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

TL;DR

This paper investigates the asymptotic behavior of mean first escape times for Brownian motion on asymptotically hyperbolic and gas giant surfaces, revealing different scaling laws and confirming results through simulations.

Contribution

It provides new asymptotic expansions for escape times on these surfaces and explains the differences using polyhomogeneous conormal functions techniques.

Findings

Escape time on hyperbolic surfaces scales as -log epsilon.

Escape time on gas giant surfaces remains bounded as epsilon approaches zero.

Monte Carlo simulations confirm the theoretical asymptotic results.

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.