Variational Solutions and Random Dynamical Systems to SPDEs Perturbed by Fractional Gaussian Noise

Caibin Zeng,Junfei Cao,Qigui Yang

doi:10.1155/2014/601327

Abstract

This paper deals with the following type of stochastic partial differential equations (SPDEs) perturbed by an infinite dimensional fractional Brownian motion with a suitable volatility coefficient Φ: dX(t) = A(X(t))dt+Φ(t)dB H(t), where A is a nonlinear operator satisfying some monotonicity conditions. Using the variational approach, we prove the existence and uniqueness of variational solutions to such system. Moreover, we prove that this variational solution generates a random dynamical system. The main results are applied to a general type of nonlinear SPDEs and the stochastic generalized p-Laplacian equation.

Highlights

Fractional Brownian motion has been used successfully to model a variety of physical phenomena such as hydrology, turbulence, economic data, telecommunications, biology, and medicine
Based on the variational approach to stochastic partial differential equation (SPDE), we prove the existence and uniqueness of a variational solution to this general type of SPDEs perturbed by an infinite dimensional fractional Brownian motion (fBm) with a suitable volatility coefficient, where the drift part is a nonlinear operator satisfying the standard monotonicity and coercivity conditions
Using the variational approach we have studied a general type of fBm-driven nonlinear SPDEs with a suitable volatility coefficient in Hilbert space

Summary

Introduction

Fractional Brownian motion (fBm) has been used successfully to model a variety of physical phenomena such as hydrology, turbulence, economic data, telecommunications, biology, and medicine. Stochastic equations driven by fBm do not generate a Markov process, which precludes the study of invariant measures for fBm-driven systems using classical tools This motivates the study that fBm-driven SPDEs generate random dynamical systems. The Scientific World Journal stochastic influence in nature or man-made complex systems can be modeled by such systems In this case, variational approach has been used to investigate nonlinear SPDEs which are not necessarily of semilinear type. We refer the readers to [17, 18] and references therein Within this framework, there seems to be only the work [19] analyzing the RDS from nonlinear SPDEs driven by infinite dimensional fBm. In [19], the existence of random attractors for a large class of SPDEs driven by general additive noise (including fBm) was established.

Stochastic Integral for fBm

Main Results

Application to Examples

Conclusion