On the uniform Besov regularity of local times of general processes
Brahim Boufoussi, Yassine Nachit
TL;DR
The paper introduces $\alpha$-local nondeterminism as a versatile replacement for Gaussian local nondeterminism to study uniform Besov regularity of local times $L(x,t)$ in the time variable. Using a Fourier/Berman representation of local times and dyadic Besov norms, it proves that under $\alpha\in(0,1/d)$ and a moment hypothesis on increments, $L(x,t)$ lies in $\mathbf{B}^{1-d\alpha}_{p,\infty}(I)$ almost surely, and that the underlying process $X(\cdot)$ cannot be in $\mathbf{B}^{\alpha}_{p,q}(I)$ for $p>1/\alpha$, extending Adler’s theorem to Besov spaces. The results are applied to Gaussian models (including bifractional Brownian motion) and to nonlinear stochastic heat equations, yielding sharp Besov regularity for local times and Besov irregularity for sample paths. This framework unifies the treatment of local times across Gaussian and non-Gaussian settings and provides sharp characterizations of fine-scale path regularity with potential implications for stochastic analysis and SPDE theory.
Abstract
Our main purpose is to use a new condition, $α$-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.