Continuous Gaussian multifractional processes with random pointwise H\"older regularity

TL;DR

This paper investigates the variability of pointwise H"older regularity in Gaussian processes and constructs counterexamples showing it can differ across trajectories, unlike global and local regularities.

Contribution

It introduces a family of multifractional Gaussian processes demonstrating that pointwise H"older regularity can vary among trajectories, contrary to global and local regularities.

Findings

Pointwise H"older regularity can differ across trajectories.

Global and local H"older exponents are almost surely constant.

Counterexamples show variability in pointwise regularity.

Abstract

Let X be an arbitrary centered Gaussian process whose trajectories are, with probability one, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability one, the trajectories of X have the same global H\"older regularity over any compact interval, that is the uniform H\"older exponent does not depend on the choice of a trajectory. A similar phenomenon happens with their local H\"older regularity measured through the local H\"older exponent. Therefore, it seems natural to ask the following question: does such a phenomenon also occur with their pointwise H\"older regularity measured through the pointwise H\"older exponent? In this article, using the framework of multifractional processes, we construct a family of counterexamples showing that the answer to this question is not always positive.