Irreducibility of stochastic complex Ginzburg-Landau equations driven by pure jump noise and its applications

TL;DR

This paper proves the irreducibility of stochastic complex Ginzburg-Landau equations driven by pure jump noise, establishing conditions for ergodicity that are dimension-free and applicable even in weakly dissipative cases.

Contribution

It introduces a novel irreducibility criterion for stochastic equations driven by pure jump noise and applies it to demonstrate ergodicity of complex Ginzburg-Landau equations.

Findings

Irreducibility established for equations driven by pure jump noise

Results are dimension free and hold under mild conditions

Ergodicity proven even in weakly dissipative cases

Abstract

Considering irreducibility is fundamental for studying the ergodicity of stochastic dynamical systems. In this paper, we establish the irreducibility of stochastic complex Ginzburg-Laudau equations driven by pure jump noise. Our results are dimension free and the conditions placed on the driving noises are very mild. A crucial role is played by criteria developed by the authors of this paper and T. Zhang for the irreducibility of stochastic equations driven by pure jump noise. As an application, we obtain the ergodicity of stochastic complex Ginzburg-Laudau equations. We remark that our ergodicity result covers the weakly dissipative case with pure jump degenerate noise.