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

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 $ρ$, the mean first escape time $u_ε(x)$ from the truncated Riemannian surface with an asymptotically hyperbolic metric $(M_ε,\bar{g}