Topological equivalence for discontinuous random dynamical systems and applications

TL;DR

This paper establishes topological equivalence for certain discontinuous stochastic differential equations driven by non-Gaussian Lévy processes, leading to a stochastic Hartman-Grobman theorem and results on global random attractors.

Contribution

It introduces topological conjugacy for SDEs with jump discontinuities and proves a stochastic Hartman-Grobman theorem, extending classical results to non-Gaussian Lévy-driven systems.

Findings

Topological equivalence is established for SDEs with Lévy jumps.

A stochastic Hartman-Grobman theorem is proved for linearization.

Existence of global random attractors for Marcus SDEs is demonstrated.

Abstract

After defining non-Gaussian L\'evy processes for two-sided time, stochastic differential equations with such L\'evy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an It\^o stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman-Grobman theorem is proved for the linearization of the It\^o stochastic differential equation. Furthermore, for Marcus stochastic differential equations,this topological equivalence is used to prove existence of global random attractors.