From the long jump random walk to the fractional Laplacian

TL;DR

This paper explains how a simple random walk with long jumps relates to fractional Laplacian operators, providing an elementary and self-contained exposition of this mathematical connection.

Contribution

It offers an accessible, self-contained explanation of the relationship between long jump random walks and fractional Laplacians, clarifying a complex mathematical concept.

Findings

Long jump random walks correspond to fractional Laplacian operators.

Elementary and self-contained exposition makes the concept accessible.

Clarifies the mathematical relationship between stochastic processes and differential operators.

Abstract

This note illustrates how a simple random walk with possibly long jumps is related to fractional powers of the Laplace operator. The exposition is elementary and self-contained.