On Besov regularity and local time of the stochastic heat equation

TL;DR

This paper investigates the precise Besov regularity and local time properties of the stochastic heat equation driven by space-time white noise, providing new insights into its regularity, local times, and level set dimensions.

Contribution

It establishes sharp Besov regularities, existence and regularity of local times, and Hausdorff dimensions of level sets for the stochastic heat equation's solution.

Findings

Sharp Besov regularity results in time and space

Existence and regularity of local times for the solution

Hausdorff dimensions of level sets of the solution

Abstract

Sharp Besov regularities in time and space variables are investigated for $\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, H\"{o}lder continuity, and Besov regularity of local times are established for $u(t,x)$ viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained.