Stochastic McKean-Vlasov equations

Abstract

We prove the existence and uniqueness of solution to the nonlinear local martingale problems for a large class of infinite systems of interacting diffusions. These systems, which we call the stochastic McKean-Vlasov limits for the approximating finite systems, are described as stochastic evolutions in a space of probability measures on T~ d and are obtained as weak limits of the sequence of empirical measures for the finite systems, which are highly correlated and driven by dependent Brownian motions. Existence is shown to hold under a weak growth condition, while uniqueness is proved using only a weak monotonicity condition on the coefficients. The proof of the latter involves a coupling argument carried out in the context of associated stochastic evolution equations in Hilbert spaces. As a side result, these evolution equations are shown to be positivity preserving. In the case where a dual process exists, uniqueness is proved under continuity of the coefficients alone. Finally, we prove that strong continuity of paths holds with respect to various Sobolev norms, provided the appropriate stronger growth condition is verified. Strong solutions are obtained when a coercivity condition is added on to the growth condition guaranteeing existence.

Superprocesses with Dependent Spatial Motion and General Branching Densities

“…The SDSM contains as special cases several models arising in different circumstances such as the one-dimensional super Brownian motion, the molecular diffusion with turbulent transport and some interacting diffusion systems of McKean-Vlasov type; see e.g. Chow (1976), Dawson (1994), Dawson and Vaillancourt (1995) and Kotelenez (1992Kotelenez ( , 1995. It is thus of interest to construct the SDSM under reasonably more general conditions and formulate it as a diffusion processes in M(IR).…”

Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics

“…When there is no control, the dynamics (1.1) is sometimes called stochastic McKean-Vlasov equation (see [21]), where the term "stochastic" refers to the presence of the random noise caused by the Brownian motion W 0 w.r.t. a McKean-Vlasov equation when σ 0 = 0, and for which coefficients depend on the (deterministic) marginal distribution P Xt .…”

Superprocesses arising from interactive stochastic flows

“…refs. [6,[9][10][11][12][13]. Ma and Xiang [8] constructed the superprocesses of stochastic flows (SSF) which are SAISF with parameters (a p,q , c p,q , b p,q , 0, 0) under the assumption that c p,q (x) = a p,q (x, x) for any x ∈ R d and 1 p, q d. It is very interesting that their superprocesses determine the stochastic coalescence and probably have very strange properties if the stochastic flow is replaced by different stochastic flows.…”