ABSTRACT

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.

1. Introduction

For a Banach space B we denote by B^* the dual Banach space of B. The pairing between and is denoted by. The norm in B^* is defined by

. For a, we denote by

the Banach space of continuous curves equipped with the norm.

A deterministic dynamic in can be naturally specified by a vector-valued ordinary differential equation

(1)

with a given initial value, where the mapping is from to. More generally, one often meets the situations when does not belong to, but to some its extension. Namely, let be a dense subset of, which is itself a Banach space with the norm. A deterministic dynamic in can be specified by Equation (1), where the mapping is from to. Written in weak form, Equation (1) means that, for all,

(2)

In many applications, Equation (2) appears in the form

(3)

where the mapping is from to bounded linear operators such that, for each pair, generates a strongly continuous semigroup in. Of major interest is the case when is the space of measures on a locally compact space. It turns out that, in this case and under mild technical assumptions, an evolution (2) preserving positivity has to be of form (3) with the operators generating Feller processes, see Theorems 6.8.1 and 11.5.1 from [1].

Equation (3) contains most of the basic equations from non-equilibrium statistical mechanics and evolutionary biology, see monograph [1] for an extensive discussion.

In this paper we are mostly interested in yet more general equation. Namely, let be a closed convex subset of, which is also closed in. For a, let denote a closed convex subset of

consisting of curves with values in, and a closed convex subset of

, consisting of curves with initial data.

The main object of this paper is a “path-dependent” version of Equation (3), that is

(4)

where

maps to bounded linear operators. We refer to equation (4) as the general path-dependent kinetic equation. It should hold for all test functions. Compared to Equation (4), Equation (3) is often referred to as a path-independent case.

When the operators only depend on the history of the trajectory of, that is

(5)

we call (5) an adapted kinetic equation, where is a shorthand for. Adapted kinetic equations can be seen as analytic analogs of stochastic differential equations with adapted coefficients, and their well-posedness can be obtained by similar methods. When the generators only depend on the future of the trajectory of, that is

(6)

we call (6) an anticipating kinetic equation, where is a shorthand for.

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 [2-4].

Assume that the objective of an agent described by a controlled stochastic process (passing through x at time t), given an evolution of the empirical distributions of a large number of other players, is to maximize (over a suitable class of controls) the payoff

By dynamic programming the optimal payoff of such an agent, which equals

should satisfy certain HJB equation (backward evolution). On the other hand, when all optimal controls are found, the empirical measure of the resulting process satisfies the controlled kinetic equation of type (3) (forward equation), that is

(7)

The main consistency condition of MFG is in the requirement that the initial coincides with the resulting. Equalizing in (7) clearly leads to anticipating kinetic equation of type (6).

Our main results concern the well-posedness of adaptive kinetic Equations (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. This yield an improved version of the existence results of the unpublished preprint [4].

2. 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 from S to itself, parametrized by the pairs of numbers (resp.) from a given finite or infinite interval is called a (forward) propagator (resp. a backward propagator) in S, if is the identity operator in S for all t and the following chain rule, or propagator equation, holds for (resp. for):

A backward propagator of bounded linear operators on a Banach space B is called strongly continuous if the operators depend strongly continuously on t and r.

Suppose is a strongly continuous backward propagator of bounded linear operators on a Banach space with a common invariant domain. Let, be a family of bounded linear operators that are strongly continuous in outside a set of zero-measure in. Let us say that the family generates on if, for any, the equations

(8)

hold for all s outside S with the derivatives taken in the topology of B. In particular, if the operators depend strongly continuously on, equations (8) hold for all s and, where for (resp.) 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 (8) is called the backward Kolomogorov equation.

We can now formulate our main results.

Theorem 2.1 (local well-posedness for general “pathdependent” case) Let be a bounded convex subset of with, which is closed in the norm topologies of both and. Suppose that 1) the linear operators are uniformly bounded and Lipschitz in, i.e. for any

(9)

(10)

for a positive constant;

2) for any, let the operator curve generate a strongly continuous backward propagator of bounded linear operators in, , on the common invariant domain, such that

(11)

for some positive constants, and with their dual propagators preserving the set.

Then, if

(12)

the Cauchy problem (4) is well posed, that is for any, it has a unique solution (that is (4) holds for all) that depends Lipschitz continuously on time t and the initial data in the norm of, i.e.

(13)

and for

(14)

Theorem 2.2 (global wellposedness for an “adapted” case) Under the assumptions in Theorem 2.1, but without the locality constraint (12), the Cauchy problem (5) is well posed in and its unique solution depends Lipschitz continuously on initial data in the norm of.

Theorem 2.3 (global existence of the solution for general “path dependent”case) Under the assumptions in Theorem 2.1, but without the locality constraint (12), assume additionally that for any from a dense subset of, the set

(15)

is relatively compact in. Then a solution to the Cauchy problem (4) exists in.

In Proposition 4.3 in Section 4, we give the conditions under which the compactness assumption (15) holds.

3. Proofs of the Main Results

Proof of Theorem 2.1

By duality, for any

By (8),

Then, together with assumptions (9) and (11),

(16)

Consequently, if (12) holds, the mapping

is a contraction in

. Hence by the contraction principle there exists a unique fixed point for this mapping and hence a unique solution to Equation (4).

Inequality (13) follows directly from (4). Finally, if and, then

From (11) and (16),

(17)

implying (14).

Proof of Theorem 2.2

For a, let us construct an approximating sequence, by defining for and then recursively

. By non-anticipation, arguing as in the proof of (16) above, we first get the estimate

and then recursively

that implies (by straightforward induction) that, for all,

Hence, the partial sums on the r.h.s. of the obvious equation

converge, and thus the sequence converges in . The limit is clearly a solution to (5).

Finally, let us assume that and are some solutions with the initial conditions and respectively. Instead of (17), we now get

By Gronwall’s lemma, this implies that

does not exceed

yielding uniqueness and Lipchitz continuity of solutions with respect to initial data.

Proof of Theorem 2.3

Since is convex, the space

is also convex. Since the dual operators preserve the set, for anythe curve belongs to as a function of. Hence, the mapping

is from to itself. Moreover, by (16), this mapping is Lipschitz continuous.

Denote

.

Together with (13), the assumption that set (15) is compact in for any from a dense subset of implies that the set is relatively compact in

(by the Arzela-Ascoli Theorem).

Finally, by Schauder fixed point theorem, there exists a fixed point in, which gives the existence of a solution to (4).

4. 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 solving kinetic Equation (3) in the case when is the Banach space of bounded continuous functions f on with, equipped with sup-norm and is the set of probability measures on, so that is the space of signed Borel measures on

and. As a consequencewe shall present a simple criterion for the main compactness assumption of Theorem 2.3.

We shall denote the Banach space of bounded Lipschitz continuous functions f on with the norm

and (resp.) the Banach space of continuously differentiable functions f on such that f and the derivative belongs to, equipped with the norm

resp. twice continuously differentiable with and the norm

.

Let be a family of operators in of the Lévy-Khintchin type, that is

(18)

where denotes the gradient operator; for , is a symmetric non-negative matrix, is a vector, is a Lévy measure on, i.e.

(19)

depending measurably on, and denotes, as usual, the indicator function of the unit ball in. Assume that each operator (18) generates a Feller process with one and the same domain such that .

Proposition 4.1 Suppose the assumptions of Theorem 2.2 are fulfilled with generators of “path-independent” type (3) and a probability measure is given. Then there exists a family of processes defined on a certain filtered probability space such that solves the Cauchy problem for Equation (3) with initial condition and solves the nonlinear martingale problem, specified by the family, that is, for any,

(20)

is a martingale.

By the assumptions of Theorem 2.2, a solution of Equation (3) with initial condition specifies a propagator, of linear transformations in, solving the Cauchy problems for equation

(21)

In its turn, for any, Equation (21) specifies marginal distributions of a usual (linear) Markov process in with the initial measure. Clearly, the process 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.

Using martingales allows us to prove the following useful regularity property for the solution of kinetic equations.

Proposition 4.2 Suppose the assumptions of Theorem 2.2 are fulfilled for a kinetic equation of “path-independent” type (3) with generators of type (18). Let denote a nonlinear Markov process constructed from the family of generators by Proposition 4.1. Assume, for and, the following boundedness condition holds:

(22)

and for the initial measure

.

Then the distributions, solving the Cauchy problem for Equation (3) with initial condition have uniformly bounded pth moments, i.e.

(23)

and are -Hölder continuous with respect to t in the space, i.e.

(24)

with a positive constant.

Proof. For a fixed trajectory with initial value, one can consider as a usual Markov process. Using the estimates for the moments of such processes from formula (5.61) of [5] (more precisely, its straightforward extension to time non-homogeneous case), one obtains from (22) that

(25)

This implies (23) and the estimate

(26)

where constants can have different values in various formulas above.

Since is the distribution law of the process,

(27)

From (26), (27) and Markov property, we get (24) as required.

Remark 4.1 For diffusions with, (24) was proved in [3].

Our main purpose for presenting Proposition 4.2 lies in the following corollary that follows from (23) and an observation that a set of probability laws on with a bounded th moment, , is tight.

Proposition 4.3 Under the assumptions of Theorem 2.1 for generators of Lévy-Khintchin type (18), but without locality condition (12), suppose the boundedness condition (22) holds for some and. Then the compactness condition from Theorem 2.3 (stating that set (15) is compact in) holds for any initial measure with a finite moment of th order.

5. Basic Examples of Operators

We present here basic examples of generators that fit to our main Theorems and are relevant to the study of mean field games. The most nontrivial condition of Theorem 2.1 is 2).

The simplest examples are McKean-Vlasov diffusions defined by SDE

with corresponding generator

where the condition of Theorem 2.1 2) is known to follow, for Lipshitz continuous coefficients, from Ito’s calculus.

Another example is supplied by nonlinear Lévy processes that are specified by generators of type (18) such that all coefficients do not depend on z, i.e.

It follows from Proposition 7.1 of [1] that if the coefficients are continuous in t and Lipschitz continuous in in the norm of Banach space

, i.e.

with a positive constant c, then condition 2) of Theorem 2.1 holds with. Notice also that the usual examples of a functional F on measures given by monomials are Lipschitz continuous (or even smooth) in space whenever g is sufficient smooth.

Another example is supplied by processes (describing lots of models including spatially homogeneous and mollified Boltzmann equation and interacting -stable laws with) with generators of order at most one

with the Lévy measures having finite first moment. As is established in Theorem 4.17 of [1], if are continuous in t, Lipshitz continuous in and Lipschitz continuous in, that is, , ,

then condition 2) of Theorem 2.1 is satisfied with .

As other examples let us mention pure jump processes with bounded rates, where the conditions of Theorem 2.1 are satisfied with, and nonlinear stablelike processes (see [1]).

Let us note finally that not all interesting evolution of type (3) satisfy our Lipschitz continuity assumption. A different type of continuity should be applied for coefficients depending on measures via quantiles, e.g. value at risk (VAR), which is analyzed in [6] inspired by preprint [7].

REFERENCES

NOTES

^*Supported by IPI RAN grants RFBR 11-01-12026 and 12-07-00115, and by the grant 4402 of the Ministry of Education and Science of Russia.