Stability theory of stochastic evolution equations with multiplicative fractional Brownian motions in Hilbert spaces

Exponential stability is often used to barricade stochastic disturbance in some practical problems. However, the stability theory of mild solution of infinite-dimensional stochastic differential equations (SDEs) with multiplicative fractional Brownian motions (fBms) is still an unsolved problem of great concern until now. In this paper, we try to address this problem by considering a class of semilinear stochastic evolution equations with multiplicative fBms for H is an element of (1/2, , 1) . Firstly, we impose some natural assumptions on the nonlinear term multiplied by fBms, and then use the assumptions to obtain two crucial estimates of p th mean of stochastic integral for general integrand. The proof of the estimates is technical and delicate. With the help of the obtained estimates of p th mean of the stochastic integral, we give the sufficient conditions for the exponentially asymptotic stability and almost sure asymptotic stability of the mild solution. The obtained results are new and innovative in this field. Finally, we give numerical simulations to verify the theoretical results.