Exponential stability for neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion

摘要

In this paper, we study the exponential stability in the pth moment of mild solutions to neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion: $$ d igl[x(t)+g(t,x_{t}) igr]= igl[Ax(t)+f(t,x_{t}) igr] ,dt+h(t,x_{t}),dW(t)+sigma(t),dB^{H}(t), $$ where (Hin(1/2,1)). Our method for investigating the stability of solutions is based on the Banach fixed point theorem. The obtained results generalize and improve the results due to Boufoussi and Hajji (Stat. Probab. Lett. 82:1549a??1558, 2012), Caraballo et al. (Nonlinear Anal. 74:3671a??3684, 2011), and Luo (J. Math. Anal. Appl. 355:414a??425, 2009).KeywordsNeutral stochastic functional partial differential equation??Mild solution??Exponential stability??Brownian motion??Fractional Brownian motion??MSC34k40??35B35??47D03??60H15??1 IntroductionMany dynamical systems not only depend on present and past states but also involve derivatives with delays. Neutral stochastic functional partial differential equations (NSFPDEs) are often used to describe such kind of systems. In recent years, NSFPDEs have been extensively studied in the literature, we can refer to [6, 9, 12, 13, 14, 19] for those only driven by Brownian motion and also refer to [1, 2, 4, 5, 11] for those only driven by fractional Brownian motion (fBm). For example, Luo [13] studied the exponential stability in mean square of mild solution for NSFPDE only driven by Brownian motion; Boufoussi and Hajji [2] discussed the exponential stability in mean square of mild solution for NSPDE only driven by fBm with finite delay. Furthermore, the stochastic processes in hydrodynamics, telecommunications, and finance demonstrate the availability of random noise that can be modeled by Brownian motion and also the so-called long memory that can be modeled with the help of fBm with Hurst index (1/2 H1). Since the seminal paper [7], mixed stochastic models containing both standard Brownian motion and fBm have gained a lot of attention. Very recently, there has been considerable interest in studying this class of SDEs (see [3, 10, 16, 17, 20, 21]).