On the uniform Besov regularity of local times of general processes

Abstract

The $\alpha$-local nondeterminism notion ($\alpha$-LND, for short) has been introduced by the authors [Stoch. Partial Differ. Equ. Anal. Comput. 11 (2023), 388–425] to investigate the existence, joint continuity, and uniform Hölder continuity, i.e., Hölder continuity in the time variable $t$ uniformly in the space variable $x$, for local times $L(x,t)$ of general processes. In the present paper, we aim to use the $\alpha$-LND property to improve this uniform Hölder continuity of local times to a uniform Besov regularity, i.e., Besov regularity in the time variable $t$ uniformly in the space variable $x$ and in $p$ (the index of the Besov space ${\mathbf{B}}_{p,\infty }^{\nu }(I;\mathbb{R})$). The Besov regularity of local times, in the time variable $t$ for fixed space variable $x$, has never been treated in the literature even for Gaussian or stable processes. We also extend the classical Adler’s theorem, i.e., Theorem 8.7.1 [in: The Geometry of Random Fields (1981)] to the Besov spaces case. These results are then exploited to study the uniform (in $p$) Besov irregularity of the sample paths of the underlying processes. As applications, we get sharp uniform Besov irregularity results for a class of Gaussian processes and the solutions of systems of non-linear stochastic heat equations. The uniform Besov regularity of their corresponding local times is also obtained.