Principal Eigenvalue for Brownian Motion on a Bounded Interval with Degenerate Instantaneous Jumps

TL;DR

This paper analyzes the principal eigenvalue of a Brownian motion with spatially dependent jumps on a bounded interval, providing sharp asymptotic bounds as the jump rate parameter increases.

Contribution

It offers the first sharp asymptotic bounds for the principal eigenvalue in a model with degenerate jump rates, answering a question posed by previous researchers.

Findings

Sharp asymptotic bounds on the principal eigenvalue as the jump rate parameter tends to infinity

The jump rate function's positivity and boundary behavior critically influence eigenvalue estimates

Provides insights into the spectral properties of jump-diffusion processes with spatially varying jump intensities

Abstract

We consider a model of Brownian motion on a bounded open interval with instantaneous jumps. The jumps occur at a spatially dependent rate given by a positive parameter times a continuous function positive on the interval and vanishing on its boundary. At each jump event the process is redistributed uniformly in the interval. We obtain sharp asymptotic bounds on the principal eigenvalue for the generator of the process as the parameter tends to infinity. Our work answers a question posed by Arcusin and Pinsky.