A note on decompositions of the stochastic convolution driven by a white-fractional Gaussian noise

Abstract

Let $u=\{u(t,x);(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R}\}$ be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameter $H\in (0,1)$. For any given $x\in \mathbb{R}$ (resp. $t\in {\mathbb{R}}_{+}$), we show a decomposition of the stochastic process $t\mapsto u(t,x)$ (resp. $x\mapsto u(t,x)$) as the sum of a fractional Brownian motion with Hurst parameter $H/2$ (resp. $H$) and a stochastic process with $C^{\infty }$-continuous trajectories. Some applications of those decompositions are discussed.