Existence of Solutions to Path-Dependent Kinetic Equations and Related Forward-Backward Systems ()
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 any
the 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].
NOTES