Invariant measures of stochastic Schrödinger delay lattice systems

Abstract

In this paper, we investigate stochastic Schrödinger lattice systems with time delay, whose drift and diffusion coefficientsare locally Lipschitz continuous. Firstly, some uniform estimates of solutions areestablished which include higher-order moment estimates and uniform tail-ends estimates. Then the tightness of a family ofprobability distributions of solutions in $C([-\\rho,0];l^2)$ is proved by the Arzel${\\rm~\\grave{a}}$-Ascoli theorem and the technique of diadicdivision. Finally, the existence of invariant measures for the Markov semigroup of the system is provedby the Krylov-Bogolyubov method.