Non-local Boundary Value Problems, stochastic resetting and Brownian motions on graphs

TL;DR

This paper explores non-local boundary conditions for Brownian motions on graphs, linking them with stochastic resetting, and provides new characterizations of boundary behaviors in complex graph structures.

Contribution

It introduces novel characterizations of boundary behavior for Brownian motions with non-local boundary conditions on graphs, including cases with resetting and trapping points.

Findings

Characterization of non-local boundary operators and their relation to random times

Extension of results from the real line to star graphs with special vertices

Analysis of boundary behavior for Feller-Wentzell diffusions with jumps

Abstract

We consider dynamic boundary conditions involving non-local operators. Our analysis includes a detailed description of such operators together with their relations with random times and random (additive) functionals. We provide some new characterizations for the boundary behaviour of the Brownian motion based on the interplay between non-local operators and boundary value problems. Our main focus is on Feller-Wentzell diffusions with jumps (resetting/restart). We first consider the instructive case of the real line, then we extend our results on star graphs with trapping points or repulsive vertices.