Itô-Wentzell-Lions Formula for Measure Dependent Random Fields under Full and Conditional Measure Flows

We present several Itô-Wentzell formulae on Wiener spaces for real-valued functional random field of Itô type that depend on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games. Derivatives with respect to the measure components are understood in the sense of Lions.

More recently, there has been an explosion of new literature around mean-field models where, thinking of stochastic differential equations, maps now contain a dependency on the measure of the solution map.These are the so-called mean-field equations or McKean-Vlasov equations and have appeared as models in many different sub-fields.A critical tool here is the so-called Itô-Lions formula that extends the classical Itô formula and allows one to produce a dynamics for functionals of measure flows [11,12].The construction relies on the notion of Lions derivative for measure functionals.
Regarding measure derivatives, we point out that the earlier (to Lions [36]) notion of intrinsic (measure) derivative was introduced in [2], and used for stochastic analysis on the configuration space over Riemannian manifolds -see additionally [40] for characterisations on the relations of different type derivatives in measures.The more recent Lions derivative is a stronger notion than the intrinsic derivative of [2] in the sense that, over the same space, the intrinsic derivative is a Gateaux derivative while the Lion derivative is a Fréchet derivative.We remark that the Lions derivative concept is not an intuitive notion especially when seen from the lens of geometric analysis (but the intrinsic is); arguably, the intrinsic derivative is not the intuitive concept when seen from the lens of Mathematical finance.Lastly, we emphasize that both these derivative concepts are different from the "linear functional derivative" one usually sees in optimal transport e.g., [3] -see "Otto Calculus" in [45,Ch 15] and for a comparison [11,Section 5.2].
For deterministic functionals of measures, extending the classical Itô formula to the so-called Itô-Lions formula, there are several approaches and results available in the literature.The classical difference of increments approach is used in [7] under a strong regularity assumption of existence of second order Fréchet derivatives.In [15] an approach using projections over empirical measures is used allowing for minimal regularity assumptions.Both approaches are neatly reviewed in [11,Chapter 5].Linked to the existence of a regular solution to the master equation for mean-field games with common noise is the approach by [10,Appendix 6].Their proof is carried out using Itô-Taylor type expansions (similar to [7]) and requiring the involved maps to be twice Fréchet differentiable.Lastly, another approach is to use a semi-group type approach to describe the flow of measures and obtain the necessary infinitesimal expansions see [9, Appendix A].More recently [14] present such Itô-Lions formula for maps belonging to Sobolev spaces, [27,42] provide also such formula for semimartingales -these three works leave out the conditional measure-flow case.An Itô-type formula for measure-valued diffusion processes as conditional distributions of image dependent SDEs has been proved in [46].To the best of our knowledge we have found only one Itô-Wentzell-Lions type formula in the literature, [9, Appendix A].Their approach is set in relation to an existing regular solution to a certain master equation for mean-field control games with common noise.Their proof is carried out via expansions of the densities of the underlying (conditional) measure flow but where the involved diffusion components are constants.
Our contribution.In this manuscript we propose Itô-Wentzell formulae for random fields that embed measure-functionals in a way that is amenable to an analysis in the sense of Lions derivatives.We establish two formulae, and two further corollaries, all decoupled from the applications either in mean-field game theory in finance [11,12,27,42], fluid mechanics [6,28,30], neuroscience modeling [24], population dynamics models [4] or further related stochastic analysis problems [20,35] albeit motivated by them.
Our first result is for the full flow of measures (the measure is deterministic) while the second is for a partial flow of measures (the measure is random).Each result is then further extended to a full joint chain rule allowing for the an additional driving stochastic processes (X t ) t≥0 having a semi-martingale expansion.In particular, we recover the results in [9, Appendix A] while finessing their assumptions, see our Remark 3.6 below.A by-product of our results is a clarification on the necessity of the assumptions on the classical Itô-Wentzell formula [34,Theorem 3.3.1](see our Theorem 2.2 below).Namely, we prove that one can require one order of regularity less from the drift and diffusion coefficient of the random vector field to which the Itô-Wentzell formula is applied to (see our Theorem 2.3).This smaller result is of its own interest.
The usefullness of these result is manyfold.Direct applications within mean-field optimal control could be envisaged in neuroscience modeling [24]; extending the contribution of [22,37], where the classical Itô-Wentzell formula is used to develop a consistent forward utilities of investment and consumption -introducing the relative performance concerns (as in [18,19]).Also building from [4], where a mean-field games with Fisher-Wright common noise is discussed.This model is used in the evolution of population genetics and where it would be natural to update the model to support the distributional component, making use of the results we provide in order to establish the verification procedure.In fluid dynamics these formulae would allow to expand the dynamics of driving signals against the underlying vector field [6,28,30].
Lastly, our work can be extended in several directions to include anticipative processes [38], general semimartingale dynamics [27,42], path dependent functionals in combination with functional Itô calculus [16], or extensions to K-forms for SPDEs in fluid dynamics [5].
Methodological perspective: from Itô-Lions and Itô-Wenzell to Itô-Wenzell-Lions.Our proofs combine two techniques, the projection over empirical measures approach of [11,15], which have the benefit of yielding lower regularity requirements on the underlying coefficients and Taylor-like expansion arguments in the vein of [33] -we argue next that this is the suitable methodology for this result.
The chain rule in the measure component first appears in [7] making use of the telescopic summation technique and building on a strong assumption of a second order Fréchet differentiability of the lifting map.To overcome the requirement of a second Fréchet derivative and reduce it to just first order Fréchet derivative (in fact the so-called Partial-C 2 regularity) for full measure case, the approach of empirical projection was introduced [11,12,15]: this is the approach we follow.In [9] the Itô-Wentzell-Lions formula is shown under the constant diffusion of the random field.The authors follow the semi-group approach and require the existence of the density.
Recently, [27] introduced the use of cylindrical polynomials approximation to build a measure chain rule for the measure flow of semimartingales, i.e., an Itô-Lions formula for semimartingales.Finally, [42] shows an Itô-Lions formula for semimartingales with jumps (the exact same result of [27]) but using the mechanisms of [7].Concretely, they make use of a telescopic summation technique building on the functional linear derivative instead of the Lions one.This approach relies on the assumptions of growth and boundedness of the functional linear derivative and its partial derivative with respect to new spatial variable.For both [27,42] the conditional measure flow case is left unaddressed.
We already argued that neither the proof techniques of [9] or [7] are appropriate as proofs for our results.The former requires constant diffusion coefficients to ensure existence of densities while the latter requires higher Fréchet regularity than needed.Hence the reason we follow [11,12,15].Two recent works [27,42], posterior to ours, use new techniques to prove the Itô-Lions formula for general semimartingales (for deterministic fields) -it is not clear if those techniques can be adapted to prove the Itô-Wentzell-Lions formulae we present in this manuscript under the same minimal regularity constraints we impose.The difficulty stems from our use of random fields while in [27,42] the fields are deterministic.
Concretely, to prove the Itô-Wentzell-Lions formula of this manuscript with the same methodology of [27] one would require a Leibniz rule to interchange the Fréchet derivative symbol with the stochastic integral one and thus would demand further regularity assumptions on top of the existing ones -a general Leibniz rule within this framework is presently an open question 1 .Moreover, such a result is not needed in [27,42] due to their use of deterministic fields!The telescopic summation approach from [42] has another limitation in the context of proving an Itô-Wentzell-Lions formula.One needs to expand the local difference of integrands by the application of the classical Itô-Wentzell formula which requires the existence and well-definiteness of the random fields spanned by the differentiation in measure (in sense of Lions or linear functional; see our Theorem 2.2).This limitation could be avoided by proving the aforementioned Leibniz rule which in turn would demand stronger regularity for the random field and its characteristics as mentioned earlier.
For these reasons, we argue that the 'empirical projection' technique [11,12,15] is the suitable methodology.
Organisation of the paper.In Section 2 we set notation and review a few concepts necessary for the main constructions.In Section 3 we state the full measure flow results.While Section 2 builds towards Section 3, we will need to reframe some notation for Section 4 where we present the conditional flow results.
Acknowledgements.The authors would like to thank François Delarue (Université de Nice Sophia-Antipolis, FR) for the helpful discussions.

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Notation and Spaces
Let N be the set of natural numbers starting at 1, R denotes the real numbers.For collections of vectors in {x l } l ∈ R d , let the upper index l denote the distinct vectors, whereas the lower index the vector components, i.e.
are once continuously differentiable in the first variable, twice so in the second variable (as in C 2 (R d , R m )) and jointly continuous across the several derivatives.
We say that the function is locally bounded, when its restriction to the compact set is bounded.

Spaces
We introduce over R d the space of probability measures P(R d ) and its subset P 2 (R d ) of those with finite second moment.The space P 2 (R d ) is Polish under the Wasserstein distance where Π(µ, ν) is the set of couplings for µ and ν such that π ∈ Π(µ, ν) is a probability measure on Throughout set some 0 < T < +∞ and we work the finite time interval [0, T ].Let our probability space be a completion of (Ω, F, F, P) with F = (F t ) t∈[0,T ] carrying a d-dimensional Brownian motion W = (W 1 , • • • , W d ) generating the probability space's filtration, augmented by all P-null sets, and with an additionally sufficiently rich sub σ-algebra F 0 independent of W .Let our probability space be an atomless Polish.We denote by E[ • ] = E P [ • ] the usual expectation operator wrt to P.
We adopt the following convention, that for d-dimensional random vector ).The convenience of this notation will become apparent in the later Section 4.
Lastly, for convenience we choose to work over 1-, dand d × d-dimensional spaces.This is particularly helpful in lowering complexity of the presentation of the later sections where many sequences of approximating vector-valued stochastic processes are pushed through the Itô and Itô-Wentzell formula.The generalisation to different dimensions is straightforward from our text.

The Itô-Wentzell formula (classic)
We first introduce the stochastic process (X t ) t∈[0,T ] satisfying the dynamics dX t = β t dt + γ t dW t , and initial condition X 0 , where W is a d-dimensional Brownian motion.The involved parameters satisfy the next condition.
x ∈ R d is a random field that admits the Itô dynamics where Then (V t (X t )) t∈[0,T ] is an Itô process and it satisfies P-a.s. the following expansion The first two terms correspond to dynamics of the field V t (•) within installed X t -trajectories.The next three terms correspond to the usual Itô formula.The last term is a cross-variation of the diffusion factor of the process with the same nature noise induced by the stochastic field V t (•) which we write using a matrix-trace notation, this is a short notation to describe the sum over i ∈ 1, . . ., N of the cross variations where γ •,i stands for the i-th row of γ and ∂ x ψ •,i (•) stands for the gradient (in x) of the i-th entry of ψ.
Proof.In this formulation, we state conditions on the differentiability of φ, ψ directly as opposed to the original formulation by [34,Theorem 3.3.1]where conditions over the characteristics of the driving semimartingale were given, [34, Exercise 3.1.5]closes the gap.
A close inspection of Theorem 2.2 and its proof ( [33], [34]) reveals that the theorem holds under reduced regularity requirements.We explore this observation with our next result.

Theorem 2.3 (Itô-Wentzell under reduced regularity). The conclusion of Theorem 2.2 still holds for
) the constraints on φ, ψ are replaced by: Proof.The arguments we use are classical.We mollify V, φ, ψ in their spatial components by convolution with a smoothing kernel and obtain a sequence respectively (in fact even more due to the mollification), such that for any compact Lastly, P-a.s. for t ∈ [0, T ] a.e.we have that φ n t , ψ n t , ∂ x ψ n t converge to φ t , ψ t , ∂ x ψ t uniformly (in n) on compact sets.It is clear that V n retains the properties of V , uniformly over n for the 0-th, 1-st and 2-nd derivative.In particular, P-a.s.
is an Itô process satisfying the expansion given.
The passage to the limit as n → ∞ is also argued in a classical way.First we make use of a localizing sequence (τ m ) m∈N over X defined as τ m := inf{t > 0 : |X t | > m}, m ∈ N which in turn allows us to make use of the uniform convergence over compacts for the maps' sequence (in n) and (2.2)-(2.3)repeatedly, i.e. we can assume that X is bounded.Arguing convergence of the Lebesgue integrals follows via continuity of the maps, integrability of the coefficients (see Assumption 2.1) and dominated convergence theorem taking advantage of uniform convergence over compacts given that X is assumed to take values in a bounded set.The stochastic integral terms requires an additional argument which we provide for the 2nd integral (the 1st is handled similarly), Since ∂ x V, ∂ x V n are jointly continuous in their variables and converge uniformly over compacts, X is assumed to take values in a bounded set and γ satisfies Assumption 2.1, then the RHS converges to zero as n → ∞.

The Lions derivative and notational conventions
To consider the calculus for the mean-field setting one requires to build a suitable differentiation operator on the 2-Wasserstein space.Among the several notions of differentiability of a functional u defined over P 2 (R d ) we follow the approach introduced by Lions in his lectures at Collège de France [36] and further developed in [8].A comprehensive presentation can be found in the joint monograph of Carmona and Delarue [11], [12].
Remark 2.4 (The intrinsic and the Lions derivative).We follow the measure derivative approach by Lions.We point that this notion of derivative is a stronger notion of derivative than the intrinsic measure derivative concept introduced in [2] (see also the Appendix in [39]).See [40] for further details and characterisations on the relations of different types of derivatives in measures.
We consider a canonical lifting of the function u : 1) .Denoting the gradient by Dũ and using a Hilbert structure of the L 2 space, we can identify Dũ as an element its dual, L 2 itself.It was shown in [8] that Dũ is a σ(X)-measurable random variable and given by the function Du(µ, •) : R d → R d , depending on the law of X and satisfying We always denote ∂ µ u as the version of the L-derivative that is continuous in the product topology of all components of u.Moreover, let ∂ 2 µ denote second derivative in measure and ∂ v ∂ µ u denote the derivative with respect to new variable arisen after applying derivative in measure.The notion of ∂ 2 µ is chosen in favour of ∂ 2 µµ , as the latter may be hinting at the linear nature of L-derivative, that is not the case at all.
When we do the lift ξ and ξ are the lifted random variables defined over the twin stochastic spaces ( Ω, F , P) and ( Ω, F , P) respectively, having the same law µ.We form a new probability space (Ω, F, P) × ( Ω, F , P) and consider random variables ξ(ω, ω) = ξ(ω).Since this procedure is valid for the stochastic processes on respective stochastic bases ( Ω, F , F = (F t ) t∈[0,T ] , P) and ( Ω, F , F = (F t ) t∈[0,T ] , P), one can consider (X t , Xt , Xt ) as a triple of independent identically distributed processes.The same applies to a finite amount of copy spaces (Ω l , F l , F l = (F l t ) t∈[0,T ] , P l ), 1 ≤ l ≤ N ∈ N to form a new product space and the respective tuple (X t , Xt , Xt , X 1 t , . . ., X N t ) remains mutually independent.
We will add the bases ( Ω, F , F, P) and ( Ω, F , F, P) and further use them as an environment for model representatives of the mean-field (each living in the distinct respective space), whereas sampling from the mean-field will give us N particles living within respective spaces (Ω l , F l , F l = (F l t ) t∈[0,T ] , P l ), 1 ≤ l ≤ N, to be used within the propagation of chaos procedures below.Hereinafter Ẽ denotes the expectation acting on the model twin space Ω.
Over the present work we omit the re-notation after adding some new probability spaces, but will assume that adding a copy processes automatically intimates the procedure described above.The common noise setting given in Section 4 requires a slightly variation of this approach which we disclose in the proof of Theorem 4.6.

Regularity in the measure argument
In this section we recall several spaces of measure-regularity arising in the literature on Wasserstein calculus.Definition 2.5.We say the functional u : We next restrict the regularity with respect to the space variable arising after taking measure derivative to the Supp(µ), since in our probabilistic setting the process sitting there obviously will not escape this set.This restriction comes from the interplay with the Partial-C 2 -regularity of [11,Chapter 5.6.4].Definition 2.6.We say the function u : This regularity level does not require a second Frechét derivative of the lift to exist.Looking ahead, we do not expect to receive any second-order terms in the expansion of the measure component, hence it is quite essential not to demand such a regularity (see Theorem 3.9 or Theorem 3.4 below).
For the purpose of Theorem 2.12 we require the regularity in all components, and we introduce the following definition.

Definition 2.7. A function
is joint-continuous and locally bounded at every quadruple (t, x, µ, v), with (t, x, µ) ) is continuous and locally bounded at every quadruple (t, x, µ, v),

The Empirical projection map
We recall the concept of empirical projection map given in [15] which will be one of the main workhorses throughout our work.

Definition 2.8 (Empirical projection of a map). Given u
We recall [11, Proposition 5.91 and Proposition 5.35] which relates the spatial derivative of u N with the L-derivative of u.Proposition 2.9.Let u :

Itô-Lions chain rule along a full flow of measures (classic)
Alongside (X t ) t∈[0,T ] given by (2. The requirements of higher integrability of the involved coefficients stems from the proof methodology we implement.Namely, the convergence of the formula for the mollified version (see Step 1 of the Proof of 3.4) follows from these higher moment bounds (for more details see [11, Proof of Lemma 5.95]).At the same time we emphasise that one can reduce the integrability of b and σ at the expense of asking for higher moments for measure derivative terms.
Remark 2.11.One can take "closed-loop" type dependence for the coefficients, i.e. coefficients of the form bt := b t (Y t , µ t ) and σt := σ t (Y t , µ t ), since our setting covers all the special cases.In fact, an existence & uniqueness result for the SDE for Y allows to freeze the components inside the coefficients and with sufficient integrability the "frozen" SDE follows the dynamics (2.4).

Itô-Wentzell-Lions chain rule with a full flow of measures
As it was shown in [15], one can apply an approach based on empirical projections to built the chain rule.This approach very convenient since with it we are able require (loosely) the same regularity as in Theorem 2.12 above.One can notice that the second measure derivative term of the formulae appearing within measure argument expansion vanishes when applying the limit procedure.Nonetheless, in order to argue via Taylor expansions the second derivative in measure has to exist which is a very strong assumptions.We can avoid this requirement using this technique. Let where f (x, µ) : Throughout we will work with the law (µ t ) t∈[0,T ] of the process (Y t ) t∈[0,T ] given in (2.4) under Assumption 2.10.In the second portion of the section, we additionally work with (X t ) t∈[0,T ] solution to (2.1) under under Assumption 2.1.

Itô-Wentzell-Lions formula for measure functionals
We start by discussing the measurability of the involved structures and for which the following remark addresses the issue for the whole manuscript.[11,Remarks 5.101 and 5.103].Within the present work we are interested in conditioning on the field noise, the matter of which is discussed in [12,Section 4.3].We refer the reader to this monograph for comprehensive and detailed approach.

Itô-Wentzell expansion
In this subsection we work with the Itô random field (3.1) and we keep x ∈ R d at some fixed value for the whole subsection and hereinafter we will omit its presence within u, φ and ψ, i.e. we set Similarly to the full-and partial-C 2 maps concept in Definition 2.5 and 2.6, we introduce the concept of a partially-C 2 Itô random field, describing the field's regularity in the measure component and we coin it RF-Partially C 2 .

Remark 3.3.
In contrast with [11,12], where the local boundedness condition is present in the regularity conditions, we restrict ourselves to the continuous version of the Lions derivative from the beginning, hence local boundedness is automatically implied by the continuity.
The main proof mechanics relies on the projection over empirical distributions technique as explored in [11,15].Recall that Ẽ denotes the expectation acting on the model twin space ( Ω, F, P) and let the processes ( Ỹt , bt , σt ) t∈[0,T ] be the twin processes of (Y t , b t , σ t ) t∈[0,T ] respectively living within (see Section 2. Let (µ t ) t∈[0,T ] be the law of the solution to (2.4) satisfying Assumption 2.10.Then (u t (µ t )) t∈[0,T ] is an Itô process P-a.s.satisfying the expansion Remark 3.5.Following from Theorem 2.12 we have that for fixed r ∈ [0, T ], t → u(r, µ t ) P-a.s.satisfies the expansion Proof of Theorem 3.4.For this proof we follow as guideline the proof of Theorem 5.99 in [11].Let throughout t ∈ [0, T ].Recall that E 1,...,N denotes an expectation with respect to the product of sample twin spaces We again underline that we act on an atomless Polish space.
Step 1: Mollification & compactification.If the desired expansion holds true for any u -RF-Partially C 2 , bounded and uniformly continuous (in space and measure arguments), then the formula holds for u satisfying the conditions of the theorem.This fact is straightforward by applying a two-step mollification procedure in the vein of [11,Theorem 5.99] and which we introduce next.
Defining for any t ∈ [0, T ] the (u ⋆ ρ) t (µ) := u t (µ • ρ −1 ) with ρ : R d → R d smooth function with compact support, the P-a.s.boundedness of In order to obtain continuity over the whole space we smooth out the distribution by convolution with a Gaussian density, i.e. considering µ Now we introduce φ ε,G -Gaussian densities N (0, εI d ).Letting ε ց 0 one can see convergence of φ ε,G to Dirac measure at 0 for the W 2 distance and thus convergence of µ, v) P-a.s..One should notice that all the conditions in the theorem hold true while doing mollification.Thus we can assume that u and its first and partial second order derivatives are P-a.s.uniformly bounded and uniformly continuous, and Y is a bounded process.Now we are to show the well-posedness of the mollification scheme, i.e. that chain rule applied to u n := u ⋆ ρ n converges to the one for u.It is straightforward to verify that u n satisfies P-a.s.(3.2) uniformly in n ≥ 1.We apply the dominated convergence theorem twice to conclude the P-a.s.convergence for all the terms but the stochastic integral.To handle the latter one additionally requires an argument across the quadratic variation as written in Theorem 2.3 and localisation.
Step 2. Wellposedness and approximation.For a smooth compactly supported density ρ on R d we define, for n ∈ N, the mollified version u N,n of u N (introduced in Definition 2.8) for any t ∈ [0, T ], where ρ is a smooth and compactly supported density.We define φ N,n , ψ N,n , in the same way as u N,n .One can notice that u N,n t , φ N,n t , ψ N,n t are maps in C 2 (R d ) N and thus all derivatives up to second order exist and are regular.Furthermore, to u N,n one can apply the standard Itô-Wentzell formulae, since it satisfies all the conditions of Theorem 2.2 (verified below).Now we describe the approximation procedure.From the properties of the Wasserstein metric for finitely supported measures with uniformly bounded second moments, we have where C depends on the support of ρ.
The technique is as follows: we mollify the empirical projection u N t and obtain u N,n t , this way we can take second-order derivatives and afterwards apply the "propagation of chaos" argument to approximate u t by u N t , namely for any t ∈ [0, T ] one have P-a.s.
where (ε k ) k≥1 is a sequence of random variables P-a.s.converging to 0, as k → ∞ uniformly in time, this is seen via a propagation of chaos argument, continuity of u, dominated convergence theorem and the fact that convergence in Wasserstein metric only depends on the moments of the distribution.
By the P-a.s.boundedness of u one can get for any p ≥ 1 sup where (ε k ) k∈N is a sequence converging P-a.s. to 0. Now we use the Proposition 2.9 to get for any t ∈ [0, T ], P-a.s.
Applying the same argument as above we get P-a.s., p ≥ 1 sup Now we differentiate once again with respect to y i The procedure to deal with T 12,N n,i also applies to T 2,N n,i and yields P-a.s.
with an additional multiplicative factor n appearing after differentiating the regularisation kernel.We say that φ t (•), ψ t (•) = 0 for all other t, where φ, ψ are not defined.Now the same technique is valid to φ N,n , ψ N,n to get P-a.s for almost all t Hence, P-a.s., p ≥ 1 ) Without loss of generality we pick the (ε k ) k∈N the same as for u.One can notice that ψ N,n , φ N,n satisfy condition (2.2) of Theorem 2.3, due to mollification and the identification from Proposition 2.9.
Step 3: Applying the classical Itô-Wentzell to the approximation.Under our assumptions and the mollification argument in combination with Proposition 2.9, we have sufficient regularity that we can apply the standard Itô-Wentzell formula (see Theorem 2.2 and Theorem 2.3) to u N,n and obtain Note two important simplifications.Firstly, one would expect the second-derivative term to contain a Hessian, but for independent processes ) ⊺ dt and hence only diagonal terms appear.Secondly, no cross-variation term d ∂ µ u N,n , Y l t appears, this is due to the independence of the field's noise W t and noise of the particles {W l t } l=1,...,N within empirical approximation (this will not be the case in the next section).Now we can proceed with the expected result.Define ∆ N,n as the difference between the RHS of (3.10) and the RHS of the below equation, we then have for any t ∈ [0, T ] P-a.s.(the tautology) We let N → ∞ to get by Fatou's lemma, the law of large numbers and the joint-continuity of all derivatives with localisation argument for stochastic integral term, P-a.s. that sup 0≤t≤T |∆ n t | ≤ 3ε n , where P-a.s.
where ∆ n t := lim n→∞ ∆ N,n t , and we applied Fubini's theorem to interchange the Lebesgue integral with the expectation.Note that to handle the stochastic integral we apply the localisation technique and use dominated convergence theorem once more.Letting n → ∞ in the equation above, we conclude that ∆ n t → ∆ ≡ 0, P-a.s., which finishes this part of the proof.The measurability of the involved coefficients follows the guidelines set in Remark 3.1.

The joint chain rule
Now we are ready to provide a joint chain rule formula expanding the nature of the random field to support a space variable dependence, i.e. the case t → u t (X t , µ t ) for µ the law of (2.4) and X solution to (2.1).Let us start by inheriting the structure and properties of the setup of Theorem 3.4.

Definition 3.8. We say the random field
) is P-a.s.continuous in the Wasserstein metric and L-differentiable P-a.s. at every Assume that for any compact and sup Let (µ t ) t∈[0,T ] be the law of the solution to (2.4) satisfying Assumption 2.10.Let (X t ) t∈[0,T ] be the solution process to (2.1) under Assumption 2.1.
Then the process (u t (X t , µ t )) t∈[0,T ] is an Itô process P-a.s.satisfying the dynamics Observe that the terms of the first and the last line on the RHS of the formula are the ones from our Theorem 3.4, whereas the middle two arise from the standard Itô-Wentzell formulae.
Proof.In view of the proof of Theorem 3.4 we assume a compactification/mollification argument in the measure component has been applied.In this way we avoid a repetition of arguments.
We start by fixing a time T and let with the filtration F obtained by augmenting the product filtration F 0 ⊗ F 1 in a right-continuous way and by completing it.In the vein of Section 2.3.1 let E 0 and E 1 taking the expectation on the first and second space respectively.
Let u : Ω × [0, T ] × R d × P 2 (R d ) → R be a random field, satisfying the dynamics where f (x, µ) : Take measurable (b, σ 0 , σ 1 ) : We name (W 0 t ) t∈[0,T ] as a common noise affecting the whole setting, whilst (W 1 ) t∈[0,T ] is the idiosyncratic chaos for the random field and all processes within.For the purposes of the present section we fix the common noise and derive the dynamics of the random field by conditioning on W 0 .Once again, all measurability issues are discussed at Remark 3.1.

Itô-Lions chain rule along a conditional flow of measures (classic)
We recall the Itô-Lions formula for the flow of marginals [12,Theorem 4.17].First, we provide the regularity assumption as given in [12,Subsection 4.3.4].
We highlight the slight abuse of notation in the way point i) in the above Definition 4.5 is formulated.This avoids re-stating a full assumption that is nonetheless clear to understand.Theorem 4.6.Let u be RF-Generally-C 2 Itô random field (4.1) (where x ∈ R d is fixed and omitted throughout, also for φ and ψ).Assume for any compact K ⊂ P 2 (R d ) we have For almost all ω 0 ∈ Ω 0 take (µ t ) t∈[0,T ] := Law(Y t (ω 0 , •)) t∈[0,T ] , with Y solution to (4.2) under Assumption 4.1.Then (u t (µ t )) t∈[0,T ] is an Itô process P-a.s.satisfying the expansion where the formula above Ẽ and Ê denote the expectation acting on the model twin spaces ( Ω, F, P) and ( Ω, F, P) respectively, and let the processes ( Ỹt , bt , σt ) t∈[0,T ] and ( Ŷt , bt , σt ) t∈[0,T ] be the independent twin processes of (Y t , b t , σ t ) t∈[0,T ] respectively living within.
with O(1/N ) standing for the Bachmann-Landau big-O notation (sequence bounded by C N , for some C ≥ 0) which appears from the second 1/N 2 summation term (notice the sum is over only one index).
Lifting to L 2 -space and using continuity of the underlying process, as in [12, Theorem 4.14], we conclude that P 0 ⊗ P 1 -a.s.
Now due to the continuity in the measure-component and dominated convergence theorem we can pass to the limit in (4.6) (as N → ∞) to conclude the formula.The convergence of stochastic integral is secured by localisation and arguing across quadratic variation.We swap the integral and expectation by stochastic Fubini theorem.Finally we rewrite the expectations in the RHS upon dependance on two model particles (living on (Ω 0 × Ω1 , F 0 ⊗ F1 , F 0 ⊗ F1 , P 0 ⊗ P1 ) and (Ω 0 × Ω1 , F 0 ⊗ F1 , F 0 ⊗ F1 , P 0 ⊗ P1 ) respectively).Measurability is again secured by Remark 3.1.
We again underline that we do not have additional ∂ xy l u and ∂ y l ψ terms due to the fact that W 1 , W 1,l t = 0, l = 1, . . ., N , at the same time diagonally summing one of ∂ 2 µ u, due to mutual independence of W i , W j , i, j ∈ 1, . . ., N, i = j.Now we transform the equation according to Proposition 2.9, and applying E 1,1,...,N • := E • |F 0 ⊗ F 1 , law of large numbers, Fubini theorem and boundedness of ∂ 2 µ u we get P-a.sE 1,1,...,N u(X T , μN T ) − E 1,1,...,N u N (X 0 , μN 0 ) = x l j denotes the j-th component of l-th vector.For x, y ∈ R d denote the scalar product by x • y = d j=1 x j y j ; and |x| = ( d j=1 x 2 j ) 1/2 the usual Euclidean distance; and x ⊗ y denotes the tensor product of vectors x, y ∈ R d .Let ½ A be the indicator function of set A ⊂ R d .For a matrix A ∈ R d×n we denote by A ⊺ its transpose and its Frobenius norm by |A| = Trace{AA ⊺ } 1/2 .Let I d : R d → R d be the identity map.We denote by C(A, B) for A, B ⊆ R d , d ∈ N, the space of continuous functions f : A → B. In terms of derivative operators and differentiable functions, ∂ t denotes the partial differential in the time parameter t ∈ [0, T ]; ∂ x denotes the gradient operators in the spatial variables x in R d while ∂ 2 xx , ∂ 2 yy the Hessian operator in x or y

Definition 4 . 5 .
We say the random field u
a continuous adapted process taking values over R d and (φ t (µ)) t∈[0,T ] , (ψ t (µ)) t∈[0,T ] are F-progressively measurable processes with values in R and R d respectively;ii) For almost all t ∈ [0, T ], the maps µ → φ t (µ), µ → ψ t (µ) are P-a.s.continuous in the topology induced by the Wasserstein metric for any µ ∈ P 2 (R d );iii) For any t ∈ [0, T ] the map µ → u t (µ) is P-a.s.continuous in topology, induced by Wasserstein metric and L-differentiable P-a.s. at every µ ∈ P 2 [11,ighlight the requirement of the square integrability on ∂ µ u and∂ v ∂ µ u in (3.2) which is not present in [9, Appendix A].The requirement is necessary for the intermediary step of W 2 -convergence of the empirical measure appearing in those terms.Here we write Trace within last term assuming the symmetry of respective matrix holding P-a.s. for any t ∈ [0, T ].One can see that within the approximating procedure, i.e. the distance between the Hessian of the mollified empirical projection and the ∂ v ∂ µ u-term is controlled through the decreasing sequence ε N ց 0, thus the symmetry follows by approximation.See[11, Remark 5.98] for details.