One dimensional Brownian motion with holding and jumping boundary

Abstract

Let a particle start at some point in the unit interval and undergo Brownian motion in until it hits one of the end points. At this instant, the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside with a probability density or parametrized by the boundary point it was at. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper by Peng and Li listed in the reference. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process. In particular, we answer a question on the spectral gap of the DHJ process raised at the end of the paper by Peng and Li.