On the Lifetime of a Conditioned Brownian Motion in Domains Connected Through Small Gaps

Abstract

For Ω a domain in $\mathbb{R}^n$ the expression $\mathbb{E}_{x}^{y}\left(\tau _{\Omega}\right)$ denotes the expected lifetime of Brownian motion starting at $x\in\Omega$, conditioned to converge to $y\in \Omega$ and with paths that are killed when touching $\partial\Omega$ before reaching (an ϵ-neighborhood of) y. This expectation coincides with the integral expression in the so-called 3G-theorem of Cranston et al. [7]. In general, it is hard to find sharp estimates for the bound in the 3G-theorem. In this chapter, we study the special case that the domain Ω consists of two parts which are connected through small gaps. We will derive a formula for $\mathbb{E}_{x}^{y}\left(\tau _{\Omega}\right)$, that depends on the expected times to reach each individual gap and which involves weights that depend asymptotically on the Poisson kernels at these gaps for the two subdomains. Since conformal mappings between Jordan domains are used, we restrict ourselves to domains in $\mathbb{R}^2$.