Poisson stability of solutions for stochastic evolution equations driven by fractional Brownian motion

Abstract

In this paper, we study the problem of Poisson stability of solutions for stochastic semi-linear evolution equation driven by fractional Brownian motion \mathrm{d} X(t)= \left( AX(t) + f(t, X(t)) \right) \mathrm{d}t + g\left(t, X(t)\right)\mathrm{d}B^H_{Q}(t), where A is an exponentially stable linear operator acting on a separable Hilbert space \mathbb{H}, coefficients f and g are Poisson stable in time, and B^H_Q (t) is a Q-cylindrical fBm with Hurst index H. First, we establish the existence and uniqueness of the solution for this equation. Then, we prove that under the condition where the functions f and g are sufficiently "small", the equation admits a solution that exhibits the same character of recurrence as f and g. The discussion is further extended to the asymptotic stability of these Poisson stable solutions. Finally, we include an example to validate our results.