Polynomial Mixing for a Weakly Damped Stochastic Nonlinear Schr\"{o}dinger Equation

This paper is devoted to proving the polynomial mixing for a weakly damped stochastic nonlinear Schr\"{o}dinger equation with additive noise on a 1D bounded domain. The noise is white in time and smooth in space. We consider both focusing and defocusing nonlinearities, respectively, with exponents of the nonlinearity $\sigma\in[0,2)$ and $\sigma\in[0,\infty)$ and prove the polynomial mixing which implies the uniqueness of the invariant measure by using a coupling method.Comment: 21 pages, no figur

Ergodicity of inhomogeneous Markov processes under general criteria

This paper is concerned with ergodic properties of inhomogeneous Markov processes. Since the transition probabilities depend on initial times, the existing methods to obtain invariant measures for homogeneous Markov processes are not applicable straightforwardly. We impose some appropriate conditions under which invariant measure families for inhomogeneous Markov processes can be studied. Specifically, the existence of invariant measure families is established by either a generalization of the classical Krylov-Bogolyubov method or a Lyapunov criterion. Moreover, the uniqueness and exponential ergodicity are demonstrated under a contraction assumption of the transition probabilities on a large set. Finally, three examples, including Markov chains, diffusion processes and storage processes, are analyzed to illustrate the practicality of our method.Comment: 22 pages, no figure

Existence, uniqueness and ergodicity for McKean-Vlasov SDEs under distribution-dependent Lyapunov conditions

In this paper, we prove the existence and uniqueness of solutions as well as ergodicity for McKean-Vlasov SDEs under Lyapunov conditions, in which the Lyapunov functions are defined on $\mathbb R^d\times \mathcal P_2(\mathbb R^d)$, i.e. the Lyapunov functions depend not only on space variable but also on distribution variable. It is reasonable and natural to consider distribution-dependent Lyapunov functions since the coefficients depends on distribution variable. We apply the martingale representation theorem and a modified Yamada-Watanabe theorem to obtain the existence and uniqueness of solutions. Furthermore, the Krylov-Bogolioubov theorem is used to get ergodicity since it is valid by linearity of the corresponding Fokker-Planck equations on $\mathbb R^d\times \mathcal P_2(\mathbb R^d)$. In particular, if the Lyapunov function depends only on space variable, we obtain exponential ergodicity for semigroup $P_t^*$ under Wasserstein quasi-distance. Finally, we give some examples to illustrate our theoretical results.Comment: 20 page