On the law of the solution to a stochastic heat equation with fractional noise in time

TL;DR

This paper investigates the probabilistic properties of solutions to a stochastic heat equation driven by fractional Gaussian noise in time, revealing a decomposition involving bifractional Brownian motion.

Contribution

It introduces a novel decomposition of the solution in terms of bifractional Brownian motion, advancing understanding of stochastic heat equations with fractional noise.

Findings

Solution decomposes into bifractional Brownian motion components

Provides insights into the law of the solution with fractional temporal noise

Enhances theoretical understanding of stochastic PDEs with fractional noise

Abstract

We study the law of the solution to the stochastic heat equation with additive Gaussian noise which behaves as the fractional Brownian motion in time and is white in space. We prove a decomposition of the solution in terms of the bifractional Brownian motion.