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

TL;DR

This paper investigates the decomposition of solutions to a stochastic heat equation driven by a Gaussian noise with fractional spatial properties, revealing a sum of fractional Brownian motion and smooth processes, with applications discussed.

Contribution

It provides a novel decomposition of the stochastic heat equation solution into fractional Brownian motion and smooth components, extending understanding of such stochastic processes.

Findings

Decomposition of $u(t,x)$ into fractional Brownian motion and smooth process.

Identification of Hurst parameters for the components.

Applications demonstrating the utility of the decomposition.

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.