On the non-local boundary value problem from the probabilistic viewpoint

TL;DR

This paper explores the relationship between non-local operators and irregular domains, analyzing boundary value problems and their anomalous behaviors to better understand motion on complex geometries from a probabilistic perspective.

Contribution

It introduces new insights into non-local boundary value problems and connects probabilistic methods with irregular domain analysis.

Findings

Non-local operators relate to irregular domain behaviors.

Boundary problems exhibit anomalous bulk and surface phenomena.

Probabilistic viewpoint aids in understanding motion on complex geometries.

Abstract

We provide a short introduction of new and well-known facts relating non-local operators and irregular domains. Cauchy problems and boundary value problems are considered in case non-local operators are involved. Such problems respectively lead to anomalous behavior on the bulk and on the surface of a given domain. Such a behavior can be considered (in a macroscopic viewpoint) in order to describe regular motion on irregular domains (in the microscopic viewpoint).