On the Poisson equation and diffusion approximation 3

We study the Poisson equation Lu+f=0 in R^d, where L is the infinitesimal generator of a diffusion process. In this paper, we allow the second-order part of the generator L to be degenerate, provided a local condition of Doeblin type is satisfied, so that, if we also assume a condition on the drift which implies recurrence, the diffusion process is ergodic. The equation is understood in a weak sense. Our results are then applied to diffusion approximation.Comment: Published at http://dx.doi.org/10.1214/009117905000000062 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Poisson equations with a potential in the whole space for ergodic'' generators

In earlier works Poisson equation in the whole space was studied for so-called ergodic generators L corresponding to homogeneous Markov diffusions (Xt, t⩾0) in Rd. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenization. Here a similar equation with a potential is considered, first because it is natural for PDEs, and second with a hope that it may also be useful for some extensions related to homogenization and averaging

Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations

New weak and strong existence and weak and strong uniqueness results for the solutions of multi-dimensional stochastic McKean–Vlasov equation are established under relaxed regularity conditions. Weak existence requires a non-degeneracy of diffusion and no more than a linear growth of both coefficients in the state variable. Weak and strong uniqueness are established under the restricted assumption of diffusion, yet without any regularity of the drift; this part is based on the analysis of the total variation metric

Sub-exponential mixing rate for a class of Markov chains

We establish sub-exponential bounds for the β-mixing rate and for the rate of convergence to invariant measures for discrete time Markov processes under recurrence type conditions weaker than used for exponential inequalities and stronger than for polynomial ones

On improved convergence conditions and bounds for Markov chains

Improved rates of convergence for ergodic Markov chains and relaxed conditions for them, as well as analogous convergence results for non-homogeneous Markov chains are studied. The setting from the previous works is extended. Examples are provided where the new bounds are better and where they give the same convergence rate as in the classical Markov -- Dobrushin inequality.Comment: 33 pages, 27 reference

Differentiability of solutions of stationary Fokker–Planck–Kolmogorov equations with respect to a parameter

We obtain sufficient conditions for the differentiability of solutions to stationary Fokker-Planck-Kolmogorov equations with respect to a parameter. In particular, this gives conditions for the differentiability of stationary distributions of diffusion processes with respect to a parameter