Abstract
We construct functions and stochastic processes for which a functional relation holds between amplitude and local regularity, as measured by the pointwise or local Holder exponent. We consider in particular functions and processes built by extending Weierstrass function, multifractional Brownian motion and the Levy construction of Brownian motion. Such processes have recently proved to be relevant models in various applications. The aim of this work is to provide a theoretical background to these studies and to provide a first step in the development of a theory for such self-regulating processes.
Highlights
Background and MotivationsLocal regularity of functions and stochastic processes has long been a topic of interest both in Analysis and Probability Theory, with applications in PDE/SPDE, approximation theory or numerical analysis to name a few
It is of interest to investigate the construction of functions and processes with everywhere prescribed local regularity
This may be done in various ways, for instance by generalizing Weierstrass function in the deterministic frame, or fractional Brownian motion in a stochastic setting
Summary
Local regularity of functions and stochastic processes has long been a topic of interest both in Analysis and Probability Theory, with applications in PDE/SPDE, approximation theory or numerical analysis to name a few. It is of interest to investigate the construction of functions and processes with everywhere prescribed local regularity This may be done in various ways, for instance by generalizing Weierstrass function in the deterministic frame (see [11] or Section 2.1), or fractional Brownian motion in a stochastic setting (see [1, 3, 12] or Section 3.1). In [16, 17, 18], we have reported on experimental findings indicating that, for certain natural phenomena such as electrocardiograms or natural terrains, there seems to exist a link between the amplitude of the measurements and their pointwise regularity This intriguing fact prompts for the development of new models, where the regularity would be obtained in an endogenous way: in other words, the Hölder exponent at each point would be a function of the value of the process at this point.
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