On the Spectral Gap of Brownian Motion with Jump Boundary

Abstract

Abstract. In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are different and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap on the jump distribution in a multi-dimensional setting.

Spectral analysis of diffusions with jump boundary

“…Then it is shown in [1,6,7] via different methods that there exists an invariant distribution π L,ν and that…”

“…Similar to [7] we construct two suitable versions of diffusions with jump boundary and jump distribution δ x 0 corresponding to different initial distributions simultaneously and control the tails of the coupling time. The following relations for the total-variation-distance are well-known and will be used throughout this section.…”

“…for τ I < t τ II , jumps occur only via the boundary point b. For τ I t τ II let 6 , · · · · · · a+b 2 + μ(t − τ I,n−2 ) − σ (B t − B τ I,n−2 ): τ I,n−2 t < τ I,n , · · · · · · and similarly 7 , · · · · · · a+b 2 + μ(t − τ I,n−1 ) + σ (B t − B τ I,n−1 ): τ I,n−1 t < τ I,n+1 , · · · · · · where τ I,2 = inf t > τ I :…”

“…Different versions of proofs of this formula can be found e.g. in [1] and [7]. According to general theory (see e.g.…”

“…Concerning the general diffusion process (X t ) t 0 with jump boundary, four open questions are listed in [1]; Question 4 concerning the continuous dependence of the spectral gap on the jump distribution has been answered affirmatively in [7]. From the remaining three, Question 1 and Question 2 are highly correlated.…”