Almost sure existence of global solutions for general initial value problems

This article is concerned with the almost sure existence of global solutions for initial value problems of the form $\dot{\gamma}(t)= v(t,\gamma(t))$ on separable dual Banach spaces. We prove a general result stating that whenever there exists $(\mu_t)_{t\in \mathbb{R}}$ a family of probability measures satisfying a related statistical Liouville equation, there exist global solutions to the initial value problem for $\mu_0$-almost all initial data, possibly without uniqueness. The main assumption is a mild integrability condition of the vector field $v$ with respect to $(\mu_t)_{t\in \mathbb{R}}$. As a notable application, we obtain from the above principle that Gibbs and Gaussian measures yield low regularity global solutions for several nonlinear dispersive PDEs as well as fluid mechanics equations including the Hartree, Klein-Gordon, NLS, Euler and modified surface quasi-geostrophic equations. In this regard, our result generalizes Bourgain's method as well as Albeverio&Cruzeiro's method of constructing low regularity global solutions, without the need for local well-posedness analysis.

Initial value problems, including ODEs, PDEs and stochastic PDEs, are of major interest to both applied and fundamental Mathematics.There is an abundant literature for this broad field of research, covering often important evolution equations in science, see e.g.[10,27,47].From a theoretical point of view, one can mainly recognize two qualitative approaches: -A specific analysis that relies on the exact or almost exact form of initial value problems using particular features of given equations (e.g.exactly solvable equations, dispersive, hyperbolic, parabolic and to some extent semilinear equations).-A more general analysis that ignores the exact form of initial value problems and instead focuses on finding general criteria that ensure uniqueness, local and global existence of solutions (e.g.Carathéodory, Cauchy-Lipschitz, Peano theorems and to some extent fixedpoint theorems).
Of course, these two perspectives complement each other.On the other hand, there is a sharp distinction between initial value problems over finite and infinite-dimensional spaces.For instance, in infinite dimensions it is known that the Peano theorem is in general not true and that there exist finite lifespan solutions to initial value problems without blowup.However, the Cauchy-Lipschitz theorem still holds true on Banach spaces, thus indicating that certain results could indeed survive in infinite dimensions.The present article is concerned with the second approach.
In the past few decades, there have been significant advances in the field of dispersive PDEs to construct almost sure global solutions with low regularity, with methods stemming from the combination of probability theory, harmonic analysis and quantum field theory.These advances were inspired by the pioneering work of Bourgain [13,14,15].The latter papers continued the line of research initiated by Lebowitz-Rose-Speer [45] and Zhidkov [71], as well as McKean-Vaninsky [49,50,51].There have been many subsequent contributions on this subject, see [18,19,20,23,24,25,32,33,34,36,41,42,44,53,55,56,60,61,62,69] and the works quoted there.For an overview, we also refer the reader to the expository works [21,57,58] and the references therein.One of the main ideas of the aforementioned works is that Gibbs measures for some Hamiltonian PDEs are well-defined over Sobolev spaces with sufficiently low regularity exponent and that they are formally invariant under the flow.However, the other known conservation laws are available only at higher regularity.Consequently, the Gibbs measures can be used as a substitute for a conservation law.When combined with local well-posedness theory, they can be used to construct low regularity global solutions for almost every initial data.On the other hand, in the field of fluid mechanics, there is a long-standing interest in constructing invariant measures and global solutions using probabilistic methods (see e.g.[1,3,26]).In particular, in the nineties, Albeverio and Cruzeiro proved the almost sure existence of global solutions to the Euler equation on the two-dimensional torus [2].More recently, there has been a renewed interest for such invariant measures and stochastic flows in fluid mechanics (see e.g.[38,39,40]).In the work of Nahmod-Pavlović-Staffilani-Totz [54], the authors extended the result of Albeverio-Cruzeiro [2] to the modified quasi-geostrophic (mSQG) equations which interpolate between the Euler and the SQG equation.Again the idea behind the construction of these global flows is the invariance of a given well-understood (Gaussian) measure combined with probabilistic compactness arguments in the spirit of Prokhorov's or Skorokhod's theorems.Our main claim here is that the previous circle of ideas is quite general and robust and could be formulated as a general principle for abstract initial value problems.
Our aim in this article is to address the question of almost sure existence of global solutions from a more general perspective.More precisely, we consider an abstract initial value problem of the form γ(t) = v(t, γ(t)).
(1.1) over a separable dual Banach space 1 (B, • ) with v : R × B → B a Borel vector field.We assume that there exists a narrowly continuous 2 probability measure solution (µ t ) t∈R to the statistical Liouville equation with the same vector field.Here •, • denotes the duality bracket between the topological dual space B = E * and its predual E, while the F range over a class of smooth test functions with ∇F denoting their Fréchet differentials.The statistical Liouville equation (1.2) is explained in detail in Section 1.1.The considered class of smooth test functions is given in Definition 1. 3. In this framework, we show that for µ 0 -almost all initial data in B there exist global solutions to the initial value problem (1.1), if the vector field satisfies the integrability condition Remarkably, such a result depends neither on the shape of the vector field, nor on a suitable local well-posedness theory.Moreover, the method applies equally to finite or infinite-dimensional spaces and the vector field v is not required to be continuous.In practice, our result reduces the problem of constructing global solutions of ODEs or PDEs to finding probability measure solutions for the statistical Liouville equation (1.2).The latter problem is sometimes more tractable.In many of the cases which we study later, one can directly construct global probability measure solutions (µ t ) t∈R in such a way that they are either stationary or stationary modulo a pushforward.Generally speaking, recall that for Hamiltonian systems the Liouville theorem ensures the existence of invariant measures while for dynamical systems the Krylov-Bogolyubov theorem ([68, Lecture 2] and [31]) is an efficient tool for constructing invariant measures.Note that our understanding of invariance here is in terms of stationary solutions for the statistical Liouville equation (1.2) instead of the invariance with respect to the flow, since the latter may not exist in general.
On the other hand, in the recent work [9], the Kubo-Martin-Schwinger (KMS) equilibrium states were introduced for Hamiltonian PDEs (like Hartree, NLS, Wave equations).In particular, it was proved in this context that Gibbs measures are KMS equilibrium states satisfying the stationary statistical Liouville equation (1.2) with an appropriate choice of the vector field v in accordance with the given PDE.Hence, as a consequence of the above principle and the stationarity of Gibbs measures one deduces straightforwardly the existence of low regularity global solutions for almost all initial data for several Hamiltonian PDEs.It is also worth highlighting here the two features of our approach: -No dispersive properties are needed.
We refer the reader to Section 1.4 for more details on the examples of nonlinear PDEs considered here (including Hartree, NLS and Wave equations on the flat torus T d , d = 1, 2, 3 (see [21]) and Euler, modified surface quasi-geostrophic (mSQG) equations on the 2-dimensional torus (see [2,54])).Of course, we do not prove here global well-posedness for such PDEs.Instead, we show almost surely the existence of global solutions for such PDEs.Potentially, one can try to combine our method with a local posedness theory in order to prove global well-posedness (see [22,23]).In contrast, uniqueness generally speaking depends more on the particular properties of the considered initial value problem.Another aspect that we did not address is global solutions for random systems.Instead, we focus here on deterministic equations.In particular, it makes sense to study the problem for stochastic PDEs and random dynamical systems with widespread applications in fluid mechanics and stochastic quantization for instance (see [70] and [30]).The analysis for these random systems will be addressed elsewhere.
Techniques: Our approach is quite related to statistical physics in spirit and consists of studying the evolution of ensembles of initial data through statistical Liouville equations.However, the key argument comes from transport theory via the superposition principle (or probabilistic representation) proved for instance in [6,11,48] (see also [12]).The superposition principle shows in particular that if one has a probability measure solution (µ t ) t∈[0,T ] to the statistical Liouville equation (1.2) on the finite time interval [0, T ], then there exists a probability path measure η concentrated on the set of local solutions of the initial value problem (1.1) such that the image measure of η under the evaluation map at each fixed time is equal to µ t (see Proposition 2.1).Such a result is extended to infinite dimensions and adapted to PDE analysis in [8].Here, we extend such a principle to separable dual Banach spaces and more importantly to global solutions of initial value problems in such a way that this tool yields a powerful globalisation argument.In particular, by using the measurable projection theorem and the properties of the path measure η, we are able to find a universally measurable subset G of the Banach space B such that µ 0 (G ) = 1 and such that for each x ∈ G there exists a global solution to the initial value problem (1.1).Furthermore, we construct a measurable flow in Theorem 1.11 if in addition we assume that there exists at most one global mild solution of the initial value problem (1.4) for each initial condition.Moreover, in Theorem 1.13 we prove that if the initial value problem (1.4) admits a measurable flow (See Definition 1.9) then (µ t ) t∈R satisfies an appropriate statistical Liouville equation.
In conclusion, the globalization result (Theorem 1.7) and Theorems 1.11-1.13proved here by measure theoretical techniques are quite general and, to the best of our knowledge, new.They formalize and unify some of the deep ideas in the topic of constructing almost sure low regularity global solutions to dispersive PDEs and to fluid mechanics equations.As an application, we are able to recover several known results and to obtain new ones (see Sections 1.3-1.4).The article also connects the problem of constructing global flows for PDEs with the topic of continuity and transport equations (see [4,6]).
1.1.General framework.Let (B, || • ||) be a real separable dual Banach space.This means that there exists (E, • E ) a real Banach space such that B is the topological dual of E (i.e.[17,Theorem III.23]).When there is no possible confusion, we will denote the duality bracket •, • E * ,E simply by •, • .In all the sequel, I denotes a closed unbounded time interval (e.g.I = R, I = R − or I = R + ).We denote by t 0 ∈ I any initial time if I = R.If I is bounded from below or above, then we denote by t 0 ∈ I its endpoint.Our main purpose is the study of the initial value problem where v : I × B → B is a Borel vector field.Generally speaking, there are several notions of solutions to (1.4).A strong solution is a curve γ belonging to C 1 (I; B) and satisfying (1.4) for all times t ∈ I.However, to study such curves one usually requires v to be at least continuous in order to have a consistent equation.Instead, we focus on mild solutions of (1.4) which are continuous curves γ ∈ C (I; B) such that v(•, γ(•)) ∈ L 1 loc (I, dt; B) and for all t ∈ I the following integral formula is satisfied. (1.5) Here, the integration in the right hand side is a Bochner integral and the function s Since separable dual Banach spaces satisfy the Radon-Nikodym property (see [65]), the functions in the space AC 1 loc (I; B) are continuous and almost everywhere differentiable on I with a derivative u(•) ∈ L 1 loc (I, dt; B).Hence, a curve γ : I → B is a mild solution of (1.4) if and only if γ ∈ AC 1 loc (I; B), γ(t 0 ) = x and for almost all t ∈ I γ(t) = v(t, γ(t)).
Statistical Liouville equation: When studying the statistical Liouville equation (1.2), the following notion will be useful.
Definition 1.1 (Fundamental strongly total biorthogonal system).We say that the families {e k } k∈N and {e * k } k∈N in E, E * respectively form a fundamental strongly total biorthogonal system if the following properties hold.
We note that such an object exists in our framework.
Here, S (R n ) denotes the Schwartz space and we have the inclusions Let us note that any F ∈ C ∞ b,cyl (B) is Fréchet differentiable with a differential DF (u) ∈ B * = E * * identified with an element of E. Hence, we denote simply ∇F for the differential of F so that ∇F (u) ∈ E for all u ∈ B. In particular, we have (1.7) Let B(B) and P(B) denote respectively the Borel σ-algebra and the space of Borel probability measures on the Banach space (B, || • ||).We endow P(B) with the narrow topology.Definition 1.4 (Narrow continuity).We say a curve (µ t ) t∈I in P(B) is narrowly continuous if for any bounded continuous real-valued function F ∈ C b (B, R), the map We say that a narrowly continuous curve (µ t ) t∈I in P(B) satisfies the statistical Liouville equation with respect to the Borel vector field v : in the sense of distributions on the interior of I. On the right hand side of (1.9), the quantity v(t, u), ∇F (u) refers to the duality bracket E * , E. In order for the above statistical Liouville equation to make sense, one needs a further assumption on the vector field v which ensures the integrability of the right hand side of (1.9).
Assumption 1.5 (Assumption on the vector field v).We assume that v : I × B → B is a Borel vector field such that In particular, it follows that the duality pairing v(t, u), ∇F (u) ≡ v(t, u), ∇F (u) E * ,E makes sense and satisfies for some constant C > 0 depending on F .Using (1.11) and Assumption 1.5, it follows that the right-hand side of (1.9) is finite for almost every t ∈ I. Then there exists a universally measurable subset G of B of total measure µ t 0 (G ) = 1 such that for any x ∈ G there exists a global mild solution to the initial value problem (1.4).
Remark 1.8.The following comments are useful.
• The above theorem extends straightforwardly to any Banach space that is isometrically isomorphic to a separable dual Banach space.• The assumption on narrow continuity of (µ t ) t∈I can be relaxed to weak narrow continuity given in Definition 2.3.
• The notions of mild solutions and the statistical Liouville equation are explained in Section 1.1.• A universally measurable set is a subset of a Polish space (here B) that is measurable with respect to every complete probability measure.In particular, G is µ t 0 -measurable.• The above theorem provides no information about uniqueness of mild solutions.
Next, we introduce the notion of measurable flow.In the sequel, it is convenient to consider I = R. Definition 1.9 (Measurable flow).Let B be a separable dual Banach space and v : R × B → B a Borel vector field.We say that the initial value problem (1.4) admits a measurable flow φ t t 0 with respect to a narrowly continuous curve (µ t ) t∈R of probability measures in P(B) if for all are Borel sets of total measure µ t 0 (G t 0 ) = 1 and the map φ t t 0 is Borel measurable and satisfies for all s, t, t 0 ∈ R : Remark 1.10.It is worth noticing that in ergodic theory there exists a similar notion of measurable flow (i.e. a one parameter group of bijective measure-preserving transformations T t : X → X on a measure space (X, µ) such that (t, x) ∈ R × X → T t (x) ∈ X is measurable).Here, our Definition 1.9 is slightly different (see [43] and Remark 3.6).
We give below two further results: ) with respect to (µ t ) t∈R as in Definition 1.9.Then the curve (µ t ) t∈R satisfies the statistical Liouville equation (1.9).Remark 1.14 (Stationary measure).In particular, in Theorem 1.13 above, if (µ t ) t∈R is stationary (i.e. for all t ∈ R, µ t = µ 0 ), then µ 0 is a stationary solution of the statistical Liouville equation (1.9).1.3.Application to ODEs.Consider the euclidean space R 2d endowed with a symplectic structure given by a skew-symmetric matrix J satisfying J 2 = −I 2d .Let h : R 2d → R be a Borel function in the local Sobolev space W 2,2 loc (R 2d , R).Furthermore, suppose that there exists a nonnegative where L denotes the Lebesgue measure over R 2d .Consider the initial value problem given by the Hamiltonian system: ) is a stationary solution to the statistical Liouville equation (1.9) with the vector field v = J∇h, yields the following consequence of Theorem 1.7.
Corollary 1.15.Assume that h ∈ W 2,2 loc (R 2d , R) and the condition (1.13) is satisfied.Then the Hamiltonian system (1.14) admits a global mild solution for Lebesgue-almost any initial condition u 0 ∈ R 2d .
As an illustrative example for d ≥ 5, one can take F (t) = e −βt , t ∈ R + , for some β > 0 and Counter-example: We recall a counter-example from the work of Cruzeiro [29], which shows that Assumption 1.5 cannot be omitted.Indeed, consider the time-independent and the stationary family of probability measures µ t = µ 0 where µ 0 is the standard centered Gaussian measure on R 2 .Then, (µ t ) t∈R satisfies the statistical Liouville equation (1.9) with the above vector field v.Moreover, the initial value problem (1.4) with v as in (1.15) leads to the ODE q(t) = q(t) 2 , which has non global (unique) solutions for any initial condition q(0) = 0. On the other hand, one checks that v does not satisfy Assumption 1.5.This shows the existence of a C ∞ -vector field and a stationary probability measure solving the statistical Liouville equation, but for which the conclusion of Theorem 1.7 does not hold because Assumption 1.5 is not satisfied.In this respect, one can interpret the integral condition (1.10) or (1.12) as an almost sure non-blow up assumption.(1.17) Then, one can define Sobolev spaces with positive exponent r > 0 as and Sobolev spaces with negative exponent as From now on, we regard H, H r , H −r as real Hilbert spaces endowed respectively with the scalar products and denote them respectively by H R , H r R , H −r R (note that Re(•) refer to the real part).Then it is well-known that there exists a unique centred Gaussian probability measure ν 0 on the Sobolev space H −s with s ≥ 0 satisfying (1.17) and such that for all ξ ∈ H −s , Once we have such a centred Gaussian measure ν 0 , one can define the Gross-Sobolev space where here ∇F is the Malliavin derivative of F (see for instance [59] or [9] for brief details).In particular, D 1,2 (ν 0 ) is a Hilbert space when endowed with the inner product Our purpose is to prove that the initial value problem (1.4) admit global solutions for ν 0 -almost any initial condition x ∈ H −s (here B = H −s ) when the vector field v : R × H Thanks to the above assumptions, the following Gibbs measure is well-defined.
Then within the above framework, we prove in Section 3 the following result.
Proposition 1.16.Consider the time-dependent push-forward Gibbs measures Then for all t ∈ R and any This implies that there exists a narrowly continuous curve (µ t ) t∈R in P(H −s ), given in (1.22), satisfying the statistical Liouville equation (1.23) and the integrability condition )), which corresponds to Assumption 1.5 (equivalently to (1.12) with ω(t) = t 2 ).Thus, Theorem 1.7 yields the following statement.
Corollary 1.17.For any nonlinear functional h N L : admit global mild solutions for ν 0 -almost any initial data in Such a result is new to the best of our knowledge.It is a straightforward consequence of the Theorem 1.7 and Proposition 1.16.The main point in Corollary 1.17 is that the existence of welldefined Gibbs measure provides a global solution to the statistical Liouville equation and hence by Theorem 1.7, one deduces the almost sure existence of global solutions.It is clear therefore that there is a tight connection between Theorem 1.7 and the subject of Gibbs measures and low regularity solutions of dispersive PDEs.In fact, several examples of concrete PDEs like NLS, Hartree and Wave equations can be recast as the above initial value problem (1.24).One needs only to specify the Hilbert space H, the operator A and the nonlinear functional h N L .
Gibbs measures for nonlinear dispersive equations are well-studied and the literature on the subject is quite large as was summarised above.To highlight the connection of our results with this topic, we provide here some applications of Corollary 1.17 to concrete examples.Our aim is not to give all the possible applications, but rather to illustrate our method.
Hartree and NLS equations: Let H be the Hilbert space L 2 (T d ) with T d = R d /(2πZ d ) the flat d-dimensional torus.Take the operator A as where ∆ is the Laplacian on T d .So, the family {e k = e ikx } k∈Z d forms an O.N.B of eigenvectors for the operator A which admits a compact resolvent.Now, consider an exponent s ≥ 0 such that and define the Sobolev space H −s accordingly.Denote by ν 0 the well-defined centred Gaussian measure on H −s given by (1.18).Then, we list some nonlinear functionals, h N L : H −s → R, for which Corollary 1.17 applies; specified according to the dimension d and the type of equation.
• The Hartree equation on T: Let V : T → R be a nonnegative even L 1 function and ) be even and of positive type such that there exist ǫ > 0 and C > 0 with the property that for all (1.28) Then take • The NLS equation on T: Let r ∈ N and let • The NLS equation on Here, the notation : : refers to Wick ordering with respect to the Gaussian measure ν 0 .See for instance [61] for a self-contained construction of these Wick ordered nonlinearities (1.29)-(1.31).
Wave equations: Consider the Hilbert space R is the space of real-valued square integrable functions.For s ∈ R satisfying (1.26), define the Sobolev space The nonlinear wave equation takes the form (1.33) The Gaussian measure ν 0 in the case of the wave equation is defined as the product measure where ν 1 0 and ν 2 0 are Gaussian measures on the distribution space D ′ (T d ) with covariance operators (−∆ + 1) −1 and 1 respectively.Moreover, one can define rigorously the Gibbs measure for the nonlinear wave equation as We recall the following result, proved in [9].Before proceeding with the proof, we note a few applications of our results.
Nonlinear (an)-harmonic equations: One can consider nonlinear Schrödinger type equations with (an)-harmonic oscillators on R d , i.e. with for α > 0. In this case, the Hilbert space we consider is H = L 2 (R d ).The Gaussian measure ν 0 on H −s R is defined as in (1.18) with A as in (1.34).The probability measure ν 0 is well-defined provided that the assumption (1.17) on the eigenvalues of A is satisfied.Let us now verify the range of s for which the latter is true.We first recall the Lieb-Thirring inequality [35], which states that for γ > d 2 , we have By (1.35), we deduce that for the assumption (1.17) is satisfied.Hence, we have all the ingredients to apply Corollary 1.17 and obtain the following statement.
Corollary 1.20.Assume (1.36) and consider any nonlinear functional h N L : Then the nonlinear (an)-harmonic equation on R d , admits global mild solutions on H −s almost surely with respect to the Gaussian measure ν 0 in (1.18), with A as in (1.34).
Note that such a result still holds true under the perturbation of the (an)-harmonic oscillator (1.34) by a potential (see for instance the spectral asymptotics in [37] and the references therein).
Nonlinear dispersive equations on bounded domains or manifolds: Instead of working on the torus T d or on the whole space R d , it is possible to consider the Hartree, NLS and Wave equations on bounded domains or on compact Riemannian manifolds without boundary.In particular, Corollary 1.17 holds true in the following two cases.
• Let Ω be a bounded open domain in R d .Take the Hilbert space H = L 2 (Ω) and the operator A = −∆ Ω + c , (1.37) with Dirichlet or Neumann boundary conditions.Here, ∆ Ω is the Laplace operator on Ω and c > 0 is a constant chosen such that A is positive.Thanks to Weyl's law (see [72,Chapter 14]), the assumption (1.17) on the eigenvalues of the Laplacian −∆ Ω is satisfied whenever s > d 2 − 1 .
• Let (M, g) be a d-dimensional compact Riemannian manifold without boundary.Take the Hilbert space H = L 2 (M) and consider A as the Laplace-Beltrami operator Thanks to Weyl's law, the assumption (1.17) on the eigenvalues of the Laplace-Beltrami operator −∆ g is satisfied whenever As Fluid mechanics equations: In this paragraph, we follow the work of [54] and refer the reader to the references therein for more details on the Euler and modified SQG equations.Indeed, the mSQG equation takes the form: for δ > 0.Here θ : . The case δ = 1 in the above equation corresponds to the 2D Euler equation.The streamline formulation (u = |D| −1 θ) of the above mSQG equation yields so that the original mSQG equation is rephrased as an initial value problem with an autonomous vector field given by v Consider now the centred Gaussian measure ν 0 defined on the negative the Sobolev space H −s , s > 0, as in the previous section (with A = −∆ and the Hilbert space L 2 0 (T 2 ) of mean zero square integrable functions).There are nice results on one hand by Albeverio and Cruzeiro [2] for the Euler equation and on the other by Nahmod-Pavlović-Staffilani-Totz [54] for the mSQG equation (0 < δ ≤ 1), establishing the existence of solutions for arbitrarily large lifespan and for almost initial data in the spaces H s , s < −2, with respect to the Gaussian measure ν 0 .Actually, thanks to preliminary results in [54], one can apply our Theorem 1.7 to these equations too.Indeed, take µ t = ν 0 , B = H −s , s > 2, and the vector field v as in (1.41), then in [54, Proposition 4.1] it is proved that v ∈ L 2 (H −s , ν 0 ; H −s ) for all s > 2. This implies that the integrability condition (1.12) is satisfied with any ω such that ω −1 ∈ L 1 (R + , dt).Moreover, the stationary Liouville equation (1.9) is satisfied by the Gaussian measure ν 0 thanks to the proof of [54,Lemma 5.1].Hence, as a consequence of Theorem 1.7, we obtain ν 0 -almost surely the existence of global solutions to the mSQG equation (1.39)-(1.40) in C (R + , H −s ) for all s > 2. In particular, our application of Theorem 1.7 yields an improvement of the result [54] as it gives almost sure global solutions instead of arbitrarily large lifespan solutions.

Global superposition principle
Our purpose in this part is to state and prove the global superposition principle (Proposition 2.1).For that, we need to introduce the path space Then, we accordingly define a distance d over the product space X = B × C (I; B), Furthermore, we define, for each t ∈ I, the evaluation map Now, we are in position to state the global superposition principle.Recall that I is an unbounded closed interval and B is a separable dual Banach space equipped with a biorthogonal system satisfying (a)-(c) given in Definition 1.1.
Proposition 2.1 (Global superposition principle).Let v : I × B → B be a Borel vector field.Let (µ t ) t∈I be a narrowly continuous curve in P(B) satisfying the integrability condition (1.10) and the statistical Liouville equation (1.9).Then, there exists a Borel probability measure η ∈ P(X) such that: (i) η concentrates on the set of pairs (x, γ) such that γ ∈ AC 1 loc (I; B) is a mild solution of the initial value problem (1.4) for a.e.
The remaining part of this section is dedicated to the proof the above proposition.In subsection 2.1, we introduce convenient weaker topologies on the space B and the path space X.In Subsection 2.2, we set up a finite-dimensional projection argument.In Subsection 2.3, we prove Proposition 2.1 when B = R d .Then, we extend such a result to infinite-dimensional separable dual Banach spaces in Subsection 2.4.

Weak topologies.
The following topological and measure theoretical aspects will be very useful in the proofs of our main Theorem 1.7 and the global superposition principle (Proposition 2.1).
It is useful to introduce a norm over B that ensures relative compactness of bounded sets, It is useful to distinguish two narrow topologies over P(B).Namely, the (strong) narrow topology when B is equipped with the original norm • and the "weak" narrow topology when B is endowed with the norm • * .Definition 2.3 (Weak narrow topology).We say that a sequence (µ n ) n∈N of Borel probability measures in P(B) converges weakly narrowly to µ ∈ P(B) if for every bounded continuous function In such a case, we denote Accordingly, a curve (µ t ) t∈I in P(B) is said to be weakly narrowly continuous if the real-valued map Remark 2.4.Note that in finite dimensions the weak narrow and (strong) narrow topologies coincide.
On the other hand, we define similarly new distances on C (I; B) and the path space X = B × C (I; B) given respectively by and Recall that a Polish space is a Hausdorff topological space homeomorphic to a separable complete metric space while a Suslin space is a Hausdorff topological space which is the image of a Polish space under a continuous map.In particular, in our case, (B, • * ) is a Suslin space.Proof.In fact, (X, d) is a metric space which is the product of two separable complete metric spaces.Note that C (I; B) is separable and complete with respect to the compact-open topology because I is a hemicompact space (see e.g.[52,67]).On the other hand, the identity map It is also useful to stress the following result.
Hence, as a consequence, the set of Borel probability measures on (X, d) and on (X, d * ) coincide.
2.2.Projective argument.We introduce the finite rank linear operators: x, e k e * k . (2.9) Lemma 2.7.The operators T n : B → B satisfy the following properties: Proof.Recall that the families {e k } k∈N and {e * k } k∈N define a fundamental and strongly total biorthogonal system satisfying (a)-(c) in Definition 1.1.(i) For all x ∈ B, Then using (i) and the density of Span{e * k , k ∈ N} in (B, • ), we prove (ii) by an approximation argument.
For n ∈ N, let B n = T n (B) = Span(e * 1 , . . ., e * n ) ⊂ B. Denote by Define also the maps (2.10) Remark that we have the following relations Let (µ t ) t∈I be a family of Borel probability measures on B satisfying the assumptions in Proposition 2.1.Then consider the following image measures in the distribution sense over the interior of I and for some Borel vector field v n : Proof.Notice first that the curve (µ n t ) t∈I is narrowly continuous.Indeed, let ϕ t∈I is weakly narrowly continuous as it is (strongly) narrowly continuous (see Definitions 1.4 and 2.3).We then consider the statistical Liouville equation (1.9) and select The left part of (1.9) is transformed to: (2.17) )) e j ∈ E, the right part of (1.9) is transformed to: Remark that since (B, • * ) is a separable Radon space, we can apply the disintegration Theorem F.1 (see Appendices C and F).In particular, there exists a µ n t -a.e.determined family of measures {µ n t,y , y ∈ R n } ⊂ P(B) such that µ n t,y B \ (π n ) −1 (y) = 0 and applying formula (F.1) with So, we get where we have introduced the vector field v n as follows: π n • v(t, x) µ n t,y (dx), for t ∈ I and µ n t -a.e.y ∈ R n .

Analysis in finite dimensions.
In this part, we restrict our selves to the case B = R d .So, we aim to prove the global superposition principle (Proposition 2.1) when B = R d .In fact, similar results are already known in finite dimensions and proved in the book of Ambrosio et al. [6] and in the work of Maniglia [48].The main difference here is that we consider times in unbounded intervals like the half-line R + while in the latter references it is restricted to [0, T ].Unfortunately, we could not deduce directly Proposition 2.1 in the case B = R d from the result of Maniglia [48] or even from [6].In fact, one needs to go through the main ideas and adjust some topological arguments which are behind the compactness properties that lead to the construction of the probability measure η on the path space X.We first discuss in Subsection 2.3.1 the case where the vector field is locally Lipschitz in the second variable, then we consider the Borel case in Subsection 2.3.2.

2.3.1.
The Lipschitz case.Let • R d be any norm on R d .We denote often v t = v(t, •).In this paragraph, we impose the following local Lipschitz condition.
Assumption 2.9 (Lipschitz condition).For every compact set where lip(•, K) denotes the Lipschitz constant on K.  Let which is a Borel subset of R d .Remark that by construction {G d,T } T ∈N is a decreasing sequence of Borel sets.Then, by monotone convergence theorem, we have Recall that in the Lipschitz case, we have uniqueness of solutions for the initial value problem (1.4).So, we conclude that for each x ∈ G d there exists a global unique solution of (1.4).Thus, we construct a well-defined global flow where γ x is the global solution of (1.4) with the initial condition γ(t 0 ) = x.Note that Φ is the flow with prescribed initial conditions at time t 0 .Moreover, using the identity (B.1), we have for all t ∈ I, µ t = (Φ t ) ♯ µ t 0 .Now, we construct the measure η as where Id × Φ • is the map given by Next, we want to prove that η satisfies the conditions (i) and (ii) of Proposition 2.1.

For (i):
We have to prove that η(F d ) = 1 where which implies that we have For each t j ∈ Q ∩ I, a rational number, there exists a null set N t j with η(N t j ) = 0 and such that (2.22) holds true on X \ N t j .Then taking with η(N ) = 0 such that for all (x, γ) / ∈ N and for all rational numbers in {t j } j = Q ∩ I, we have Now, using the continuity of the curves γ and since v(•, γ(•)) ∈ L 1 loc (I, dt) η−almost surely (see condition (1.12)), the identity (2.23) is well-defined and moreover we get (2.22) for all times t ∈ I and for all (x, γ) / ∈ N .

The Borel case.
In this section, we prove Proposition 2.1 when B = R d and v is a Borel vector field.Now since v is no more assumed to be Lipschitz in the second variable, we have to take into account that the characteristics may not be unique.Indeed, the potential lack of uniqueness of solutions to (1.4) on finite intervals makes it impossible to follow the same strategy as before.Proof.The proof of Proposition 2.11 is based on the three lemmas stated below.The scheme goes as follows.We apply first the regularization Lemma 2.12 to get an approximating family of probability measures (µ ε t ) t∈I which satisfies a statistical Liouville equation similar to (1.9) with a locally Lipschitz vector field v ε satisfying Assumption 2.9.Then, we can apply Proposition 2.10 to the couple (v ε , (µ ε t ) t∈I ) and get a corresponding probability measure η ε ∈ P(X).Hence, we apply Lemma 2.13 to prove that the family {η ε } ε is tight in P(X).Therefore, there exists η ∈ P(X) such that η ε ⇀ ε→0 η weakly narrowly (at least for a subsequence).Finally, by Lemma 2.14, we check that the constructed measure η satisfies (i) and (ii) in Proposition 2.1.
We provide here the above mentioned technical Lemmas 2.12, 2.13 and 2.14.
Lemma 2.12 (Regularization).Consider B = R d and v, (µ t ) t∈I as in Proposition 2.1 satisfying the same hypotheses.Then, the regularized vector field v ε t and the measures µ ε t given in (B.2) satisfy a statistical Liouville equation as in (1.9) over the interval I.Moreover, define where

25)
Then for all t ∈ I,

.26)
Proof.We apply the regularization Lemma B.3, finding the approximation µ ε t and v ε t in (B.2) satisfying the Liouville equation (1.9).In particular, the vector field v ε t is locally Lipschitz as in Assumption 2.9 and satisfies (1.12) as a consequence of (B.3).Thus, we can apply Proposition 2.10 to get the unique global solution to (2.25) for all x ∈ G d,ε , where we have denoted by G d,ε the set of all initial data where (2.25) admits a unique global solution.More precisely, we define here We have then The identity (2.26) follows from (2.24).Lemma 2.13 (Tightness).The family {η ε } ε , defined in (2.24), is tight in P(X).
Proof.We use here Lemma C.3 with X ≡ X, X 1 ≡ R d and X 2 ≡ C (I; R d ).The latter spaces are separable metric spaces.Recall that C (I; R d ) is endowed with the compact-open topology (see the metric d in (2.2)-(2.3)).Define the homeomorphism map r := r 1 × r 2 : X → X by It is obvious that r is proper.To prove the tightness of {η ε } ε , it suffices to prove: (1) The family of measures {(r (2) The family of measures Radon space, we get by Lemma C.2, that the family The proof of ( 2) is more complicated to handle.For that, we apply Lemma C.1.In fact, by Lemma E.1, we get the existence of a non-decreasing superlinear function θ : R + → [0, +∞] satisfying the inequality (E.1).Then we introduce g : In order to obtain the tightness of the family {(r 2 ) ♯ η ε } ε , it is enough according to Lemma C.1 to prove the following points: (b) For all c ≥ 0, the sublevel sets {γ ∈ C (I; R d ); g(γ) ≤ c} are relatively compact in the space C (I; R d ) endowed with the compact-open topology.For (a), let ε > 0. We have for F d,ε and G d,ε as defined in (2.27) and (2.28) Note that for the above inequality, we used Fubini's theorem and in the last step, we used [48,Lemma 3.10] which generalizes (B.3).The above inequality holds uniformly in ε > 0. For (b), thanks to Lemma E.3 and Remark E.4, the sublevels are relatively compact in the separable metric space (C (I; R d ), d 0 ) with the distance d 0 inducing the compact-open topology given in (2.3).Hence, thanks to Lemma C.3, we conclude that {η ε } ε is tight in P(X).
Lemma 2.14 (Concentration and lifting properties).The subsequential limit η (in the sense of narrow convergence) of the family {η ε } ε satisfies (i) and (ii) in Proposition 2.1.
Proof.The existence of η is guaranteed by Lemma 2.13.We show that η satisfies (i) and (ii) in Proposition 2.1.Remark that we do not have the explicit expression for η in this case.And thus, we cannot proceed as before to prove first (i).Then, we start with proving (ii).For ϕ ∈ C b (R d ; R), we have where η ε is as in (2.24).Hence, we can let ε → 0 in the above equation and deduce The above equality is true for all ϕ ∈ C b (R d ; R).This gives (Ξ t ) ♯ η = µ t , for all t ∈ I.And thus, condition (ii) is satisfied.Finally, we check condition (i).Let w : I × R d → R d be a bounded continuous vector field.We write w(t, x) ≡ w t (x) and introduce the regularized vector field w ε τ := (w τ µ τ ) * ρ ε µ τ * ρ ε (for ρ ε as in Lemma B.3).For T > 0, we have for all t For the above inequality, we used Fubini's theorem as well as the inequality (B.3).On the other hand, where ρ ε (x) = 1/ε d ρ(x/ε).Moreover, by the Lebesgue dominated convergence theorem, the above expression tends to zero as ε → 0. We then deduce that Follow the same argument as in the proof of Lemma E.1 to define a measure ν as in (E.2) on the product space J × R d .Then using Lemma G.1, the space (2.30) At the end, we use the triangle inequality, and apply (2.29) with w ≡ w m together with (2.30) to obtain We take m → +∞ to deduce that: Hence, for all t ∈ I, we get (2.31).This implies that for each t ∈ I, we have Now, due to the continuity of the curves γ as well as v(•, γ(•)) ∈ L 1 loc (I; dt) η−almost surely, using the same arguments as in the proof of Proposition 2.10, we can find by density arguments an η−null set N such that the Duhamel formula (2.32) holds true for all times t ∈ I and for all (x, γ) / ∈ N .
2.4.Analysis on Banach spaces.We want to complete the proof of the global superposition principle of Proposition 2.1 by applying the results in the previous Section 2.3.

Proof of Proposition 2.1:
The strategy of the proof is similar to the finite-dimensional case in Proposition 2.11.Consider (B, • ) to be an infinite-dimensional separable dual Banach space.Let v and (µ t ) t∈I as in Proposition 2.1 satisfying the same hypotheses.Recall the image measures µ n t ∈ P(R n ) and μn t ∈ P(B n ) given in (2.12) and the subspace B n = Span(e * 1 , . . ., e * n ) ⊂ B as well as ||| • ||| R d in (2.14).We apply then the projection argument in Lemma 2.8.Hence, we conclude that there exists a Borel vector field v n : I × R n → R n given in (2.18) such that the probability measures (µ n t ) t∈I satisfy the estimate (2.16) and the statistical Liouville equation (2.15).Therefore, we have all the ingredients to apply Proposition 2.11 for the couple (v n t , (µ n t ) t∈I ) and get the existence of the path measure η n ∈ P(R n × C (I; R n )) so that η n satisfies the concentration and lifting properties in Proposition 2.11 for each n ∈ N. We then define ηn : where πn is introduced in (2.10) and . Thanks to Lemma 2.15 given below, we obtain that the sequence {η n } n is tight in P(X) ⊃ P(X n ) (since X n is a Borel subset of X).So, there exist an η ∈ P(X) and a subsequence that we still denote by (η n ) n such that ηn ⇀ n→∞ η weakly narrowly.Finally, by Lemma 2.16, we conclude that the constructed path measure η satisfies (i) and (ii) in Proposition 2.1.
We are now going to state and prove the aforementioned technical Lemmas 2.15 and 2.16 used in the proof of Proposition 2.1.
Lemma 2.15 (Tightness in Banach spaces).The family of path measures {η n } n given in (2.33), is tight for the weak narrow topology of P(X).
Proof.We use here Lemma C.3 with X ≡ (X, d * ) defined in (2.1), X 1 ≡ B w = (B, • * ) and X 2 ≡ (C (I; B), d 0, * ).The latter spaces are separable metric spaces.Define the homeomorphism map r := r 1 × r 2 : X → X by According to Lemma C.3, to prove the tightness of {η n } n , it suffices to show: (1) The family of measures {(r 1 ) ♯ ηn } n is tight in P(X 1 ).(2) The family of measures {(r 2 ) ♯ ηn } n is tight in P(X 2 ).For (1), we have (r 1 ) ♯ ηn = μn t 0 for all n ∈ N. Indeed, let ϕ ∈ C b (B w ; R), we have where (π n ) ♯ µ n t 0 = μn t 0 is given in (2.12).Remark also μn t 0 ⇀ µ t 0 as n → +∞.And since B w is a separable Radon space, we get by Lemma C.2, that the family {(r 1 ) ♯ ηn } n is tight in P(B).The proof of (2) follows the same strategy as in the finite-dimensional case.Using Lemma E.1, there exists a non-decreasing super-linear continuous convex function θ : R + → [0, +∞] such that We want to apply Lemma C.1.To this end, we introduce g : for all c ≥ 0. For (a), let n ∈ N. We have where we used the definition of v n as in (2.18), Jensen's inequality and the arguments as for (2.19).For the last line, we used which follows from (2.14), the second equality in (2.11), and Lemma 2.7 (i).Then by using the estimate (2.34), the monotonicity of θ, and v(t, x) * ≤ v(t, x) (which follows from (2.5)), we conclude from the above calculation that sup n∈N C (I;B) For (b), we apply Lemma E.3 and conclude that the sublevels A c := {γ ∈ C (I; B); g(γ) ≤ c} are relatively compact in C (I; B w ), d 0, * for all c ≥ 0. However, we still need to check that A c is relatively compact in C (I; B), d 0, * .Let (γ n ) n be a sequence in A c .Then there is a subsequence (γ n k ) k and γ ∈ C (I; B w ) such that d 0, * (γ n k ; γ) −→ 0 k→∞ .Hence, we just need to prove γn (s) B ds.
Assume |t| ≤ T and consider the set Remark that Then by Lemma D.1, F is equi-integrable.And thus by the Dunford-Pettis theorem D.2, F is relatively sequentially compact in the topology σ(L 1 , L ∞ ).More precisely, this means there exists And thus, for all .
Thus, we conclude that γ ∈ AC 1 loc (I; B) ⊂ C (I; B).We give now the proof of the concentration and lifting properties (i) and (ii) in Proposition 2.1.
Lemma 2.16.Let η ∈ P(X) be any cluster point of the tight sequence {η n } n defined in (2.33).Then η satisfies the properties (i) and (ii) of Proposition 2.1.
Proof.We start to give the proof of (ii).Then, we address to the proof of (i) which can be achieved using (ii).

For (ii):
We have, for n ∈ N, for ϕ ∈ C b (B w ), by (2.33), Proposition 2.11 and (2.12) By (2.12), we have μn t ⇀ µ t and ηn ⇀ η as n → +∞.We take limits in the above formula to get The above equality implies that Ξ t♯ η = µ t , for all t ∈ I.For (i): The above identity is true by the support property of μn t,y .To prove (2.36), we start with (2.39) Gathering (1), ( 2) and ( 3), (2.37), Lemma 2.7, then using the disintegration Theorem F.1 and the Lebesgue dominated convergence theorem, the second line in (2.39) with ε 1 (n) −→ 0 as n → +∞.Again by the Lebesgue dominated convergence theorem, the third line of (2.39) gives with ε 2 (n) −→ 0 as n → +∞.Combining the bounds on the second and third lines in (2.39) and the above calculations, we conclude Letting n → +∞ in the above equality, we get (2.36).We have then First, since we have (1.12), then by the Lebesgue dominated convergence theorem Remember that • * ≤ • .So, it remains to seek a sequence of continuous bounded functions To this end, set for a, b ∈ [t 0 , t] with and for every Borel set We have Hence, we have This implies that there exists an η null set N such that Then using a density argument and the continuity of the curves γ in B as in the finite-dimensional case, we obtain the concentration property (i) in Proposition 2.1.

The globalization argument
We give the proof of our main results.In particular, Theorem 1.7 is proved in the subsection below while the applications to ODEs and PDEs are analyzed in Subsection 3.2.
3.1.Proof of main results.In order to prove Theorem 1.7, we rely on the global superposition principle Proposition 2.1 proved in Section 2 and the measurable projection theorem recalled below (see [28,Theorem 2.12]).Theorem 3.1 (Measurable projection theorem).Let (X, T ) be a measurable space and let (Y, B) be a Polish space with Borel σ-algebra B. Then for every set S in the product σ-algebra T ⊗ B the projected set p(S), p : X × Y → X, p(x, y) = x, is a universally measurable set of X relatively to T .
We will also need a measure theoretical argument provided for instance in [7,Lemma 4.
is Borel measurable.
Define the set Proof.According to Proposition 2.1, the path measure η constructed there concentrates on the set F t 0 of global solutions with specified initial conditions given in (3.3).More precisely, the concentration property (i) says that X \ F t 0 is a η-null set.So, to prove the lemma it is enough to show that F t 0 is a Borel subset of X.Such a statement follows from Lemma 3.2.We have On the other hand, for each j ∈ N Define the set G t 0 = {x ∈ B : ∃γ a global mild solution of (1.4) s.t.γ(t 0 ) = x} . (3.9) Lemma 3.4.The set G t 0 is a universally measurable subset of (B, • ).
Proof.Take the projection map p : B × C (I; B) → B, p(x, γ) = x.Recall that according to Lemma 2.5, C (I; B) endowed with the metric d 0 of the compact-open topology is a Polish space.Then using Lemma 3.3 and the measurable projection Theorem 3.1, we obtain that p(F t 0 ) = G t 0 is a universally measurable set of (B, • ).
Proof of Theorem 1.7:The global superposition principle, Proposition 2.1, yields the existence of a probability measure η ∈ P(X) such that µ t = (Ξ t ) ♯ η, for all t ∈ I where Ξ t is the evaluation map Thanks to Lemma 3.4, we know that the set G t 0 is µ t 0 -measurable since it is universally measurable.Hence, Proposition 2.1 and Lemma 3.3 imply The last inequality is a consequence of the inclusion F t 0 ⊂ Ξ −1 t 0 (G t 0 ).Before proceeding with the proof of Theorem 1.11, we note the following measure-theoretic result, which is proved for instance in [63, Theorem 3.9].Lemma 3.5.Let X 1 and X 2 be two complete separable metric spaces with Let us now show how Theorem 1.7 and Lemma 3.5 imply Theorem 1.11.
Proof of Theorem 1.11: Let t 0 ∈ R be any initial time.Consider X 1 = (X, d) given by (2.1)-(2.4),X 2 = B and E 1 = F t 0 defined by (3.3).Recall that (X, d) is a (Polish) complete separable metric space by Lemma 2.5 and F t 0 is Borel measurable by Lemma 3.3.Let ψ be the (measurable) projection map p : F t 0 ⊂ X → B, p(x, γ) = x.Since by assumption, for any x ∈ B the initial value problem (1.4) admits at most one global mild solution, one deduces that p is a one-to-one map.Hence, according to Lemma 3.5, we have that p(F t 0 ) is Borel measurable.Moreover, we have The latter equality is a consequence of (3.10).By Lemma 3.5, we conclude that p −1 : G t 0 → F t 0 is Borel measurable.Therefore, the flow map is well-defined and Borel measurable by composition.Now, we check that φ t t 0 is a measurable flow satisfying Definition 1.9.By construction, the properties φ t 0 t 0 = Id and t → φ t t 0 (x) = γ(t) ∈ C (R; B) hold true.Thanks to the uniqueness assumption, we see that for all t, t 0 ∈ R, G t = φ t t 0 (G t 0 ) .In particular, for any x ∈ G t 0 let γ x (•) denote the (unique) global mild solution of the initial value problem (1.4) satisfying the initial condition γ x (t 0 ) = x, then we check Analysis of ODEs: Recall the initial value problem (1.14) and assume that the assumptions of Corollary 1.15 are satisfied.
Proof of Corollary 1.15: Without loss of generality, we may assume that the symplectic structure J is canonical.Precisely, the skew-symmetric matrix J satisfying J 2 = −I 2d is given by J = 0 I d −I d 0 .For any ϕ ∈ C ∞ c (R 2d ), we have by integration by parts and similarly Combining the two identities, we obtain Hence, using the symplectic structure on R 2d and the Hamiltonian character of the initial value problem, we prove .18)This shows that the measure F (h(•))dL F (h(u))L(du) satisfies the statistical Liouville equation and so we are within the framework of Theorem 1.7.The latter grants us the almost sure existence of global solutions to the ODE (1.14).

Analysis of PDEs:
In this paragraph, we provide the proof of Proposition 1.16 and Corollary 1.17.Recall that the initial value problem (1.24) can be written equivalently in the interaction representation as γ(t) = v(t, γ(t)), with a Borel vector field v : R × H −s → H −s given by First, we notice the following invariance: Proposition 3.7.Consider the Gibbs measure

.20)
Then for any F ∈ C ∞ b,cyl (H −s ), where {•, •} refers to the Poisson bracket (see [9] for more details).Take a sequence of functions (G n ) n∈N in C ∞ c,cyl (H −s ) with given fixed basis 3 .Suppose that G n → 1 pointwisely with ∂ j G n are uniformly bounded with respect to n and ∂ j G n → 0 pointwise.Then replacing G by the sequence G n and letting n → ∞ in (3.22), yields the identity (3.21).
Proof of Proposition 1.16: On the other hand, we check Proof of Corollary 1.17: In this framework: B = H −s is a separable dual Banach space with predual E = H s .The vector field v : R × H −s → H −s given by (3.19) is Borel measurable.The Gaussian measure ν 0 and the Gibbs measures µ t are well-defined Borel probabilities on B. Then, one checks thanks to Proposition 1.16 that (µ t ) t∈R is a narrowly continuous curve in P(B) satisfying the condition (1.12) and the statistical Liouville equation (1.9).Hence, applying Theorem 1.7, we obtain the µ 0 -almost sure existence of global mild solutions for the initial value problem (1.24).Taking now into account the expression (3.20) of the measure µ 0 , one deduces the ν 0 -almost sure existence of global solutions as stated in Corollary 1.17.
Let (X, Σ) be a measurable space and let µ be a finite measure on (X, Σ).We say that a family F ⊂ L 1 (X, µ) is equi-integrable if for any ε > 0 there exists δ > 0 such that: A characterization of equi-integrability is given below.

Appendix E. Compactness argument
We discuss in this paragraph the main compactness argument used throughout the text.Let v : I × B → B be Borel vector field and (µ t ) t∈I a weakly narrowly continuous curve in P(B).Proof.By the general Arzela-Ascoli theorem (see [52,Theorem 6.1]),A c is relatively compact in (C (I; B w ), d 0, * ) provided that we prove the following claims: • For all t ∈ I, the set A c (t) = {γ(t); γ ∈ A c } is relatively compact in B w .
• The set A c is equicontinuous.
• A c (t) relatively compact: In fact, remark that A c (t) is bounded.Indeed, by Jensen's inequality θ( γ(t) * ) ≤ θ( for all γ ∈ A c .Since θ is superlinear, we get A c (t) is bounded in B w .Now, since t ∈ I is fixed and the norm • * induces the weak-* topology in B on bounded sets, it follows that A c (t) is relatively compact in B w .
Hence, choosing 0 < δ ≤ ε 2L , we show that A c is equi-continuous at t 1 .Remark E.4.The above lemma applies, mutatis mutandis, to finite dimensions with any norm on R d .

Appendix F. Disintegration theorem
Let E and F be Radon separable metric spaces.We say that a measure-valued map x ∈ E → µ x ∈ P(E) is Borel if x ∈ F → µ x (B) is a Borel map for any Borel set B of E. We recall below the disintegration theorem (see [6,Theorem 5.3.1]).
Theorem F.1.Let E and F be Radon separable metric spaces and µ ∈ P(E).Let π : E → F be a Borel map and ν = π ♯ µ ∈ P(F ).Then, there exists a ν-a.e.uniquely determined Borel family of probability measures {µ y } y∈F ⊂ P(E) such that µ y (E \ π −1 (y)) = 0 for ν-a.e.y ∈ F and  Let f ∈ L 1 (X, µ; R n ), then f = (f i ) n i=1 .Now for every 1 ≤ i ≤ n, f i ∈ L 1 (X, µ).This implies for all 1 ≤ i ≤ n there exists a sequence of Lipschitz bounded functions (f i k ) k∈N such that Let f k = (f i k ) n i=1 a bounded Lipschitz function i.e. f k ∈ Lip b (X; R n ), we have And thus L 1 (X, µ; R n ) = Lip b (X; R n ).This implies that s)) is strongly measurable and satisfies for all a, b ∈ I, a < b, b a v(s, γ(s)) ds < +∞.Equivalently, we define the space of locally absolutely continuous curves AC 1 loc (I; B) to be the space of all functions u : I → B such that there exists m ∈ L 1 loc (I, dt) satisfying ∀s, r ∈ I, s < r : ||u(s) − u(r)|| ≤ r s m(t)dt .

1. 4 .
Application to PDEs.Consider a complex Hilbert space (H, • H ) and a self-adjoint operator A : D(A) ⊂ H → H such that there exists a constant c > 0, A ≥ c 1, and A having a compact resolvent.So, there exist a sequence of eigenvalues {λ k } k∈N and an O.N.B of eigenvectors Ae k = λ k e k (1.16) such that Ae k = λ k e k for all k ∈ N. Furthermore, assume that there exists s ≥ 0 such that k∈N λ −(s+1) k < +∞.

. 1 )
composed of pairs (x, γ) where x ∈ B and γ is a continuous curve in (B, • ).Denote, for any γ ∈ C (I; B) and m ∈ N, γ m = sup [−m,m]∩I γ(t) , (2.2) with the convention γ m = 0 if [−m, m] ∩ I is the empty set.Since the interval I is unbounded and closed, it is convenient to equip the space C (I; B) with the compact-open topology which is metrizable in our case with the metric:

. 5 )Lemma 2 . 2 .
Here •, • is the duality pairing of E * , E and {e k } k∈N , {e * k } k∈N is the fixed biorthogonal system in Definition 1.1.Actually, the norm • * yields a distance on B = E * which metrizes the weak-* topology σ(E * , E) on bounded sets.For convenience, we denote by B w the space B endowed with the above norm (2.5) and remark that B w = (B, • * ) is separable.Recall that P(B) denotes the space of Borel probability measures on (B, • ).The following lemma shows that P(B) is unchanged as a set, if we equip the space B with the norms • or• * .The σ-algebras of Borel sets of (B, • ) and (B, • * ) coincide.Proof.See[7, Lemma C1].
) where γ * ,m = sup [−m,m]∩I γ(t) * , (2.8) with the convention γ * ,m = 0 if [−m, m] ∩ I is the empty set.It is clear that the topology induced by d * is coarser than the one induced by the distance d given in (2.4).Moreover, d * is the induced metric on X corresponding to the product topology between (B, • * ) and the space C (I; B w ) endowed with the compact-open topology.
is continuous with (X, d) a Polish space and hence its image (X, d * ) is a Suslin space.The spaces (C (I; B), d 0 ) and (C (I; B), d 0, * ) are treated similarly.
1 and B.2 to show that there exists a Borel set G d,T ⊂ R d such that µ t 0 (G d,T ) = 1 and for all x ∈ G d,T , there is a unique solution γ ∈ AC 1 (I ∩ [−T, T ]; R d ) to the initial value problem (1.4) on I ∩ [−T, T ] while we take T > ±t 0 so that I ∩ [−T, T ] has non empty interior.

Proposition 2 . 11 (
The global superposition principle in the Borel case).Consider B = R d and v, (µ t ) t∈I as in Proposition 2.1 satisfying the same hypotheses.Then the conclusion of Proposition 2.1 holds true.
1, we have to prove the following points: (a) sup n∈N C (I;B) g(γ) (r 2 ) ♯ ηn (dγ) < +∞.(b) The sublevel sets A c := {γ ∈ C (I; B); g(γ) ≤ c} are relatively compact in C (I; B), d 0, * 3 and Lemma C.2] and in the PhD thesis of C. Rouffort [64, Lemma 3.A.1].Lemma 3.2 (see [64, Lemma 3.A.1]).Let (M, d M ) be a metric space, let a, b ∈ R, a < b.Then, for any Borel measurable function f : [a, b] × M → R such that for all u ∈ M, f (•, u) ∈ L 1 ([a, b]), the mapping given by )where {e k } k∈N is the elements of the biorthogonal system in B (see Definition 1.1).Hence, it is enough to show E j,k are Borel sets.Let L(I, B) denote the set of curves γ ∈ C (I;B) such that v(•, γ(•)) ∈ L 1 loc (I; B).Taking the functions Λ T : C (I; B) −→ R γ −→ [−T,T ]∩I v(s, γ(s)) ds = lim N →+∞ [−T,T ]∩I min(N, v(s, γ(s)) ) ds, we prove using Lemma 3.2 with M = C (I; B) and the monotone convergence theorem that Λ T are Borel measurable for all T ∈ N. Hence, we conclude that L(I, B) = ∩ T ∈N Λ −1 T (R) is a Borel subset of C (I; B).In particular, Borel sets of (L(I, B), d 0 ) equipped with the induced metric d 0 in (2.2) coincide with Borel sets of C (I; B) which are in L(I, B).Now, using again Lemma 3.2 with M = (L(I, B), d 0 ), we show that the map
[46,a 1.2 is proved in[46, Proposition 1.f.3].We henceforth fix a system {e k } k∈N , {e * k } k∈N as in Definition 1.1, whose existence is guaranteed by Lemma 1.2 above.This allows us to define a convenient class of cylindrical test functions.
Lemma 1.2.Let (B, • ) be a separable dual Banach space.Then a fundamental strongly total biorthogonal system {e k } k∈N , {e * k } k∈N as in Definition 1.1 exists.Definition 1.3 (Cylindrical test functions).A function Theorem 1.13 (Liouville principle).Let B be a separable dual Banach space and v : R × B → B a Borel vector field.Let (µ t ) t∈R be a narrowly continuous curve in P(B) such that (1.10) holds true.Assume that the initial value problem (1.4) admits a measurable flow (φ t t 0 [62,xample of a nonlinear functional h N L in this framework, one can consider the Wickordered nonlinearity (1.31) on bounded domains in R 2 or on 2-dimensional compact Riemannian manifolds without boundary.We refer the reader to the discussions and full details given in[61,  Proposition 4.3, 4.5 and 4.6]and[62, Section 1.2]for the rigorous construction of this type of functionals.
Lemma 2.8 (Projection to finite dimensions).Assume (1.12).For each n ∈ N, the curve (µ n t ) t∈I given by (2.12) is narrowly continuous and satisfies the statistical Liouville equation, n→∞ µ t (2.13) for all t ∈ I.In our case, it is useful to consider R n with the norm |||y||| R n := πn y * .(2.14) be any bounded continuous function.Then, we claim that for all t ∈ I Using the disintegration Theorem F.1 with the projection T n : B → B n , and since μn t = (T n ) ♯ µ t ∈ P(B n ), there exists a μn t -a.e.uniquely determined family of Borel probability measures {μ n t,y } y∈Bn ⊂ P(B) such that μn t,y (B \ T −1 n (y)) = 0 for μn t -a.e.y ∈ B n andB f (x) µ t (dx) = Remark that B n can be identified with R n .Then, applying Lemma G.1, one obtains a sequence of continuous bounded functions (h n ) n∈N from [t 0 , t] × B to B n ⊂ B such that global mild solution of (1.4) s.t.γ(t 0 ) = x} .Lemma 3.3.The set F t 0 is a Borel subset of (X, d) satisfying η(F t 0 ) = 1 where η is the Borel probability measure on X provided by the global superposition principle in Proposition 2.1.