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On ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations

We study ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations. For nonlinear Markov chains we obtain sufficient conditions for existence and uniqueness of an invariant measure and uniform ergodicity. We also prove optimality of these conditions. For stochastic McKean-Vlasov equations we establish exponential convergence of their solutions to stationarity in the total variation metric under Veretennikov-Khasminskii-type conditions.


Introduction
In this paper we investigate ergodic properties of nonlinear Markov processes with discrete time (nonlinear Markov chains) and ergodic properties of solutions of nonlinear stochastic differential equations (stochastic McKean-Vlasov equations).
Recall that nonlinear Markov processes are stochastic processes whose transition functions may depend not only on the current state of the process but also on the current distribution of the process. These processes were introduced by H.P. McKean [11] to model plasma dynamics. Later nonlinear Markov processes were studied by a number of authors, let us mention here the books of V.N. Kolokoltsov [9] and of A.-S. Sznitman [16]. These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes ( [2], [5], [19]) and have a wide range of applications, including financial mathematics, population dynamics, neuroscience (see, e.g., [4] and the references therein).
As shown in [14], nonlinear Markov processes may have peculiar ergodic properties. For instance, an irreducible nonlinear Markov chain may have infinitely many invariant measures. Clearly, for standard homogeneous Markov chains this is impossible ([12, Proposition 10.1.1 and Theorem 10.4.9]). This paper extends the recent work of the author [1] and consists of two parts. The first part of the paper (Section 2) is devoted to ergodic properties of nonlinear Markov chains. We establish sufficient conditions for existence and uniqueness of an invariant measure and uniform ergodicity of a nonlinear Markov process. These conditions are optimal in a certain sense. It is interesting to note that in contrast to the Markovian case, positivity of the elements of the one-step transition matrix does not imply even weak convergence to the invariant measure, see Example 2.1 below.
The second part of the paper (Section 3) deals with stochastic McKean-Vlasov equations (SMVEs). Recall that SMVE is a stochastic differential equation (SDE) whose drift and diffusion coefficients may depend on the current distribution of the process. To study convergence of solutions of SMVE to an invariant measure one usually considers associated nonlinear partial differential equation or investigates the mean-field limit. Using and developing these ideas, P. Cattiaux, A. Guillin and F. Malrieu ( [2]) and A. Ganz ([5]) estimated convergence rate of strong solutions of SMVE to an invariant measure in the Wasserstein metric. However these methods can not be applied to obtain such estimates in the total variation metric (recall that this metric is stronger than the Wasserstein metric). To study convergence in total variation we develop a new approach. This approach uses the ideas of M. Hairer and J. Mattingly [7].

Convergence of nonlinear Markov chains
First of all, let us introduce some notation. We assume that all random objects are defined on a common probability space (Ω, F , P). Consider a measurable space (E, E) and let P(E) be the class of all probability measures on this space. The Dirac delta measure concentrated at a point x ∈ E is denoted by δ x .
Let X = (X µ n ) n∈Z + be a nonlinear Markov process with the state space (E, E), initial distribution Law(X µ 0 ) = µ, µ ∈ P(E) and transition probabilities where x ∈ E, B ∈ E, n ∈ Z + and µ n := Law(X µ n ). Further, by Law Q ξ we denote the distribution of the random vector ξ under the measure Q. If Q = P, then the subscript is omitted.
Note that if the function P ν (x, B) does not depend on the measure ν, then the process X is Markov (in this case the transition probability of X is denoted by P (x, B) and the argument ν is dropped).
For probability measures µ, ν ∈ P(E) and a measurable function f : E → [0, ∞), introduce the weighted total variation distance by the following formula: In particular, if the function f is identically equal to 1, then the weighted total variation distance coincides with the (unweighted) total variation distance; the latter is denoted by For a transition kernel Q : E × E → [0, 1], a measurable function ϕ : E → R, and a probability measure ν ∈ P(E), define We say that a transition kernel Q satisfies the local Dobrushin condition on a set A ⊂ E if there exists α > 0 such that for any x, y ∈ A d T V (Q(x, ·), Q(y, ·)) 2(1 − α). (2.1) If the kernel Q satisfies the local Dobrushin condition on the whole space E, then we say that Q satisfies the global Dobrushin condition.
A process X is called uniformly ergodic (see, e.g., [12, p. 393]) if it has a stationary distribution π and there exist C > 0, θ > 0 such that Recall that we denoted µ n = Law(X µ n ). If the process X is Markov, then the global Dobrushin condition is sufficient for the existence and uniqueness of an invariant measure ( [3]). Moreover, this condition implies uniform ergodicity of X and guarantees the following convergence rate: for any µ, ν ∈ P(E) one has The following condition is a natural generalization of the global Dobrushin condition for nonlinear Markov processes: there exists α > 0 such that for any x, y ∈ E sup µ,ν∈P(E) However, it turns out that in contrast to the Markov case, for any 0 < α < 1 this condition may be insufficient even for a weak convergence of µ n to the stationary measure. Let us give a corresponding example.
Thus, the global Dobrushin condition (2.3) does not guarantee uniform ergodicity of a nonlinear Markov process. Furthermore, as explained below, (2.3) does not imply even existence of a stationary measure. Let us show how this condition can be extended in such a way that the new condition is sufficient for uniform ergodicity.
(i) If there exists λ ∈ [0, α] such that for all x ∈ E and µ, ν ∈ P(E) one has then the process X has a unique invariant measure π. Moreover, if λ < α, then for any 3) and (2.4), and measure µ ∈ P(E), such that the process X has more than one stationary measure, the process Y has no stationary measures, Now we pass on to the study of nonlinear processes that do not satisfy the global Dobrushin condition, but satisfy the local Dobrushin condition on a certain "good" set.

Consider a multidimensional stochastic McKean-Vlasov equation in
We say that a function h : Here ·, · is a standard scalar product in R d .
If the drift coefficient b does not depend on the measure µ (and satisfies certain conditions), then (3.1) is a stochastic differential equation and its strong solution is a Markov process. Ergodic properties of this Markov process were studied by many authors. As shown in [17], if the function b satisfies inequality (3.2), then the strong solution of this equation has a unique invariant measure. Moreover, Law(X t ) converges exponentially to this measure in total variation as t → ∞.
Let us extend this result to the case of McKean-Vlasov equations. Assume that the drift b consists of two parts b 1 and εb 2 , where the function b 1 does not depend on the measure, and εb 2 is a small nonlinear perturbation. In other words, where ε > 0. We also assume that the functions b 1 and b 2 are Lipschitz, i.e., there exists a positive L > 0 such that where ρ 2 (µ, ν) is the 2-Wasserstein distance between the measures µ and ν. Recall that this distance is defined by the following formula is the set of all probability measures on (R d ×R d ) with marginals µ and ν.
As shown in [8,Proposition 1.2], under these conditions for any ε 0 equation (3.1) has a unique strong solution (X ε t , µ ε t ) t 0 . If the initial distribution µ = µ ε 0 is fixed (and hence the distribution µ ε t is also fixed), then the process (X ε t ) t 0 is a nonhomogeneous Markov process (however its transition probabilities are different for different initial distributions µ). We will denote P t 2) the function b 2 is uniformly bounded, i.e.
Thus, Theorem 3.1 shows that if a drift that satisfies Veretennikov-Khasminskii condition undergoes a small nonlinear perturbation, then ergodic properties of the strong solution of McKean-Vlasov equation remain the same.
While the proof of the theorem is postponed to Section 4, we outline now the main steps. Note that for a fixed ε > 0 the discretized process (X ε T n ) n∈Z + , where T > 0, is a nonlinear Markov chain. However this process satisfies condition (2.3) only on certain subsets of R d (for instance, on all compact sets) but not on the whole space R d . Therefore Theorem 2.2 cannot be applied here.
Sketch of the proof of Theorem 3.1. To prove the theorem we develop the Hairer-Mattingly technique ( [6], [7]) of constructing auxiliary contraction mappings. Namely, we show that for all sufficiently "small" ε and a certain "nice" function f : R d → R + we have the following contraction inequality in the weighted total variation metric d f : where λ ∈ (0, 1) and λ does not depend on measures µ, ν. This inequality yields the existence and uniqueness of an invariant measure as well as exponential convergence to stationarity.
The right-hand side of the above inequality tends to 0 as n → ∞. Hence P π π = π and the measure π is invariant. The uniqueness of an invariant measure follows from (4.1). Indeed, if a measure ζ is also invariant for the process X and π = ζ, then d T V (π, ζ) = d T V (P π π, P ζ ζ) < d T V (π, ζ), which is impossible. Substituting π for ν in (4.4) and (4.5), we get (2.5) and (2.6).
(ii) Now let us prove that the condition λ α is optimal. Assume that this condition does not hold and 0 < α < λ 1. First, we give an example of the process that has more than one stationary measure.
Consider a nonlinear Markov chain X taking values in the state space (E, E) = ({1, 2}, 2 {1,2} ). Define the transition probability matrix of the chain by the following formula: .
Let us verify that X satisfies conditions (2.3) and (2.4). Indeed, for any i, j ∈ E and µ, ν ∈ P(E) we have P µ (i, j) α/2. Consequently, P µ (i, 1) ∧ P ν (j, 1) + P µ (i, 2) ∧ P ν (j, 2) α, and condition(2.3) holds. Similarly, for any i ∈ E, µ, ν ∈ P(E) and condition (2.4) is also met. On the other hand, it is easy to see that for any a ∈ I := [α/(2λ), 1 − α/(2λ)] the measure µ(a) := aδ 1 +(1−a)δ 2 is stationary for X. Hence, X has more than one stationary measure (actually, continuum of stationary measures). Moreover, d T V (X µ(a 1 ) n , X µ(a 2 ) n ) → 0 as n → ∞ for a 1 , a 2 ∈ I, a 1 = a 2 . Now we give an example of the process, which has no stationary measures. To construct the example one should consider processes taking values in infinite state spaces. Indeed, if a process has a finite state space and satisfies (2.3) and (2.4), then, by Brouwer fixed-point theorem, it has (at least one) stationary measure.
Consider the following auxiliary SDE: where the function b 1 is defined in (3.3). The lemma below is well-known to the specialists working in this area. For example, one can find the statement of the lemma (without proof) in [18, p. 317] or in [10, p. 603] and the idea of the proof (to apply the Harnack inequality) was suggested to the author by A.Yu. Veretennikov and A.M. Kulik. Nevertheless, the author were not able to find the full proof of the lemma in the literature. Therefore we give it here for the completeness of exposition.
Proof. Let us use the Harnack inequality for diffusion processes [20, Theorem 1.1 (2)]. This inequality can be written in the following form: for any t > 0 there exists C = C(t) > 0 such that for any x, y ∈ R d , A ∈ B(R d ). We claim that for any A ∈ B(R d ) we have the following estimate Indeed, suppose without loss of generality that P(Y . Then by (4.11), we have This inequality implies (4.10).
Let us introduce twice continuously differentiable function V : R d → R + such that V (x) = e |x|(r/4∧1) for |x| M. Let us check that for some T > 0 and small ε the transition kernels P T µ (ε), µ ∈ P(R d ) satisfy conditions (2.1) and (4.8).
Moreover for any measure µ ∈ P(R d ) we have the following estimate Proof. First, let us prove inequality (4.13). Denote G(x) := V 2 (x) and set κ := (r/4) ∧ 1. It follows from the definition of V that G(x) = exp(2κ|x|) for |x| M. Fix a measure µ and let Z ε = (Z ε t ) t 0 be a strong solution of the SDE where (X ε t , µ ε t ) t 0 is a strong solution of SMVE (3.1) with the initial distribution µ ε 0 = µ. The definition of the process Z ε yields Apply Ito's formula to the function e θt G(Z ε t ), θ ∈ R. We derive where C = C(θ) > 0, and in the second inequality we made use of condition (3.2) and the boundedness of b 2 . By taking θ = 2κ(r − εD − κ), we obtain This yields (4.12)-(4.14) with T = 1 and ε 0 = α(R,1) 2D ∧ r 2D . To complete the proof, it remains to note that estimate (4.15) directly follows from (4.17): Lemma 4.4. Suppose the conditions of Theorem 3.1 are satisfied. If a measure π is invariant for nonlinear operator P T q (ε) q (that is P T π (ε)π = π), then π(V ) K. Here ε ∈ [0, ε 0 ]; the constants T , ε 0 , and K are the same as in Lemma 4.3.
Proof. Fix S > 0. Since the measure π is invariant, we have P nT π (ε)π = π for any positive integer n. Therefore, using (4.17), we derive measure P ε,ν in a similar way (with the corresponding substitution ν ε s for µ ε s ). By the Girsanov theorem, the process (W ε,µ s ) 0 s t defined by (4.20) and the process (W ε,ν s ) 0 s t defined similarly are d-dimensional Brownian motions with respect to the measures P ε,µ and P ε,ν , correspondingly. As in the proof of Lemma 4.3, we see that P t µ 0 (ε)δ x = Law P ε,µ (Y (x) t ) and P t ν 0 (ε)δ x = Law P ε,ν (Y (x) t ). Denote ρ := dP ε,µ /dP ε,ν . By the Pinsker inequality, we get Note that the right-hand side of this inequality does not depend on x. Combining this estimate with (4.22), and using inequality (a + b) 2 2a 2 + 2b 2 , which holds for all real a, b, we obtain Now the application of Gronwall's lemma to the function ψ(t) := (d T V (µ ε t , ν ε t )) 2 yields (4.21).
where β 0; T is the same as in Lemma 4.3; and C > 0 depends only on L, T and ε 0 .
Proof. We begin by observing that Arguing as in the proof of Lemma 4.5 and using the same notation, we see where ρ := dP ε,µ /dP ε,ν . It follows from (4.13) that T )) 2 2 + 2β 2 P T ν 0 (ε)V (x) 2 2β 2 γ 2 V (x) 2 + 2 + 2β 2 K 2 . (4.26) To estimate the second factor in the right-hand side of (4.25), we use the same technique as in [13]. Denote ϕ(s) s , ν ε s ). Then, we have E P ε,ν ρ 2 = E P ε,ν exp 2ε where the second inequality follows from the Cauchy-Schwarz inequality; and in the last inequality we used Lemma 4.5. Since ε ε 0 and e a − 1 ae a for any real a, we derive Combining this with (4.25) and (4.26), we finally get where C > 0 depends only on L, T and ε 0 . This, together with (4.24), implies (4.23).
It is possible to find such ε 1 because λ < 1. Let us prove that for any 0 ε ε 1 the strong solution of SMVE (3.1) has a unique invariant measure.
Denote by P V (R d ) the space of all probability measures on (R d , B(R d )) which integrate V . Let (X ε,ν t , ν ε t ) t 0 be the strong solution of (3.1) with initial condition distributed as