Elastic Brownian motion with random jumps from the boundary

TL;DR

This paper introduces a novel elastic Brownian motion model where particles restart inside the domain after boundary interactions, providing a comprehensive mathematical characterization including SDE, generator, invariant measure, and spectral properties.

Contribution

It presents a new stochastic process with boundary restart mechanism, detailed mathematical analysis, and spectral characterization, extending classical Brownian motion theory.

Findings

Derived the SDE and generator for the process

Established the invariant probability measure

Analyzed harmonic functions on the upper half-space

Abstract

In this paper, we study elastic Brownian motion on a \(C^2\) domain. Instead of being killed at the boundary, the process restarts from a random position inside the domain. We characterize this process through its stochastic differential equation (SDE), its generator, and a description of the paths. We also derive the invariant probability measure and the spectral representation. At the end, we focus on the harmonic functions on the upper half-space to study the trace process.