1Introduction

The \(\Phi^4_3\) measure on the three dimensional torus \(\Lambda =\mathbb{T}^3 = (\mathbb{R}/ 2 \pi \mathbb{Z})^3\) is the probability measure \(\nu\) on distributions \(\mathscr{S}' (\Lambda)\) corresponding to the formal functional integral

where \(\mu\) is the law of the Gaussian free field with covariance \((1 - \Delta)^{- 1}\) on \(\Lambda\), \(Z\) a normalization constant and \(\lambda\) a coupling constant. The \(\infty\) appearing in this expression reminds us that many things are wrong with this recipe. The key difficulty can be traced to the fact that the measure we are looking for it is not absolutely continuous wrt. the reference measure \(\mu\). This fact seems part of the folklore even if we could not find a rigorous proof for it in the available literature apart from a work of Albeverio and Liang [1] which however refers to the Euclidean fields at time zero. The singularity of the \(\Phi^4_3\) measure is indeed a major technical difficulty in a rigorous study. Obtaining a complete construction of this formal object (both in finite and infinite volume) has been one of the main achievements of the constructive quantum field theory program [10, 8, 23, 9, 19, 5, 6].

In recent years the rigorous study of the \(\Phi^4_3\) model has been pursued from the point of view of stochastic quantization. In the original formulation of Parisi–Wu [22], stochastic quantization is a way to introduce additional degrees of freedom (in particular a dependence on a fictious time) in order to obtain an equation whose solutions describe a measure of interest, in this case the \(\Phi^4_3\) measure on \(\Lambda\) as in (1) or its counterpart in the full space. Rigorous analysis of stochastic quantization for simpler models like \(\Phi^4_2\) (the two-dimensional analog of eq. (1)) started with the work [17]. It has been only with the fundamental work of Hairer on regularity structures [16] that the three dimensional model could be successfully attacked, see also [7, 18]. This new perspective on this and related problems led to a series of new results on the global space-time control of the stochastic dynamics [21, 13, 2, 20] and to a novel proof of the constructions of non-Gaussian Euclidean quantum field theories in three dimensions [12].

A conceptual advantage of stochastic quantization is that it is insensitive to questions of absolute continuity wrt. to a reference measure. This, on the other hand, is the main difficulty of the Gibbisan point of view as expressed in eq. (1). In order to explore further the tradeoffs of different approaches we have recently developed a variational method [4] for the construction and description of \(\Phi^4_3\) with which we were able to provide an explicit formula for the Laplace transform of \(\Phi^4_3\) in terms of a stochastic control problem. In this control problem the controlled process represents the scale-by-scale evolution of the interacting random field.

The present paper is the occasion to explore further this point of view by constructing a novel measure via a random shift of the Gaussian free field and proving that the \(\Phi^4_3\) measure can be constructed as an absolutely continuous perturbation thereof. Without entering into technical details now let us give the broad outline of this construction. We consider a Brownian martingale \((W_t)_{t \geqslant 0}\) with values in \(\mathscr{S}' (\Lambda)\) and such that \(W_t\) is a regularization of the Gaussian free field \(\mu\) at scale \(t\). Let us denote \(\mathbb{P}\) its law, \(\mathbb{E}\) the corresponding expectation. In particular \(W_t \rightarrow W_{\infty}\) as \(t \rightarrow \infty\) and \(W_{\infty}\) has law \(\mu\). We can identify the \(\Phi^4_3\) measure \(\nu\) as the weak limit \(\nu^T \rightarrow \nu\) as \(T \rightarrow \infty\) of the family of probability measures \((\nu^T)_{T \geqslant 0}\) defined as

where \(\mathbb{P}^T\) is the measure on paths \((W_t)_{t \geqslant 0}\) with density

and

is a quartic polynomial in the field \(\varphi\) with \((a_T, b_T)_T\) a family of (suitably diverging) renormalization constants. The presence of the scale parameter \(t \in \mathbb{R}_+\) allows to introduce a filtration and a family of measures \(\mathbb{Q}^v\) defined as the Girsanov transformation

where \((\langle L^v \rangle_t)_{t \geqslant 0}\) is the quadratic variation of the (scalar) local martingale \((L^v_t)_{t \geqslant 0}\) and \((v_t)_{t \geqslant 0}\) is an adapted process with values in \(L^2 (\Lambda)\). Let

be the density of \(\mathbb{P}^T\) wrt. \(\mathbb{Q}^v\). We will show that it is possible to choose \(v\) in such a way that the family \((D_T)_{T \geqslant 0}\) is uniformly integrable under \(\mathbb{Q}^v\) and that \(D_T \rightarrow D_{\infty}\) weakly in \(L^1 (\mathbb{Q}^v)\). With particular choice of \(v\) we call \(\mathbb{Q}^v\) the drift measure: it is the central object of this paper. By Girsanov's theorem the canonical process \((W_t)_{t \geqslant 0}\) satisfies the equation

where \((\tilde{W}_t)_{t \geqslant 0}\) is a Gaussian martingale under \(\mathbb{Q}^v\) (and has law equal to that of \((W_t)_{t \geqslant 0}\) under \(\mathbb{P}\), that is is a regularized Gaussian free field). We will show also that the drift \(v_t\) can be written as a (polynomial) function of \(\)\((\tilde{W}_s)_{s \in [0, t]}\), that is \(v_t = \tilde{V}_t ((\tilde{W}_s)_{s \in [0, t]})\). Therefore we have an explicit description of the process \((W_t)_{t \geqslant 0}\) under the drift measure \(\mathbb{Q}^v\) as the unique solution of the path-dependent SDE

Let us note that this formula expresses the “interacting” random field \((W_t)_t\) as a function of the “free” field \((\tilde{W}_t)_t\). In this respect is a formula with very similar technical merits as the stochastic quantization approach.

Intuitively this new measure \(\mathbb{Q}^v\), is half way between the variational description in [4] and the (formal) Gibbsian description of eq. (1). It constitutes a measure which is relatively explicit, easy to construct and analyze and which can be used as reference measure for \(\Phi^4_3\), very much like the Gaussian free field can be used as reference measure for \(\Phi^4_2\) [11].

As an application we provide a self-contained proof of the singularity of the \(\Phi^4_3\) measure \(\nu\) wrt. the Gaussian free field \(\mu\). As we already remarked the singularity of \(\Phi^4_3\) seems to belongs to the folklore and we were not able to trace any written proof of that. However M. Hairer, during a conference at Imperial College in 2019 showed us an unpublished proof of him of singularity using the stochastic quantization equation. Our proof and his are very similar and we do not claim any essential novelty in this respect. Albeit the proof is quite straightforward we wrote down all the details in order to provide a reference for this fact. The main point of the present paper remains that of describing the drift measure as a novel object in the context of \(\Phi^4_3\) and similar measures.

Our proof of singularity, in particular also shows that the drift measure \(\mathbb{Q}^v\) is singular wrt. \(\mathbb{P}\). The intuitive reason is that the drift \((V_t)_{t \geqslant 0}\) in the SDE (3) is not regular enough (as \(t \rightarrow \infty\)) to be along Cameron–Martin directions for the law \(\mathbb{P}\) of the process \((W_t)_{t \geqslant 0}\) and therefore the Girsanov transform (2) gives a singular measure when extended all the way to \(T = + \infty\).

Acknowledgments. M.G. would like to thank S. Albeverio, D. Brydges, C. Garban and M. Hairer for interesting discussions on the topic of singularity of \(\Phi^4_3\). N.B would like to thank Bjoern Bringmann for some helpful comments. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program SRQ: Scaling limits, Rough paths, Quantum field theory during which part of the work on this paper was undertaken. This work is supported by DFG via CRC 1060 and by EPSRC via Grant Number EP/R014604/1.

Notations. Let us fix some notations and objects.

• For \(a \in \mathbb{R}^d\) we let \(\langle a \rangle := (1 + | a |^2)^{1 / 2}\).

• The constant \(\varepsilon > 0\) represents a small positive number which can be different from line to line.

• Denote with \(\mathscr{S} (\Lambda)\) the space of Schwartz functions on \(\Lambda\) and with \(\mathscr{S}' (\Lambda)\) the dual space of tempered distributions. The notation \(\hat{f}\) or \(\mathscr{F} f\) stands for the space Fourier transform of \(f\) and we will write \(g (\mathrm{D})\) to denote the Fourier multiplier operator with symbol \(g : \mathbb{R}^n \rightarrow \mathbb{R}\), i.e. \(\mathscr{F} (g (\mathrm{D}) f) = g \mathscr{F} f\).

• \(B^{\alpha}_{p, q} = B^{\alpha}_{p, q} (\Lambda)\) denotes the Besov spaces of regularity \(\alpha\) and integrability indices \(p, q\) as usual. \(\mathscr{C}^{\alpha} = \mathscr{C}^{\alpha} (\Lambda)\) is the Hölder–Besov space \(B^{\alpha}_{\infty, \infty}\), \(W^{\alpha, p} = W^{\alpha, p} (\Lambda)\) denote the standard fractional Sobolev spaces defined by the norm \(\| f \|_{W^{s, q}} := \| \langle \mathrm{D} \rangle^s f \|_{L^q}\) and \(H^{\alpha} = W^{\alpha, 2}\). The symbols \(\prec, \succ, \circ\) denotes spatial paraproducts wrt. a standard Littlewood–Paley decomposition. The reader is referred to Appendix A for an overview of the functional spaces and paraproducts.

2The setting

The setting of this paper is the same of that of our variational study [4]. In this section we will briefly recall it and also state some results from that paper which will be needed below. They concern the Boué–Dupuis formula and certain estimates which will be important also in our analysis of absolute continuity.

Let \(\Omega := C (\mathbb{R}_+ ; \mathscr{C}^{- 3 / 2 - \varepsilon} (\Lambda))\) and \(\mathscr{F}\) be the Borel \(\sigma\)–algebra of \(\Omega\). On \((\Omega, \mathscr{F})\) consider the probability measure \(\mathbb{P}\) which makes the canonical process \((X_t)_{t \geqslant 0}\) a cylindrical Brownian motion on \(L^2 (\Lambda)\) and let \((\mathscr{F}_t)_{t \geqslant 0}\) the associated filtration. In the following \(\mathbb{E}\) without any qualifiers will denote expectations wrt. \(\mathbb{P}\) and \(\mathbb{E}_{\mathbb{Q}}\) will denote expectations wrt. some other measure \(\mathbb{Q}\).

On the probability space \((\Omega, \mathscr{F}, \mathbb{P})\) there exists a collection \((B_t^n)_{n \in (\mathbb{Z})^3}\) of complex (2-dimensional) Brownian motions, such that \(\overline{B^n_t} = B^{- n}_t\), \(B^n_t, B^m_t\) independent for \(m \neq \pm n\) and \(X_t = \sum_{n \in \mathbb{Z}^3} e^{i \langle n, \cdot \rangle} B^n_t\).

Fix some \(\rho \in C_c^{\infty} (\mathbb{R}_+, \mathbb{R}_+)\),decreasing, such that \(\rho = 1\) on \(B (0, 9 / 10)\) and \(\operatorname{supp} \rho \subset B (0, 1)\). For \(x \in \mathbb{R}^3\) let \(\rho_t (x) := \rho (\langle x \rangle / t)\) with \(\langle x \rangle := (1 + | x |^2)^{1 / 2}\) and

Let \(J_s =\)\(\sigma_s (\mathrm{D}) \langle \mathrm{D} \rangle^{- 1}\) and consider the process \((W_t)_{t \geqslant 0}\) defined by

It is a centered Gaussian process with covariance

for any \(\varphi, \psi \in \mathscr{S} (\Lambda)\) and \(t, s \geqslant 0\), by Fubini theorem and Ito isometry. By dominated convergence \(\lim_{t \rightarrow \infty}\)\(\mathbb{E} [\langle W_t, \varphi \rangle \langle W_t, \psi \rangle] = \sum_{n \in \mathbb{Z}^3} \langle n \rangle^{- 2} \hat{\varphi} (n) \overline{\hat{\psi} (n)}\) for any \(\varphi , \psi \in L^2 (\Lambda)\). For any finite “time” \(T\) the random field \(W_T\) on \(\Lambda\) has a bounded spectral support and the stopped process \(W^T_t = W_{t \wedge T}\) for any fixed \(T > 0\), is in \(C (\mathbb{R}_+, C^{\infty} (\Lambda))\). Furthermore \((W^T_t)_t\) only depends on a finite subset of the Brownian motions \((B^n)_{n \in \mathbb{Z}^3}\). We write \(g (\mathrm{D})\) for the Fourier multiplier operator with symbol \(g\).

Observe that \(J_t\) has the property, that for some function \(f \in B_{p, p}^s\) or \(f \in W ^{s, p}\) with \(p \in [1, \infty]\) and \(s \in \mathbb{R}\), for any \(\alpha \in \mathbb{R}\)

We will denote by \(\mathopen{\unicode{x27E6}} W^n_t \mathclose{\unicode{x27E7}}\), \(n = 1, 2, 3\), the \(n\)-th Wick-power of the Gaussian random variable \(W_t\) (under \(\mathbb{P}\)) and introduce the convenient notations \(\mathbb{W}_t^2 := 12 \mathopen{\unicode{x27E6}} W^2_t \mathclose{\unicode{x27E7}}\), \(\mathbb{W}_t^3 := 4 \mathopen{\unicode{x27E6}} W_t^3 \mathclose{\unicode{x27E7}}\). Furthermore we will write \(\mathopen{\unicode{x27E6}} (\langle \mathrm{D} \rangle^{- 1 / 2} W_t)^n \mathclose{\unicode{x27E7}}, n \in \mathbb{N}\) for the \(n\)-th Wick-power of \(\langle \mathrm{D} \rangle^{- 1 / 2} W_t\) . It exists for any \(0 < t < \infty\) and any \(n \geqslant 1\) since it is easy to see that \(\langle \mathrm{D} \rangle^{- 1 / 2} W_t\) has a covariance with a diagonal behavior which can be controlled by \(\log \langle t \rangle\). These Wick powers converge as \(T \rightarrow \infty\) in spaces of distributions with regularities given in the following table:

We denote by \(\mathbb{H}_a\) the space of \((\mathscr{F}_t)_{t \geqslant 0}\)-progressively measurable processes which are \(\mathbb{P}\)-almost surely in \(\mathcal{H} := L^2 (\mathbb{R}_+ \times \Lambda)\). We say that an element \(v\) of \(\mathbb{H}_a\) is a drift. Below we will need also drifts belonging to \(\mathcal{H}^{\alpha} := L^2 (\mathbb{R}_+ ; H^{\alpha} (\Lambda))\) for some \(\alpha \in \mathbb{R}\) where \(H^{\alpha} (\Lambda)\) is the Sobolev space of regularity \(\alpha\). We denote the corresponding space with \(\mathbb{H}_a^{\alpha}\). For any \(v \in \mathbb{H}_a\) define the measure \(\mathbb{Q}^v\) on \(\Omega\) by

Denote with \(\mathbb{H}_c \subseteq \mathbb{H}_a\) the set of drifts \(v \in \mathbb{H}_a\) for which \(\mathbb{Q}^v (\Omega) = 1\), and set \(W^v := W - I (v)\), where

Below will need also the following operators. For all \(t \geqslant 0\) let \(\theta_t : \mathbb{R}^3 \rightarrow [0, 1]\) be a smooth function such that

for some \(T_0 > 0\). For example one can fix smooth functions \(\tilde{\theta}, \eta : \mathbb{R}^3 \rightarrow \mathbb{R}_+\) such that \(\tilde{\theta} (\xi) = 1\) if \(| \xi | \leqslant 1 / 2\) and \(\tilde{\theta} (\xi) = 0\) if \(| \xi | \geqslant 2 / 3\) , \(\eta (\xi) = 1\) if \(| \xi | \leqslant 1\) and \(\eta (\xi) = 0\) if \(| \xi | \geqslant 2\). Then let \(\tilde{\theta}_t (\xi) := \tilde{\theta} (\xi / t)\) and define \(\)

where \(\)\(\zeta (t) : \mathbb{R}_+ \rightarrow \mathbb{R}\) is a smooth function such that \(\zeta (t) = 0\) for \(t \leqslant 10\) and \(\zeta (t) = 1\) for \(t \geqslant 3\). Then eq (5) will hold with \(T_0 = 3\). We will let

for any \(f \in \mathscr{S}' (\Lambda)\)..

Our aim here to study the measures \(\mu_T\) defined on \(\mathscr{C}^{- 1 / 2 - \varepsilon}\) as

for \(\varphi \in C^{\infty} (\Lambda)\)

with suitable \(a_T, b_T \rightarrow \infty\). For convenience the measure \(\mu^T\) is not normalized and, wrt. to the notations in the introduction we have

With these notations we can recall the following results of [4].

This is a consequence of the more general Boué–Dupuis formula which can be stated as follows.

We will use several times below the Boué–Dupuis formula (9) in order to control exponential integrability of various functionals. By a suitable choice of renormalization and a change of variables in the control problem (8) we were able in [4] to control the functional in Theorem 1 uniformly up to infinity:

As a consequence we obtain the following corollary(See corollary 1 and Lemma 6 in [4])

3Construction of the drift measure

We start now to implement the strategy discussed in the introduction: construct a shifted measure sufficiently similar to \(\Phi^4_3\). Intuitively the \(\Phi^4_3\) measure should give rise to a canonical process which is a shift of the Gaussian free field with a drift of the form given by eq. (10). Indeed this drift \(u\) should be the optimal drift in the variational formula.

A small twist is given by the fact that the relevant Gaussian free field entering these considerations is not the process \(W = W (X)\) but that obtained from the shifted canonical process \(X^u_t = X_t - \int_0^t u_s \mathrm{d} s\) which we denote by

Moreover for technical reasons we have to modify the drift in large scales and add some coercive term which will allow later to prove some useful estimates. We define the functional

where \(\bar{T} > 0, n \in \mathbb{N}\) are constants which will be fixed later and where we understand all the Wick renormalizations to be given functions of \(W\). We look now for the solution \(u\) of the equation

Expanding the Wick polynomials appearing in \(\Xi (W - I (u), u)\) we obtain the equation

for all \(s \geqslant 0\). This is an integral equation for \(t \mapsto u_t\) with smooth coefficients depending smoothly on \(W\) and can be solved via standard methods. Since the coefficients are of polynomial growth we must expect explosion in finite time, so we have to be careful. Note that for any finite time the process \((u_s)_{s > 0}\) has bounded spectral support. As a consequence we can solve the equation in \(L^2\) and as long as \(\int^t_0 \| u \|^2_{L^2} \mathrm{d} s\) is finite we can see from the equation that \(\sup_{s \leqslant t} \| u_s \|^2_{L^2}\) is finite. Therefore by the existence of local solutions we have that, for all \(N \geqslant 0\), the stopping time

is strictly positive \(\mathbb{P}\)-almost surely and \(u\) exists up to the (explosion) time \(T_{\exp} := \sup_{N \in \mathbb{N}} \tau_N\). Moreover, by construction, the process \(u_t^N := \mathbb{1}_{\{ t \leqslant \tau_N \}} u_t\) satisfies Novikov's condition, so it is in \(\mathbb{H}_c\) and by Girsanov transformation we can define the probability measure on \(C (\mathbb{R}_+, \mathscr{C}^{- 1 / 2 - \varepsilon} (\Lambda))\) given by

and under which \(X^{u^N}_t = X_t - \int^t_0 u_s^N \mathrm{d} s\) is a cylindrical Brownian motion. In particular \((W^{u^N}_t)_{t \geqslant 0}\), given by \(\int^t_0 J_s \mathrm{d} X^{u^N}_s\) has under \(\mathbb{Q}^{u^N}\) the same law as \((W_t)_{t \geqslant 0}\) has under \(\mathbb{P}\). Moreover we have that \(W^{u^N}_s = W^u_s\) for \(0 \leqslant s \leqslant \tau_N\) and that \(u\) satisfies the equation

where we introduced the notations \(\mathbb{W}_s^{u, 3} := 4 \mathopen{\unicode{x27E6}} (W_s^u)^3 \mathclose{\unicode{x27E7}}\) and \(\mathbb{W}_s^{u, 2} := 12 \mathopen{\unicode{x27E6}} (W^u_s)^2 \mathclose{\unicode{x27E7}}\).

Note that here the Wick powers are still taken to be given functions of \(W\), i.e we are still taking the Wick ordering with respect to the law of \(W\) under \(\mathbb{P}\) (or the law of \(W^{u^N}\) under \(\mathbb{Q}^{u^N}\)).

If we think of the terms containing \(W^u\) as given (that is, we ignore their dependence on \(u\)), eq. (14) is a linear integral equation in \(u\) which can be estimated via Gronwall-type arguments. In order to do so, let us denote by \(U : H \mapsto \hat{u}\) the solution map of the equation

This last equation is linear and therefore has nice global solutions (let's say in \(C (\mathbb{R}_+, L^2)\)) and by uniqueness and eq. (14) we have \(u_t = U_t (W^u)\) for \(t \in [0, T_{\exp})\). From this perspective the residual dependence on \(u\) will not play any role since under the shifted measure the law of the process \(W^u\) does not depend on \(u\). By standard paraproduct estimates we have

where we have used the presence of the cutoff \(\mathbb{1}_{\{ s \geqslant \bar{T} \}}\) to introduce the small factor \(T^{- \varepsilon}\) and we have employed the notation

Therefore, by Gronwall's lemma

Under \(\mathbb{Q}^{u^N}\), the terms in \(\tilde{H}_{\tau_N}\) are in all the \(L^p\) spaces by hypercontractivity and moreover for any \(p \geqslant 1\) one can choose \(\bar{T}\) large enough to that also the exponential term is in \(L^p\). Using eq. (14) it is then not difficult to show that \(\mathbb{E}_{\mathbb{Q}^{u^{N_1}}} [\| u^{N_2} \|^p_{\mathcal{H}^{- 1 / 2 - \varepsilon}}] < \infty\) for any \(p > 1\) (again provided we take \(\bar{T}\) large enough depending on \(p\)) as long as \(N_1 > N_2\). By the spectral properties of \(J\) and the equation for \(u\), the process \(t \mapsto \mathbb{1}_{\{ t \leqslant T \}} u_t\) is spectrally supported in a ball of radius \(T\), so we get in particular that

uniformly for any choice of \(N_1 \geqslant N_2 \geqslant 0\).

Proof. Consider the filtration \((\mathscr{G}_N = \mathscr{F}_{\tau_N})_N\) and observe that \(( \mathbb{Q}^{u^N} |_{\mathscr{G}_N})_N\) is a consistent family of inner regular probability distributions and therefore there exists a unique extension \(\mathbb{Q}^u\) to \(\mathscr{G}_{\infty} = \vee_N \mathscr{G}_N\). Next observe that \(\{ T_{\exp} < \infty \} = \bigcup_{T \in \mathbb{N}} \{ T_{\exp} < T \} \subset \bigcup_{T \in \mathbb{N}} \bigcap_{N \in \mathbb{N}} \{ \tau_N < T \}\) and that for any \(N, T < \infty\), we have

On the event \(\{ \tau_N \leqslant T \}\) we have

and therefore we also have \(\mathbb{Q}^u (\{ \tau_N \leqslant T \}) \leqslant C T^{1 + \varepsilon} N^{- 1}\) which in turn implies \(\mathbb{Q}^u (T_{\exp} < T) = 0\). This proves that \(T_{\exp} = + \infty\) under \(\mathbb{Q}^u\), almost surely. As a consequence we can extend \(\mathbb{Q}^u\) to all of \(\mathscr{F} = \vee_T \mathscr{F}_T\) since for any \(A \in \mathscr{F}_T\) we can set

If \(A \in \mathscr{F}_T\) then

and

This establishes that \(\operatorname{Law}_{\mathbb{Q}^u} (X^u) =\operatorname{Law}_{\mathbb{P}} (X)\). On the other hand if \(A \in \mathscr{F}_T\) we have, using the martingale property of the Girsanov density,

And also

As a consequence

and therefore

as claimed.\(\Box\)

The following lemma will also be useful in the sequel and it is a consequence of the above discussion:

Proof. This follows from the bound (16), after choosing \(\bar{T}\) large enough. \(\Box\)

3.1Proof of absolute continuity

In this section we prove that the measure \(\mu_T\) is absolutely continuous with respect to the measure \(\mathbb{Q}^u\) we constructed in Lemma 5. First recall that the measures \(\mu_T\) defined on \(\Omega\) as

can be described, using Lemma 5, as a perturbation of \(\mathbb{Q}^u\) with density \(D_T\) given by

at least on \(\mathscr{F}_T\).

Proof. The proof of the first claim is given in Section 3.2 below. For the second claim fix \(\varepsilon > 0\). Our aim is to show that there there exists \(\delta > 0\) such that \(\mathbb{Q}^u (A) < \delta\) implies \(\int_A D_T \mathrm{d} \mathbb{Q}^u < \varepsilon\). \(\)From corollary 4 for any \(\varepsilon > 0\) there exists a \(K > 0\) such that

Then for any \(A \in \mathscr{F}\) such that \(\mathbb{Q}^u (A)^{(p - 1) / p} < \varepsilon / \left( 2 \sup_T \mathbb{E}_{\mathbb{Q}^u} \left[ | D_T |^p \mathbb{1}_{\{ \| W_{\infty} \|_{\mathscr{C}^{- 1 / 2 - \varepsilon}} \leqslant K \}} \right] \right)\)

Proof. We choose a subsequence (not relabeled) such that \(D_T \rightarrow D_{\infty}\) weakly in \(L^1 (\mathbb{Q}^u)\), for some \(D_{\infty} \in L^1 (\mathbb{Q}^u)\). It always exists by uniform integrability. We now claim that for any \(A \in \mathscr{F}\)

It is enough to check this for \(A \in \mathscr{F}_S\) for any \(S \in \mathbb{R}_+\) since these generate \(\mathscr{F}\). But there we have for \(T \geqslant S\),

by weak \(L^1\) convergence. \(\Box\)

Recall that the \(\Phi_3^4\) measure can be defined as a weak limit of the measures \(\tilde{\mu}_T\) on \(\mathscr{C}^{- 1 / 2 - \varepsilon}\) given by

where \(\theta_T\) is the gaussian measure with covariance \(\rho^2_T (\mathrm{D}) \langle \mathrm{D} \rangle^{- 2}\). From this together with the above considerations we see that any accumulation point \(\tilde{\mu}_{\infty}\) of \(\tilde{\mu}_T\) satisfies

for some \(D_{\infty} \in L^1 (\mathbb{Q}^u)\).

3.2\(L^p\) bounds

Now we will prove local \(L^p\)-bounds on the density \(D_T\). In the sequel we will denote \(\tilde{W} = W^u\), with \(u\) satisfying (13), namely \(u = U (\tilde{W})\). Before we proceed let us study how the functional \(U (\tilde{W})\) behaves under shifts of \(\tilde{W}\), since later we will want to apply the Boué–Dupuis formula and this kind of behavior will be crucial. Let \(w \in L^2 ([0, \infty) \times \Lambda)\) and denote

The process \(h^w\) satisfies

More explicitly, for all \(s \geqslant 0\) we have

Decomposing

we can write

with

The first two terms in (18) will be used for renormalization while the remainder \(r^w\) contains terms of higher regularity which will have to be estimated in the sequel.

Proof of Lemma 7. Observe that

and

Combining these two facts we have

The Boué–Dupuis formula (9) provides the variational bound

where we have set \(h^w = w + U (W + I (w))\) as above. Recall now that from Theorem 3 there exists a constant \(C\), independent of \(T\) , such that for each \(h^w\),

where

Using eq. (18) we compute

At this point we need a lower bound for

Given that we need to take \(p > 1\), this expression present a difficulty in the fact that the term \(\int^T_0 \| w_t \|^2_{L^2} \mathrm{d} t\) appears with a negative coefficient. Note that this term cannot easily be controlled via \(\int^T_0 \| r_t^w + w_t \|_{L^2}^2 \mathrm{d} t\) since the contribution \(r^w\), see eq. (19), contains factors which are homogeneous in \(w\) of order up to \(3\). This is the reason we had to localize the estimate, introduce the “good” term \(\| W_{\infty} + I_{\infty} (h^w) \|^n_{\mathscr{C}^{- 1 / 2 - \varepsilon}}\), and introduce the term \(J_s \langle \mathrm{D} \rangle^{- 1 / 2} (\mathopen{\unicode{x27E6}} (\langle \mathrm{D} \rangle^{- 1 / 2} W_s)^n \mathclose{\unicode{x27E7}})\) in (11) which will help us to control the growth of \(r^w\). Indeed in Lemma 9 below, a Gronwall argument will allow us to show that \(\int^T_0 \| w_t \|^2_{L^2} \mathrm{d} t\) can be bounded by a combination of the other “good” terms as

This implies that for \(1 < p \ll 2\),

which gives the claim. Note that here we used the bound

as well as the fact that \(\| I^{\flat}_t (h^w) \|_{\mathscr{C}^{- 1 / 2 - \varepsilon}} \lesssim \| I_{\infty} (h^w) \|_{\mathscr{C}^{- 1 / 2 - \varepsilon}}\) to conclude.\(\Box\)

The following lemmas complete the proof.

Proof. Let us introduce the notation

Write \(r_s^w = \tilde{r}_s^w +\operatorname{Aux}_s (W, w)\) and observe that

We can apply Ito formula to obtain

where

By Lemma 10 below, we have constants \(c, C\) and a random variable \(Q_T (W)\) such that

and

Integrating over space we obtain

Now by Lemma 11 below

Gathering things together we have

Then Gronwall's lemma allows to conclude

\(\Box\)

Proof. We recall that

and since \(\sup_{T < \infty} \mathbb{E} [\| \mathopen{\unicode{x27E6}} (\langle \mathrm{D} \rangle^{- 1 / 2} W_T)^i \mathclose{\unicode{x27E7}} \|^p_{\mathscr{C}^{- \varepsilon}}] < \infty\) for any \(p < \infty\) and any \(\varepsilon > 0\) it is enough to bound \(\| (\langle \mathrm{D} \rangle^{- 1 / 2} I_T (w))^{n + 1 - i} \|^q_{B_{1, 1}^{\varepsilon}}\) for some \(q > 1\) by the terms \(\| I_T (w) \|^{n + 1}_{W^{- 1 / 2, n + 1}}\) and \(\| I_T (w) \|^2_{H^1} \lesssim \int^T_0 \| w_s \|_{L^2}^2 \mathrm{d} s\). By interpolation we can estimate, for \(i \geqslant 1\),

Choosing \(q = n / \left( n - \frac{1}{(n - 1)} \right) > 1\), we have

Now for \(n\) large enough \(\frac{n}{(n - 1) n - 1} \leqslant \frac{2}{n + 1}\) and using Young's inequality we can estimate

\(\Box\)

Proof. Note that

Moreover \(h^w = u^w + w\) implies

and \(\| I^{\flat}_t (u^w) \|^4_{L^4} \lesssim \| I^{\flat}_t (h^w) \|^4_{L^4} + \| I^{\flat}_t (w) \|^4_{L^4}\) . From Lemma 18 we get

The estimation for the other terms is easy but technical and postponed until Section 5.\(\Box\)

4Singularity of \(\Phi_3^4\) w.r.t. the free field

The goal of this section is to prove that the \(\Phi_3^4\) measure is singular with respect to the Gaussian free field. For this we have to find a set \(S \subseteq \mathscr{C}^{- 1 / 2 - \varepsilon} (\Lambda)\) such that \(\mathbb{P} (W_{\infty} \in S) = 1\) and \(\mathbb{Q}^u (W_{\infty} \in S) = 0\). Together with (17), this will imply singularity. We claim that setting

for some suitable subsequence \(T_n\), does the job. Here

denotes the Wick ordering with respect to the Gaussian free field. Let us prove first that indeed \(\mathbb{P} (W_{\infty} \in S) = 1\) for some \(T_n\). For later use we define

and

Proof. Wick products corresponds to iterated Ito integrals. Introducing the notation

we can verify by Ito formula that

Since \(\theta_T J_t = 0\) for \(t \geqslant T\), Ito isometry gives

Then, again by Ito formula the expectation on the r.h.s. can be estimated as

Now recall that \(\| J_t f \|_{L^2 (\Lambda)} \lesssim \langle t \rangle^{- 3 / 2} \| f \|_{L^2 (\Lambda)}\) to conclude:

The lemma implies that \(\frac{1}{T^{(1 + \delta) / 2}} \int_{\Lambda} \mathopen{\unicode{x27E6}} (\theta_T W_{\infty})^4 \mathclose{\unicode{x27E7}} \rightarrow 0\) in \(L^2 (\mathbb{P})\). So there exists a subsequence \(T_n\) such that \(\frac{1}{T_n^{(1 + \delta) / 2}} \int_{\Lambda} \mathopen{\unicode{x27E6}} (\theta_{T_n} W_{\infty})^4 \mathclose{\unicode{x27E7}} \rightarrow 0\) almost surely.

The next step of the proof is to check that \(\mathbb{Q}^u (W_{\infty} \in S) = 0\). More concretely we will show that for a subsequence of \(T_n\) (not relabeled)

\(\mathbb{Q}^u\) almost surely. Observe that

We expect the term

to go to infinity faster than \(T^{1 - \delta}\), \(\mathbb{Q}^u\)-almost surely. To actually prove it, we start by a computation in average.

Proof. Recall that \(\mathrm{d} w_t^{\theta_T} = \theta_T J_t \mathrm{d} X_t\). With a slight abuse of notation we can write

\(\[ \begin{array}{lll} & & \int^{\infty}_0 \int_{\Lambda} (\theta_T J_t \mathbb{W}_t^{\theta_T, 3}) J_t \mathbb{W}_t^3 \mathrm{d} t\\ & = & 16 \int^{\infty}_0 \sum_k \frac{\theta_T (k) \sigma^2_t (k)}{\langle k \rangle^2} \left( \sum_{k_1 + k_2 + k_3 = k} \int^t_0 \int^{s_1}_0 \int^{s_2}_0 \mathrm{d} w_{s_1}^{\theta_T} (k_1) \mathrm{d} w_{s_2}^{\theta_T} (k_2) \mathrm{d} w^{\theta_T}_{s_3} (k_3) \right.\\ & & \times \left. \sum_{k_1 + k_2 + k_3 = k} \int^t_0 \int^{s_1}_0 \int^{s_2}_0 \mathrm{d} w_{s_1} (k_1) \mathrm{d} w_{s_2} (k_2) \mathrm{d} w_{s_3} (k_3) \right) \mathrm{d} t \end{array} \]\)

and by Ito isometry

For \(T\) large enough and since \(\sigma^2\) and \(\theta\) are positive, we have

Introduce the notation \(\mathbb{Z}_+^3 = \{ n \in \mathbb{Z}^3 : n = (n_1, n_2, n_3) \operatorname{with}n_i \geqslant 0 \}\). After restricting the sum to \((\mathbb{Z}_+^3)^3\) we get the bound

Now, for large enough \(T\) if \(k_1 + k_2 + k_3 = k\) and \(\langle k_i \rangle \leqslant T / 8\) then \(\langle k \rangle \leqslant T / 2 \times 0.9\). Furthermore if \(T\) large enough and \(k_1, k_2, k_3 \in \mathbb{Z}_+^3\) and \(k_1 + k_2 + k_3 = k\), while \(\langle k_i \rangle \geqslant (3 T / 32) \times 0.9\) (recall that if \(\langle k_i \rangle < (3 T / 32) \times 0.9\)and \(s > 3 T / 32\) then \(\sigma_s (k_1) = 0\)) we have \(\langle k \rangle \geqslant T / 8\). So for any \(k\) for which the integral is nonzero we have \(\rho_{T / 2} (k) - \rho_{T / 8} (k) = 1\) (recall that \(\rho = 1\) on \(B (0, 9 / 10)\) and \(\rho = 0\) outside of \(B (0, 1)\)). This implies

\(\Box\)

Next we upgrade this bound to almost sure divergence.

Proof. Define

We will show that \(e^{- G_T} \rightarrow 0\) in \(L^1 (\mathbb{P})\), which implies that there exists a subsequence \(T_n\) such that \(e^{- G_{T_n}} \rightarrow 0\) almost surely. From this our statement follows. By the Boué–Dupuis formula

where where have used that \(\theta_T J_t = 0\) for \(t \geqslant T\) and introduced the notations, for \(0 \leqslant i \leqslant 3\),

and

Our aim now to prove that the last three terms are bounded below uniformly as \(T \rightarrow \infty\) (while we already know that the first one diverges). For \(i \in \{ 1, 2, 3 \}\)

by Lemmas 20 and 22. Here \(Q_t (W)\) is a random variable only depending on \(W\) such that \(\sup_t \mathbb{E} [| Q_t (W) |^p] < \infty\) for any \(p < \infty\). Then

Now for the first term we obtain

For the second term we use that \(\| A_t^0 \|_{L^2} \leqslant Q_t (W)\) so

Since \(\sup_t \| I_t (v) \|^2_{H^1} \lesssim \int^{\infty}_0 \| v_t \|^2_{L^2} \mathrm{d} t\) in total we obtain for \(T\) large enough. The third term is estimated analogously.

\(\Box\)

Next we prove an estimate which will help with the proof of the main theorem.

Proof. Recall that under \(\mathbb{Q}^u\) we have \(W_t = W_t^u + I_t (u)\) where \(u\) is defined above by (13) and \(\operatorname{Law}_{\mathbb{Q}^u} (W^u) =\operatorname{Law}_{\mathbb{P}} (W)\). With this in mind we compute

where, as above,

By Lemmas 20 and 22 we have that \(\mathbb{E}_{\mathbb{Q}^u} [\| A_t^i \|^2_{L^2}] \leqslant C\) so the Cauchy–Schwartz inequality gives the result.\(\Box\)

Proof. We have

Now since \(\mathrm{d} X_t = \mathrm{d} X_t^u + u_t \mathrm{d} t\) we have

The first term goes to \(0\) \(\mathbb{Q}^u\)-almost surely by Lemma 15. To analyze the third term we estimate

By the computation from Lemma 15 we have

and \(\sup_t \mathbb{E}_{\mathbb{Q}^u} [(\| \mathbb{W}_t^{u, 2} \|_{\mathcal{C}^{- 1 - \delta / 2}} \| I_t (u) \|_{L^2})^2] < \infty\), so (20) converges to \(0\) in \(L^1 (\mathbb{Q}^u)\). For the fourth term we proceed in the same way:

which is bounded in expectation uniformly in \(T\), so the fourth term goes to \(0\) in \(L^1 (\mathbb{Q}^u)\) as well. It remains to analyze the second term. Again introducing the notation

we have

Now observe that

so the \(\limsup\)of this is \(\infty\) almost surely. To estimate the sum we again observe that for \(i \geqslant 3\) \(\mathbb{E}_{\mathbb{Q}^u} [\| A_t^i \|^2_{L^2}] \lesssim \langle t \rangle^{- 1 + \delta}\) and by Young's inequality

Taking expectation we obtain

We have deduced that

where \(R_T \rightarrow 0\) in \(L^1 (\mathbb{Q}^u)\). We can conclude by selecting a subsequence \((T_n)_n\) such that

\(\mathbb{Q}^u\)-almost surely and \(R_{T_n} \rightarrow 0\), \(\mathbb{Q}^u\)-almost surely.\(\Box\)

5Some analytic estimates

We collect in this final section various technical estimates needed to complete the proof of Lemma 11.

Proof. See [15].\(\Box\)

Proof.

So

and applying Young's inequality with the exponents \((32, 32 / 9, 32 / 22)\), we obtain

and subtracting \(\delta (\| \phi \|^4_{L^4})^{21 / 20}\) on both sides of the inequality gives the result. \(\Box\)

Proof. We observe that since \(\mathopen{\unicode{x27E6}} W_t^2 \mathclose{\unicode{x27E7}}\) is spectrally supported in a ball or radius \(\thicksim t\)

For the first estimate we know that \((1 - \theta_t) I_t (w)\) is supported in an annulus of radius \(\sim t\), so \(\| (1 - \theta_t) I_t (w) \|_{L^2} \lesssim \langle t \rangle^{- 1 + \varepsilon} \| I_t (w) \|_{H^{1 - \varepsilon}}\) and furthermore by interpolation \(\| I_t (w) \|_{H^{1 - \varepsilon}} \lesssim \| I_t (w) \|^{1 - \varepsilon}_{H^1} \| I_t (w) \|^{\varepsilon}_{L^2} \lesssim \| I_t (w) \|^{1 - \varepsilon}_{H^1} \| I_t (w) \|^{\varepsilon}_{L^4}\). By definition \(\langle t \rangle^{1 / 2} J_t\) is a uniformly bounded Fourier multiplier regularizing by 1, and putting everything together, by paraproduct estimates

For the second term in addition observe that the function \(\langle t \rangle^{1 / 2} J_t\) is spectrally supported in an annulus of radius \(\sim t\), and regularizes by \(1\) so again by estimates for the resonant product

For the third estimate again applying paraproduct estimates and the properties of \(J\),

Now, the claim follows from interpolation and Young's inequality

\(\Box\)

Proof. By the spectral properties of \(J_t\),

Applying Young's inequality with exponents \(\left( \frac{n}{2}, \frac{n / 2}{(n / 2 - 1)} \right)\) with \(n\) such that \(\frac{2 n}{(n / 2 - 1)} \leqslant 4 + \varepsilon\) where \(\varepsilon\) is chosen as in Lemma 18 we have

Now the second estimate follows from \(\operatorname{chosing}\) \(n\) large enough (depending on \(\delta\)) and using Besov embedding after taking \(f = g\).\(\Box\)

Proof. For the first estimate we again use the spectral properties of \(W, I,\) and \(J\) and obtain by paraproduct estimate

and the claim follows by Young's inequality. For the second

and the claim follows again by Young's inequality. \(\Box\)

Proof.

This proves the first estimate. For the second we continue

\(\Box\)

Proof. This follows in the same fashion as Lemma 22 .\(\Box\)

Appendix ABesov spaces and paraproducts

In this section we will recall some well known results about Besov spaces, embeddings, Fourier multipliers and paraproducts. The reader can find full details and proofs in [3, 14].

First recall the definition of Littlewood–Paley blocks. Let \(\chi, \varphi\) be smooth radial functions \(\mathbb{R}^d \rightarrow \mathbb{R}\) such that

• \(\operatorname{supp} \chi \subseteq B (0, R)\), \(\operatorname{supp} \varphi \subseteq B (0, 2 R) \setminus B (0, R)\);

• \(0 \leqslant \chi, \varphi \leqslant 1\), \(\chi (\xi) + \sum_{j \geq 0} \varphi (2^{- j} \xi) = 1\) for any \(\xi \in \mathbb{R}^d\);

• \(\operatorname{supp} \varphi (2^{- j} \cdot) \cap \operatorname{supp} \varphi (2^{- i} \cdot) = \varnothing\) if \(| i - j | > 1\).

Introduce the notations \(\varphi_{- 1} = \chi\), \(\varphi_j = \varphi (2^{- j} \cdot)\) for \(j \geqslant 0\). For any \(f \in \mathscr{S}' (\Lambda)\) we define the operators \(\Delta_j f := \mathcal{\mathscr{F}}^{- 1}_{\xi} (\varphi_j (\xi) \hat{f} (\xi))\), \(j \geqslant - 1\).

Remark 31. Proposition 30 is not proven in the above references but is quite easy and the reader can fill out a proof.

Bibliography