On the uniform Besov regularity of local times of general processes

Brahim Boufoussi, Yassine Nachit

Published 2021-10-19, updated 2024-06-04Version 2

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.