Spectral analysis of Brownian motion with jump boundary

Abstract:

Consider a family of probability measures, indexed by $\partial D$, on a bounded open region $D\subset \mathbb {R}^d$ with a smooth boundary. For any starting point inside $D$, we run a standard $d$-dimensional Brownian motion in $\mathbb {R}^d$ until it first exits $D$, at which time it jumps to a point inside the domain $D$ according to the jump measure at the exit point and starts a new Brownian motion. The same evolution is repeated independently each time the process reaches the boundary. We study the exponential rate at which the transition distribution of the process converges to its invariant measure, in terms of the spectral gap of the generator. In particular, we prove two conjectures of I. Ben-Ari and R. Pinsky for an interval (see J. Funct. Anal. 251 (2007), 122–140, and preprint (2007)) by studying when a combination of the sine and cosine transforms of probability measures on an interval has only real zeros.