A probabilistic approach to a non-local quadratic form and its connection to the Neumann boundary condition problem

TL;DR

This paper develops a probabilistic framework for non-local quadratic forms, connecting them to Neumann boundary conditions and Dirichlet-to-Neumann operators, through the construction of related Markov processes.

Contribution

It introduces a novel probabilistic approach to non-local quadratic forms and their relation to Neumann boundary problems, including the construction of associated Markov processes.

Findings

Constructed Markov processes linked to non-local quadratic forms.

Established connections between non-local forms and Neumann boundary conditions.

Analyzed the Dirichlet-to-Neumann operator for non-local operators.

Abstract

In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov process: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neuman problem. We also study the Dirichlet-to-Neumann operator for non-local operators.