On the uniform Besov regularity of local times of general processes

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}^{\nu}_{p,\infty}(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.