})\) and we have \(\varphi \in C^1 (\mathbb{R}_+ ; \mathcal{H})\) because of the relation \(\varphi (t) - \varphi (s) = \int_s^t \mathcal{L} \varphi (\tau) \mathrm{d} \tau\). We can hence compute,

by the dissipativity of the operator \(\mathcal{L}\) given by Lemma 5.8. Therefore, for any solution we have \(\| \varphi (t) \| \leqslant \| \varphi_0 \|\), which together with the linearity of the equation yields the uniqueness.\(\Box\)

6Bounds on the drift

We prove there the key bounds on the drift \(\mathcal{G}^m\).

Proof of Lemma 2.7. We start by estimating \(\mathcal{G}_+^m\). We have, by Lemma A.1 and since \(\gamma > 1 / (2 \theta)\),

Introducing the notation \(\ell_1 = \ell = k_1 + k_2\) and \(\ell_i = k_{i + 1}\) for \(i \geqslant 2\), we get

then using the symmetry of \(\hat{\varphi}_{n - 1}\) we reduce this to

from which we obtain

For \(\mathcal{G}_-^m\), note first that, by the Cauchy-Schwarz inequality and by Lemma A.1 (since \(\gamma < 1 / 2\)),

therefore,

we now let \(\ell_1 = p\), \(\ell_2 = q\), and \(\ell_i = k_{i - 1}\) for \(3 \leqslant i \leqslant n + 1\), so that

which gives the uniform bound.

Let us now discuss the \(m\)-dependent estimates, we have for \(\mathcal{G}_+^m\)

Finally, for \(\mathcal{G}_-^m\) we have,

\(\Box\)

7Stochastic Navier–Stokes on the plane

In this section we prove that the main results of the paper, namely existence and uniqueness of energy solution for the hyper-viscous Navier–Stokes eq. (1.1) extends very naturally to the setting of the whole plane \(\mathbb{R}^2\). We will discuss first existence of martingale solutions via a limiting procedure involving finite volume approximations, then we will show that the Kolmogorov equation can be solved also in the full space, which implies, as in the periodic setting, uniqueness in law.

The invariant measure

where \(\dot{H}^s (\mathbb{R}^d)\) denotes the so-called homogeneous Sobolev space of \(L^2 (\mathbb{R}^d)\) functions \(f\) having norm \(\int_{\mathbb{R}^d} | \xi |^{2 s} | \hat{f} (\xi) |^2 \mathrm{d} \xi\) finite. We denote by \(\mu_{\mathbb{R}^2}\) the energy measure in the case of the whole space.

A new approximating problem

while the inverse transform is given by

Plancherel theorem now reads as

The \(\mathcal{H}\)-norm is now given by

The Biot-Savart kernel is \(K (x) = - (2 \pi)^{- 3} \lambda^{- 2} \iota \sum_{k \in \lambda^{- 1} \mathbb{Z}^2} k^{\bot} | k |^{- 2} e^{2 \pi \iota k \cdot x}\), for \(x \in \mathbb{T}^2_{\lambda}\), since from the relation \(\omega = \nabla^{\bot} \cdot u\) we get

which gives \(\invbreve{u} (k) = - 2 \pi \iota k^{\top} | 2 \pi k |^{- 2} \cdot \invbreve{\omega} (k)\).

\(\mathcal{L}_{\theta}^{\lambda, m}\) can again be represented in Fourier terms by (2.3) for \(k_1, \ldots, k_n \in \lambda^{- 1} \mathbb{Z}^2\). The (\(\lambda\)-)Fourier transform of \(\mathcal{G}_+^{\lambda, m}\) is exactly the same as in (2.4), while the one of \(\mathcal{G}_-^{\lambda, m}\) is as in (2.5) but multiplied by a factor \(\lambda^{- 2}\) due to the convolution. Following the proof of Lemma 2.7, we get some estimates of \(\mathcal{G}_{\pm}^{\lambda, m}\) uniform both in \(m\) and in \(\lambda\) (up to the \(\lambda\)-dependence of the \(\mathcal{H}_{\lambda}\)-norm). After getting this estimate we obtain the same result as in Lemma 2.11, for every \(\lambda > 0\).

Proof of Lemma 2.7. We show here the two bounds for \(\mathcal{G}_{\pm}^{\lambda, m}\). For \(\mathcal{G}_+^{\lambda, m}\), the main difference with respect to the proof presented in Section 6 is the presence of the \(\lambda^{- 2 n}\) term in the definition of the norm (and, of course, of a different Fourier transform), the sum on \(k_1 + k_2 = \ell\) in the third step of the inequality (6.1) eats also a term \(\lambda^{- 2}\), hence thereafter we will have \(\lambda^{- 2 (n - 1)}\), which is the correct term that will enter the norm at the end of the estimate.

For the term \(\mathcal{G}_-^{\lambda, m}\) we will use the fact that, by Lemma A.1,

which implies that \(\| w (\mathcal{N}) (1 -\mathcal{L}_{\theta})^{- \gamma} \mathcal{G}_-^m \varphi \|^2\) can be bounded as in the proof in Section 6 with the extra term \(\lambda^{- 2 (n + 1)}\), that is exactly the term that will enter the norm, yielding the claimed estimate.\(\Box\)

Existence for the cylinder martingale problem

Let us therefore focus our attention on the proof of Theorem 3.3. Following Step 1, we want to show tightness of the sequence \((\omega^{\lambda, m})_{\lambda, m}\). With Step 2, we are going to conclude the existence for the martingale problem as \(\lambda, m \rightarrow \infty\). Let us assume for the moment that we can associate to each cylinder function, \(\varphi \in \operatorname{Cyl}_{\mathbb{R}^2}\), of the form \(\varphi (\omega) = \Phi (\omega (f_1), \ldots, \omega (f_n))\), where \(f_1, \ldots, f_n \in \mathcal{S} (\mathbb{R}^2)\), to \(\varphi^{\lambda} (\omega) = \Phi (\omega (f_1^{\lambda}), \ldots, \omega (f_n^{\lambda}))\) with \(f_1^{\lambda}, \ldots, f^{\lambda}_n \in \mathcal{S} (\mathbb{T}^2_{\lambda})\) in such a way that

where we are exploiting the chaos decompositions of \(\varphi\) and \(\varphi^{\lambda}\). It is then possible to recover the bound (3.4) for \(\omega^{\lambda, m}\) and therefore tightness, since \(\mathcal{L}_{\theta}^{\lambda, m}\) has \(\mathcal{L}_{\theta}^{\infty}\) as a limit of a sum converging to an integral. As regards Step 2, the crucial part is to pass from \(\mathcal{L}^{\lambda, m}\) to \(\mathcal{L}^{\infty, m}\) in (3.5) when taking the limit as \(\lambda \rightarrow \infty\). To do this, we have to show that \(\| (1 -\mathcal{L}_{\theta})^{- 1 / 2} (\mathcal{L}^{\lambda, m} \varphi -\mathcal{L}^{\infty, m} \varphi) \|_{\mathcal{H}_{\lambda}}\) tends to \(0\) when \(\lambda \rightarrow \infty\), which reduces to prove that

Comparing the explicit formulas for \(\mathcal{G}_-^{\lambda, m}\) and \(\mathcal{G}_-^{\infty, m}\), we have that the only difference between the two is the fact that in \(\mathcal{G}^{\lambda, m}_-\) the sum given by the convolution becomes an integral when taking the limit. In particular,

and the right-hand side goes to zero as \(\lambda \rightarrow \infty\).

In both cases it is important to understand the role played by the test functions with respect to the norm (which depends on \(\lambda\)) we are considering. We want in fact to take \(\varphi \in \operatorname{Cyl}_{\mathbb{R}^2}\), but we will evaluate it on \(\omega^{m, \lambda}\). This is a non-trivial step and it is worth to spend a few words on it adopting a chaos expansion point of view. To apply the Mitoma's criterion we only need to test on linear functions, see [19], while for the second step it suffices to consider functions \(\varphi\) of the form

where \(: e^{\iota g} :\) indicates the Wick exponential of \(g\), see also [13, 15]. Focusing on the latter case, we can identify \(\varphi\) with the sequence of chaoses \((\varphi_n (k_{1 : n}))_n\) where \(\varphi_n (k_{1 : n}) = \iota^n \hat{f} (k_1) \cdots \hat{f} (k_n)\) for \(k_{1 : n} \in (\mathbb{R}^2)^n\), so that, after noticing

with \(C_{\lambda} (f) \rightarrow 1\) as \(\lambda \rightarrow \infty\), we have \(\)\(\varphi^{\lambda}_n (k_{1 : n}) = \iota^n C_{\lambda} (f) \invbreve{f} (k_1) \cdots \invbreve{f} (k_n)\) for \(k_{1 : n} \in (\lambda^{- 1} \mathbb{Z}^2)^n\).

The Kolmogorov backward equation in the plane

We start by describing the homogeneous Sobolev space \(\dot{H}^1 (\mathbb{R}^2)\). As remarked in Proposition 1.34 of [4] this space is not a space of functions. Indeed consider a smooth bump function \(\theta : \mathbb{R}^2 \rightarrow \mathbb{R}\) compactly supported and such that \(\theta (0) = 1\). Let \(\theta_{\varepsilon} (x) = \theta (\varepsilon x)\), then as \(\varepsilon \rightarrow 0\) we have \(\| \theta_{\varepsilon} \|_{\dot{H}^1 (\mathbb{R}^2)} \approx 1\), \(\theta_{\varepsilon} \mathop{\rightarrow}\limits 1\) pointwise and \(\theta_{\varepsilon} \rightarrow 0\) weakly in \(\dot{H}^1 (\mathbb{R}^2)\). The elements of \(\dot{H}^1 (\mathbb{R}^2)\) consists of equivalence classes of functions modulo constants and we will have to take this into account in our analysis. We say that \(\varphi \in \dot{H}^1 (\mathbb{R}^2)\) if there exists a tempered distribution \(\tilde{\varphi} \in \mathcal{S}' (\mathbb{R}^2)\) such that

and for which

Note that the equality (7.1) implies an identification of elements whose difference is a constant. Indeed, for \(C \in \mathbb{R}\),

where \(\mathrm{D}\) denotes the derivative operator. This means that we identify \(\varphi\) with \(\tilde{\varphi} + C\) and write

As a consequence, the tensor product \((\dot{H}^1 (\mathbb{R}^2))^{\otimes n}\), understood as a tensor product of Hilbert spaces, can be described in the following way: for every \(\varphi \in (\dot{H}^1 (\mathbb{R}^2))^{\otimes n}\), there exists \(\tilde{\varphi} \in \mathcal{S}'_s ((\mathbb{R}^2)^n)\), the space of symmetric tempered distributions on \((\mathbb{R}^2)^n\), such that

and for which

that leaves the freedom to change \(\widehat{\tilde{\varphi}}\) in \(Z_n = \bigcup_{i = 1}^n \{ k_i = 0 \} \subset (\mathbb{R}^2)^n\) and defines it modulo a symmetric distribution \(\eta\) whose Fourier transform is supported in \(Z_n\). Therefore, we can identify \(\varphi\) with \(\tilde{\varphi} + \eta\), where \(\left( \prod_{i = 1}^n | k_i |^2 \right) \hat{\eta} (k_{1 : n}) = 0\).

Let us now study the operators \(\mathcal{G}_{\pm}\). With the identification above we can define

with \(\chi_m\) a smooth function such that \(\chi_m (k_1, k_2, k_3) \approx \mathbb{1}_{| k_1 |, | k_2 |, | k_3 | \leqslant m}\). These formulas have to be understood via duality

In order to check that this definition is correct, we need to make sure that whenever \(\hat{\varphi}_{n - 1}\) is supported in \(Z_{n - 1}\) or when \(\hat{\psi}\) is supported in \(Z_n\) the result is zero. This is obvious for \(\hat{\psi}\), so let us check it for \(\hat{\varphi}_{n - 1}\). Assume \(\hat{\varphi}_{n - 1}\) is supported in \(Z_{n - 1}\), then either \(k_1 + k_2 = 0\) or \(k_i = 0\) for some \(i = 3, \ldots, n\). In the first case the result is zero due to the multiplicative factor \((k_1^{\perp} \cdot (k_1 + k_2)) ((k_1 + k_2) \cdot k_2)\), while in the second the result is again zero because of the \(\)factor \(\prod_{i = 3}^n | k_i |^2\). The same works for \(\mathcal{G}_-\) since it is the adjoint of \(\mathcal{G}_+\).

The expression of the norms, and therefore the results about the estimates on the operator and so on, are exactly the same as in the periodic setting modulo changing the sums into integrals. As already shown in Section 4 for the torus case, the existence and uniqueness result for the Kolmogorov backward equation yields, via duality, uniqueness of solutions to the cylinder martingale problem also for the case of the whole space \(\mathbb{R}^2\).

Appendix ASome auxiliary results

Proof. Since \(| p |^{2 \theta} + | k - q |^{2 \theta} \gtrsim | p |^{2 \theta} + | k |^{2 \theta}\), we have

By scaling

and the integral is finite if \(\beta - 2 \theta \alpha < - d\).\(\Box\)

Proof. The proof is standard and proceeds by spectral calculus. Write \(\Psi (t) := (\partial_t - (1 -\mathcal{L}_{\theta})) \psi (t)\)

where \(\mathbb{1}_{| 1 -\mathcal{L}_{\theta} | \sim 2^i}\) denotes a dyadic partition of unity such that \(\| \varphi \|^2 \approx \sum_i \| \mathbb{1}_{| 1 -\mathcal{L}_{\theta} | \sim 2^i} \varphi \|^2\) for any \(\varphi\).

Let \(S_t = e^{- t (1 -\mathcal{L}_{\theta})}\), so that

Then, using \(\| (1 -\mathcal{L}_{\theta})^{1 + \gamma} S_{t - s} \psi \| \lesssim ((t - s)^{- 1 - \gamma} \vee 1) \| \psi \|\) and \(\| (1 -\mathcal{L}_{\theta})^{1 + \gamma} \mathbb{1}_{| 1 -\mathcal{L}_{\theta} | \sim 2^i} \| \lesssim 2^{(1 + \gamma) i}\), and letting \(\delta = 2^{- i}\), we have

and, as a consequence,

Therefore, since \(\mathcal{N}\) commutes with \(\mathcal{L}_{\theta}\), we also have

that is the claimed estimate.\(\Box\)

Acknowledgements

It is a pleasure to contribute to the proceedings for the conference celebrating the 60th birthday of Prof. Franco Flandoli. The first author of this paper has learned a great deal from him, and not only about stochastic fluid dynamics and stochastic analysis in general. The stimulating and friendly atmosphere in Pisa was fundamental for his career. The research exposed here is supported by DFG via CRC 1060.

Bibliography