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

Jesse Gell-Redman, Emanuel József Godfried, Justin Tzou, Leo Tzou

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 $ρ$, the mean first escape time $u_ε(x)$ from the truncated Riemannian surface with an asymptotically hyperbolic metric $(M_ε,\bar{g}/ρ^2) = (\{x\in M:ρ(x)\geq ε\},\bar{g}/ρ^2) \subset (M,\bar{g}/ρ^2)$ satisfies the asymptotic expansion $u_ε(x) = -\log ε+ \mathcal{O}(1)$ as $ε\to 0 $. Furthermore, we show that in the case of a gas giant metric $g = \bar{g}/ρ^α$, where $α\in (0,2)$, the mean first escape time from the surface $(M_ε,\bar{g}/ρ^α)$ satisfies $u_ε(x) = \mathcal{O}(1)$ as $ε\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.

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 from the truncated Riemannian surface with an asymptotically hyperbolic metric satisfies the asymptotic expansion as . Furthermore, we show that in the case of a gas giant metric , where , the mean first escape time from the surface satisfies as . 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.
Paper Structure (13 sections, 31 theorems, 196 equations, 2 figures)

This paper contains 13 sections, 31 theorems, 196 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notational conventions
  3. The statement of the problem and theorem
  4. Outline of the paper
  5. Boundary coordinates systems and Green's functions
  6. Boundary coordinate systems
  7. Green's functions
  8. Sobolev spaces
  9. Proof of Theorem \ref{['thm:hyperbolic']}
  10. Gas giant geometries
  11. Limiting behaviour of MFET on gas giant surfaces to asymptotically hyperbolic surfaces
  12. Numerical results
  13. Proof of Lemma \ref{['prop:double-layer-N-sharp']}

Key Result

Theorem 1.1

Assume that $u_\varepsilon(x)$ satisfies the conditions of Equation eq:bvp, then there are functions $\tilde{U} _\varepsilon(x)$ and $r_\varepsilon(x)$ such that where for each fixed $x\in M_\varepsilon$ the function $\tilde{U}_\varepsilon(x) = \mathcal{O}(1)$ as $\varepsilon \to 0$ and for each fixed $x\in M_\varepsilon$ there is a positive constant $D_x$ such that $|r_\varepsilon(x)|\leq D_x \v

Figures (2)

  • Figure 1: Required blow-up space to resolve the function $U(y,\rho_0,\beta)$ from Equation \ref{['eq:U-X0']}.
  • Figure 1: The mean first escape time of a Monte Carlo simulation of the Brownian motion starting at the origin on the unit disc $\mathbb{D}$ with the gas giant metrics from \ref{['eq:gas-giant-metric-disc']} of order $\alpha$ and the Poincaré disc as a function of $\varepsilon$ in blue based on 300 000 simulations. The square marks denote the analytically found solutions in Equation \ref{['eq:v-epsilon-def']} and in the gas giant case in Equation \ref{['eq:v-epsilon-alpha-def']}. The dashed red lines denote the value of the numerically found solution to the boundary value problem \ref{['eq:bvp']}.

Theorems & Definitions (66)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Proof 1
  • Lemma 2.3
  • Proof 2
  • Remark 2.4
  • Lemma 2.5
  • Proof 3
  • Lemma 2.6
  • ...and 56 more