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

Xinwen Zhang,Dehao Ruan

doi:10.1186/s13660-018-1793-9

Abstract

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: \t\t\td[x(t)+g(t,xt)]=[Ax(t)+f(t,xt)]dt+h(t,xt)dW(t)+σ(t)dBH(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ d \\bigl[x(t)+g(t,x_{t}) \\bigr]= \\bigl[Ax(t)+f(t,x_{t}) \\bigr] \\,dt+h(t,x_{t})\\,dW(t)+\\sigma(t)\\,dB^{H}(t), $$\\end{document} 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. 82:1549–1558, 2012), Caraballo et al. (Nonlinear Anal. 74:3671–3684, 2011), and Luo (J. Math. Anal. Appl. 355:414–425, 2009).

Highlights

Many dynamical systems depend on present and past states and involve derivatives with delays
To the best of our knowledge, there is no paper which investigates 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
The purpose of this paper is to investigate the exponential stability in the pth moment of mild solution of mixed Neutral stochastic functional partial differential equations (NSFPDEs) (1.1) by means of the Banach fixed point theory

Summary

Introduction

Many dynamical systems depend on present and past states and involve derivatives with delays. 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 refer to [1, 2, 4, 5, 11] for those only driven by fractional Brownian motion (fBm). To the best of our knowledge, there is no paper which investigates 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. Let W = {W (t), t ∈ [0, T]} be a standard Brownian motion and B = {BH(t), t ∈ [0, T]} be a fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) on the complete probability space ( , F , P). We can show the following two definitions of norms

If ξ

Chp eλs

Thus we can get Cgpeαt xt p U