MEAN FIRST ESCAPE TIMES OF BROWNIAN MOTION ON ASYMPTOTICALLY HYPERBOLIC AND GAS GIANT METRIC SURFACES

Abstract

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 ρ $\rho $ rho , 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