Asymptotic Behavior of a Class of Stochastic Differential Equations

Zaitang Huang

In this article, we address some conditions on invariant measure of Markov semigroups which ensures stochastic bifurcations of a wide class of stochastic differential equations with fractional Brownian motion. This leads to sufficient conditions on Hurst  parameter,drift and diffusion coefficients for a stochastic bifurcations of the families of random dynamical systems.According to the Hölder coefficient of the diffusion function around the singular point and Hurst  parameter, we identify different regimes. Interestingly, for the first time it is found that the Hurst parameter affects both bifurcation conclusions and large deviations which is significantly different from the classical Brownian motion process. This fact is due to the long-range dependence (LRD) property of the fBm.