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

, the mean first escape time

u ε ( x ) $u_\varepsilon (x)$ u Subscript epsilon Baseline left parenthesis x right parenthesis

from the truncated Riemannian surface with an asymptotically hyperbolic metric

( M ε , g ¯ / ρ 2 ) = ( { x ∈ M : ρ ( x ) > ε } , g ¯ / ρ 2 ) ⊂ ( M , g ¯ / ρ 2 ) $(M_\varepsilon ,\bar {g}/\rho ^2) = (\{x\in M:\rho (x)> \varepsilon \},\bar {g}/\rho ^2) \subset (M,\bar {g}/\rho ^2)$ left parenthesis upper M Subscript epsilon Baseline comma g overbar divided by rho squared right parenthesis equals left parenthesis StartSet x element of upper M colon rho left parenthesis x right parenthesis greater than epsilon EndSet comma g overbar divided by rho squared right parenthesis subset of left parenthesis upper M comma g overbar divided by rho squared right parenthesis

satisfies the asymptotic expansion

u ε ( x ) = − log ⁡ ε + O ( 1 ) $u_\varepsilon (x) = -\log \varepsilon + \mathcal {O}(1)$ u Subscript epsilon Baseline left parenthesis x right parenthesis equals minus log epsilon plus script upper O left parenthesis 1 right parenthesis

as

ε → 0 $\varepsilon \to 0 $ epsilon right arrow 0

. Furthermore, we show that in the case of a gas giant metric

g = g ¯ / ρ α $g = \bar {g}/\rho ^\alpha $ g equals g overbar divided by rho Superscript alpha

, where

α ∈ ( 0 , 2 ) $\alpha \in (0,2)$ alpha element of left parenthesis 0 comma 2 right parenthesis

, the mean first escape time from the surface

( M ε , g ¯ / ρ α ) $(M_\varepsilon ,\bar {g}/\rho ^\alpha )$ left parenthesis upper M Subscript epsilon Baseline comma g overbar divided by rho Superscript alpha Baseline right parenthesis

satisfies

u ε ( x ) = O ( 1 ) $u_\varepsilon (x) = \mathcal {O}(1)$ u Subscript epsilon Baseline left parenthesis x right parenthesis equals script upper O left parenthesis 1 right parenthesis

as

ε → 0 $\varepsilon \to 0 $ epsilon right arrow 0

. Using techniques from the theory of polyhomogeneous conormal functions, we explain this difference between the mean first escape time on gas giant metric surfaces and asymptotically hyperbolic surfaces on the unit disc.