arXiv subjects

Mingjie Liang

Publications and source records attributed to Mingjie Liang.

4 recordsLinked to original sources

arXiv2020

Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

We study stochastic differential equations (SDEs) of McKean-Vlasov type with distribution dependent drifts and driven by pure jump Lévy processes. We prove a uniform in time propagation of chaos result, providing quantitative bounds on convergence rate of interacting particle systems with Lévy noise to the corresponding McKean-Vlasov SDE. By applying techniques that combine couplings, appropriately constructed $L^1$-Wasserstein distances and Lyapunov functions, we show exponential convergence of solutions of such SDEs to their stationary distributions. Our methods allow us to obtain results that are novel even for a broad class of Lévy-driven SDEs with distribution independent coefficients.

math.PR↗

arXiv2018

A Unified Approach to Coupling SDEs driven by Lévy Noise and Some Applications

We present a general method to construct couplings of stochastic differential equations driven by Lévy noise in terms of coupling operators. This approach covers both coupling by reflection and refined basic coupling which are often discussed in the literature. As an application, we establish regularity results for the transition semigroups of the solutions to stochastic differential equations driven by additive Lévy noise.

math.PR↗

arXiv2018

Spatial regularity of semigroups generated by Lévy type operators

We apply the probabilistic coupling approach to establish the spatial regularity of semigroups associated with Lévy type operators, by assuming that the martingale problem of Lévy type operators is well posed. In particular, we can prove the Lipschitz continuity of the semigroups under Hölder continuity of coefficients, even when the Lévy kernel corresponding to Lévy type operators is singular.

math.PR↗

arXiv2018

Gradient Estimates and Ergodicity for SDEs Driven by Multiplicative Lévy Noises via Coupling

We consider SDEs driven by multiplicative pure jump Lévy noises, where Lévy processes are not necessarily comparable to $α$-stable-like processes. By assuming that the SDE has a unique solution, we obtain gradient estimates of the associated semigroup when the drift term is locally Hölder continuous, and we establish the ergodicity of the process both in the $L^1$-Wasserstein distance and the total variation, when the coefficients are dissipative for large distances. The proof is based on a new explicit Markov coupling for SDEs driven by multiplicative pure jump Lévy noises, which is derived for the first time in this paper.

math.PR↗