Distributed Parameter Systems With a Multiplicative Fractional Gaussian Noise

Distributed parameter systems provide models for partial differential equations and time delay differential equations. Often these systems are subject to perturbations or the systems have errors in the parameters either of which are effectively modeled by stochastic processes that can be described as a multiplicative Gaussian noise. Fractional Brownian motion is a family of Gaussian processes that are indexed by the Hurst parameters H ? (0, 1). These processes have been noted empirically in a wide variety of physical phenomena especially for H ? (1/2, 1). Fractional Brownian motions with values in R n were introduced by Kolmogorov and some important properties of these processes were given by Mandelbrot and van Ness. In this paper, the formal derivative of a (cylindrcal) fractional Brownian motion in a Hilbert space with H ? (1/2, 1) is used to describe a multiplicative Gaussian noise in a distributed parameter systems. This fractional Gaussian noise is used in a stochastic differential equation in a Hilbert space to model the stochastic distributed parameter system. An explicit solution is given for this stochastic differential equation.