Continuous Gaussian Multifractional Processes with Random Pointwise Hölder Regularity

Antoine Ayache ()
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Antoine Ayache: U.M.R. CNRS 8524

Journal of Theoretical Probability, 2013, vol. 26, issue 1, 72-93

Abstract: Abstract Let {X(t)} t∈ℝ 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 Hölder regularity over any compact interval, i.e. the uniform Hölder exponent does not depend on the choice of a trajectory. A similar phenomenon occurs with their local Hölder regularity measured through the local Hölder exponent. Therefore, it seems natural to ask the following question: Does such a phenomenon also occur with their pointwise Hölder regularity measured through the pointwise Hölder 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.

Keywords: Hölder regularity; Pointwise Hölder exponents; Multifractional Brownian motion; Level sets; 60G15; 60G17 (search for similar items in EconPapers)
Date: 2013
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10959-012-0418-3

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