Existence of solutions to path-dependent kinetic equations and related forward - backward systems

This paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward - forward systems of equations of mean field games. We present local well-posedness, global existence and some regularity results for these equations.


Introduction
A deterministic dynamic in B * can be naturally specified by a vectorvalued ordinary differential equatioṅ µ t = Ψ(t, µ t ) (1.1) with a given initial value µ ∈ B * , where the mapping (t, η) → Ψ(t, η) is from R + × B * to B * . More generally, one often meets the situations whenμ does not belong to B * , but to some its extension. Namely, let D be a dense subset of B, which is itself a Banach space with the norm D ≥ B . A deterministic dynamic in B * can be specified by equation (1.1), where the mapping (t, η) → Ψ(t, η) is from R + × B * to D * . Written in weak form, equation (1.1) means that, for all f ∈ D, (f,μ t ) = (f, Ψ(t, µ t )). (1.2) In many applications, equation (1.2) appears in the form where the mapping (t, η) → A[t, η] is from R + × B * to bounded linear operators A[t, η] : D → B such that, for each pair (t, η) ∈ R + × B * , A[t, η] generates a strongly continuous semigroup in B. Of major interest is the case when B * is the space of measures on a locally compact space. It turns out that, in this case and under mild technical assumptions, an evolution (1.2) preserving positivity has to be of form (1.3) with the operators A[t, η] generating Feller processes, see Theorems 6.8.1 and 11.5.1 from [7]. Equation (1.3) will be referred to as the general kinetic equation. It contains most of the basic equations from non-equilibrium statistical mechanics and evolutionary biology, see monograph [7] for an extensive discussion.
In this paper we are mostly interested in yet more general equation. Namely, let M be a closed convex subset of B * , which is also closed in The main object of this paper is a "path-dependent" version of equation we call (1.5) an adapted kinetic equation, where {µ ≤t } is a shorthand for {µ s } 0≤s≤t . Adapted kinetic equations can be seen as analytic analogs of stochastic differential equations with adapted coefficients, and their wellposedness can be obtained by similar methods. When the generators A only depend on the future of the trajectory of {µ. Remark 1.1. The terminology of adaptiveness and anticipation here should not be associated with any randomness, as in more standard usage of these words.
Equation (1.4) has many applications. Let us briefly explain the crucial role played by this equation in the mean field game (MFG) methodology, which is based on the analysis of coupled systems of forward -backward evolutions and which constitutes a quickly developing area of research in modern theory of optimization, see detail e.g. in [2,3,4,5,10].
Assume that the objective of an agent described by a controlled stochastic process X(s) (passing through x at time t), given an evolutionμ . of the empirical distributions of a large number of other players, is to maximize (over a suitable class of controls {u.}) the payoff V (t, x,μ ≥t , u ≥t ) = E T t J(s, X(s),μ s , u s ) ds + V T (X(T )) , By dynamic programming the optimal payoff of such an agent should satisfy certain HJB equation (backward evolution). On the other hand, when all optimal controls {u t = u t (μ ≥t )} are found, the empirical measure µ . of the resulting process satisfies the controlled kinetic equation The main consistency condition of MFG is in the requirement that the initial µ coincides with the resulting µ. Equalizingμ . = µ . in (1.7) clearly leads to anticipating kinetic equation of type (1.6).
Our main results concern the well-posedness of adaptive kinetic equations (1.5), the local well-posedness and global existence of anticipating and general path dependent kinetic equations and finally some regularity result for path-independent equations arising from their probabilistic interpretations.
The rest of the paper is organised as follows. In Section 2 our main results are formulated and in Section 3 they are proved. Section 4 yields some regularity results for the solutions of kinetic equations leading also to simple verifiable conditions for compactness assumption (2.12) of our main global existence result. Section 5 show some examples.

Main results
Let us recall the notion of propagators needed for the proper formulation of our results.
For a set S, a family of mappings U t,r from S to itself, parametrized by the pairs of numbers r ≤ t (resp. t ≤ r) from a given finite or infinite interval is called a (forward) propagator (resp. a backward propagator) in S, if U t,t is the identity operator in S for all t and the following chain rule, or propagator equation, holds for r ≤ s ≤ t (resp. for t ≤ s ≤ r): U t,s U s,r = U t,r .
A backward propagator U t,r of bounded linear operators on a Banach space B is called strongly continuous if the operators U t,r depend strongly continuously on t and r.
Suppose U t,r is a strongly continuous backward propagator of bounded linear operators on a Banach space with a common invariant domain D. Let A t , t ≥ 0, be a family of bounded linear operators D → B that are strongly continuous in t outside a set S of zero-measure in R. Let us say that the family A t generates U t,r on D if, for any f ∈ D, the equations hold for all s outside S with the derivatives taken in the topology of B. In particular, if the operators A t depend strongly continuously on t, equations (2.1) hold for all s and f ∈ D, where for s = t (resp. s = r) it is assumed to be only a right (resp. left) derivative. In the case of propagators in the space of measures, the second equation in (2.1) is called the backward Kolomogorov equation.
We can now formulate our main results.
for some positive constants c 2 , c 3 , and with their dual propagatorsŨ is well posed, that is for any µ ∈ M, it has a unique solution Φ t (µ) ∈ M (that is (2.6) holds for all f ∈ D) that depends Lipschitz continuously on time t and the initial data in the norm of D * , i.e.

7)
and for µ, η ∈ M (global well-posedness for general "path-independent" case) Under the assumptions in Theorem 2.1, but without the locality constraint (2.5), the Cauchy problem for kinetic equation , and the transformationsŨ t,s of M form a propagator depending Lipschitz continuously on time t and the initial data in the norm of D * , i.e.
with a constant c(T, K) depending on T and K.
is well posed in M and its unique solution depends Lipschitz continuously on initial data in the norm of D * .
Theorem 2.4 (global existence of the solution for general "path dependent" case). Under the assumptions in Theorem 2.1, but without the locality constraint (2.5), assume additionally that for any t from a dense subset of is relatively compact in M. Then a solution to the Cauchy problem In Proposition 4.3 in Section 4, we give the conditions under which the compactness assumption (2.12) holds.

Proofs of the main results
Proof of Theorem 2.1 By duality, for any {ξ 1 Next, we need to estimate the difference of the two propagators. Define an operator-valued function Y (r) : Then, together with assumptions (2.2) and (2.4), . Hence by the contraction principle there exists a unique fixed point for this mapping and hence a unique solution to equation (2.6). Inequality (2.7) follows directly from (2.6). Finally, if Φ t (µ) = µ t and Φ t (η) = η t , then From (2.4) and (3.1), Proof of Theorem 2.2 The global unique solution of (2.9) is constructed by extending local unique solutions of (2.6) via iterations, as is routinely performed in the theory of ordinary differential equations (ODE).
To prove uniqueness and continuous dependence on the initial condition, let us assume that µ t and η t are some solutions with the initial conditions µ and η respectively. Instead of (3.2), we now get By Gronwall's lemma, this implies yielding uniqueness and Lipchitz continuity of solutions with respect to initial data.

Nonlinear Markov evolutions and its regularity
This section is designed to provide a probabilistic interpretation and, as a consequence, certain regularity properties for nonlinear Markov evolution µ t solving kinetic equation (2.9) in the case when B = C ∞ (R d ) and M = P(R d ) is the set of probability measures on R d , so that B * is the space of signed Borel measures on R d and K = sup µ∈P(R d ) µ B * = 1. As a consequence, w shall present a simple criterion for the main compactness assumption of Theorem 2.4.
We will use the following notations.
is the Banach space of continuously differentiable and bounded functions f on R d such that the derivative f ′ belongs to is the Banach space of twice continuously differentiable and bounded functions f on R d such that the first derivative f ′ and the second derivative where ∇ denotes the gradient operator; for (t, z, µ) ∈ [0, T ] × R d × P(R d ), G(t, z, µ) is a symmetric non-negative matrix, b(t, z, µ) is a vector, ν(t, z, µ, ·) is a Lévy measure on R d , i.e.
depending measurably on t, z, µ, and 1 B 1 denotes, as usual, the indicator function of the unit ball in R d . Assume that each operator (4.1) generates a Feller process with one and the same domain D such that C 2 is a martingale.
Proof. By the assumptions of Theorem 2.2, a solution µ t ∈ P(R d ) of equation (2.9) with initial condition µ s = µ specifies a propagatorŨ t,r [µ . ], s ≤ r ≤ t, of linear transformations in B * , solving the Cauchy problems for equation In its turn, for any ν ∈ P(R d ), equation (4.4) specifies marginal distributions of a usual (linear) Markov process {X µ s,t (ν)} in R d with the initial measure ν. Clearly, the process {X µ s,t (µ)} is a solution to our martingale problem. We shall refer to the family of processes constructed in Proposition 4.1 as to nonlinear Markov process generated by the family A[t, µ].
Then the distributions L(X µ s,t ) = Φ t,s (µ), solving the Cauchy problem for equation (2.9) with initial condition µ s have uniformly bounded pth moments, i.e. (4.6) and are 1 2 -Hölder continuous with respect to t in the space (C Lip (R d )) * , i.e.
with a positive constant c.
Proof. For a fixed trajectory {µ t } t≥0 with initial value µ, one can consider {X µ s,t } as a usual Markov process. Using the estimates for the moments of such processes from formula (5.61) of [8] (more precisely, its straightforward extension to time non-homogeneous case), one obtains from (4.5) that E min |X µ s,t −x| 2 , |X µ s,t −x| p |X µ s,s =x) ≤ e C(T,P )(t−s) − 1. (4.8) This implies (4.6). Moreover, (4.8) implies that 10) and consequently where constants C(T, P ) can have different values in various formulas above.
Our main purpose for presenting Proposition 4.2 lies in the following corollary. Then the compactness condition from Theorem 2.4 (stating that set (2.12) is compact in P(R d )) holds for any initial measure µ with a finite moment of pth order.
Proof. It follows from (4.6) and an observation that a set of probability laws on R d with a bounded pth moment, p > 0, is tight and hence relatively compact.

Basic examples of operators A[t, µ]
In this section, we present some basic examples of generators that fit to assumptions of our main Theorems and are relevant to the study of mean field games.
Notice that the most nontrivial condition of Theorem 2.1 is (ii), as it concerns the difficult question from the theory of usual Markov process, on when a given pre-generator of Lévy-Khintchine type does really generate a Markov process. Even more difficult is the situation with time-dependent generators, as the standard semigroup methods (resolvents and Hille-Phillips-Iosida theorem) are not applicable.
Example 5.1. Nonlinear Lévy processes are specified by a families of generators of type (4.1) such that all coefficients do not depend on z, i.e.
The following statement is a consequence of Proposition 7.1 from [7].
Proposition 5.1. Supposed that the coefficients G, b, ν are continuous in t and Lipschitz continuous in µ in the norm of Banach space (C 2 ∞ (R d )) * , i.e.
Notice that the most natural examples of a functional F on measures that are Lipschitz continuous (or even smooth) in space (C 2 ∞ (R d )) * are supplied by smooth functions of monomials g(x 1 , · · · , x n )µ(dx 1 ) · · · µ(dx n ) with sufficient smooth functions g.
Example 5.2. McKean-Vlasov diffusion are specified by the following stochastic differential equation and W t is a standard Brownian motion. The corresponding generator is given by where G(t, x, µ) = tr{σ(t, x, µ)σ T (t, x, µ)}. It is well known (and follows from Ito's calculus) that if the coefficients of a diffusion are Lipshitz continuous, the corresponding SDE is well posed, implying the following.
Let us note finally that not all interesting evolution of type (1.3) satisfy the Lipschitz continuity assumption used in our main results. For instance, a different type of continuity should be applied for coefficients depending on measures via their quantiles, e.g. value at risk (VAR). This type of evolution is analyzed in [9] inspired by preprint [1].