Topological equivalence for discontinuous random dynamical systems and applications

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.