Continuous Gaussian Multifractional Processes with Random Pointwise Holder Regularity

Let {X(t)} (taa"e) be an arbitrary centered Gaussian process whose trajectories are, with probability 1, continuous nowhere differentiable functions. It follows from a classical result, derived from zero-one law, that, with probability 1, the trajectories of X have the same global Holder regularity over any compact interval, i.e. the uniform Holder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Holder regularity measured through the local Holder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Holder regularity measured through the pointwise Holder 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.