On the uniform Besov regularity of local times of general processes

TL;DR

This paper introduces a new $ ext{alpha}$-local nondeterminism condition to analyze the Besov regularity of local times of various processes, extending classical theorems and applying to Gaussian processes and stochastic heat equations.

Contribution

It develops a novel $ ext{alpha}$-local nondeterminism condition for studying Besov regularity of local times, extending Adler's theorem to Besov spaces and applying results to Gaussian processes and stochastic heat equations.

Findings

Established sharp Besov regularity for local times of classical Gaussian processes.

Extended Adler's theorem to Besov spaces.

Demonstrated applications to solutions of stochastic heat equations.

Abstract

Our main purpose is to use a new condition, $\alpha$-local nondeterminism, which is an alternative to the classical local nondeterminism usually utilized in the Gaussian framework, in order to investigate Besov regularity, in the time variable $t$ uniformly in the space variable $x$, for local times $L(x, t)$ of a class of continuous processes. We also extend the classical Adler's theorem [1, Theorem 8.7.1] to the Besov spaces case. These results are then exploited to study the Besov irregularity of the sample paths of the underlying processes. Based on similar known results in the case of the bifractional Brownian motion, we believe that our results are sharp. As applications, we get sharp Besov regularity results for some classical Gaussian processes and the solutions of systems of non-linear stochastic heat equations. The Besov regularity of their corresponding local times is also obtained.