Continuous random walks and fractional powers of operators

TL;DR

This paper presents a probabilistic representation of Fourier symbols associated with generators of certain stable processes, linking stochastic processes with fractional operator analysis.

Contribution

It introduces a novel probabilistic approach to represent Fourier symbols of generators of stable processes, enhancing understanding of fractional operators.

Findings

Derived probabilistic representations for Fourier symbols

Connected stable processes with fractional powers of operators

Provided new insights into the structure of generators

Abstract

We derive a probabilistic representation for the Fourier symbols of the generators of some stable processes.