Mean first escape times of Brownian motion on asymptotically hyperbolic and gas giant metric surfaces

Abstract

This paper deals with the mean first escape time of Brownian motion on asymptotically hyperbolic and gas giant surfaces. We show that for a boundary defining function $\rho$, the mean first escape time $u_\epsilon(x)$ from the truncated Riemannian surface with an asymptotically hyperbolic metric $(M_\epsilon,\bar{g}/\rho^2) = (\{x\in M:\rho(x)\geq \epsilon\},\bar{g}/\rho^2) \subset (M,\bar{g}/\rho^2)$ satisfies the asymptotic expansion $u_\epsilon(x) = -\log \epsilon + \mathcal{O}(1)$ as $\epsilon\to 0 $. Furthermore, we show that in the case of a gas giant metric $g = \bar{g}/\rho^\alpha$, where $\alpha\in (0,2)$, the mean first escape time from the surface $(M_\epsilon,\bar{g}/\rho^\alpha)$ satisfies $u_\epsilon(x) = \mathcal{O}(1)$ as $\epsilon\to 0 $. Using techniques from the theory of polyhomogeneous conormal functions we explain this difference between in the mean first escape time on gas giant metric surfaces and asymptotically hyperbolic surfaces on the unit disc. Finally, we confirm these results using Monte Carlo simulations and finite difference methods on the disc.