Regularity properties of the solution to a stochastic heat equation driven by a fractional Gaussian noise on ${\mathbb{S}}^2$

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to a stochastic heat equation driven by fractional Gaussian noise on the sphere, providing precise continuity properties.

Contribution

It establishes the existence, uniqueness, and sample path regularity of solutions to the stochastic heat equation on the sphere driven by fractional Gaussian noise, including the exact modulus of continuity.

Findings

Proves existence and uniqueness of solutions in Sobolev spaces.

Derives the exact uniform modulus of continuity for solutions.

Characterizes sample path regularity in time and space.

Abstract

We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.