First-Passage Time Statistics on Surfaces of General Shape: Surface PDE Solvers using Generalized Moving Least Squares (GMLS)

TL;DR

This paper introduces high-order GMLS numerical methods for computing first-passage time statistics of stochastic processes on arbitrary surfaces, accurately capturing geometry and dynamics.

Contribution

The authors develop and validate high-order GMLS surface PDE solvers for stochastic process statistics, including first-passage times, on general-shaped surfaces with complex drift and diffusion.

Findings

Methods converge with high-order accuracy.

Statistics are significantly influenced by surface shape and dynamics.

Surface geometry and diffusivity affect first-passage time distributions.

Abstract

We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics $d\mathbf{X}_t = a(\mathbf{X}_t)dt + \mathbf{b}(\mathbf{X}_t)d\mathbf{W}_t$. We formulate descriptions of Brownian motion and general drift-diffusion processes on surfaces. We consider statistics of the form $u(\mathbf{x}) = \mathbb{E}^{\mathbf{x}}\left[\int_0^\tau g(\mathbf{X}_t)dt \right] + \mathbb{E}^{\mathbf{x}}\left[f(\mathbf{X}_\tau)\right]$ for a domain $\Omega$ and the exit stopping time $\tau = \inf_t \{t > 0 \; |\; \mathbf{X}_t \not\in \Omega\}$, where $f,g$ are general smooth functions. For computing these statistics, we develop high-order Generalized Moving Least Squares (GMLS) solvers for associated surface PDE boundary-value problems based on Backward-Kolmogorov equations. We focus particularly on the mean First Passage Times (FPTs) given by the case $f = 0,\, g = 1$ where $u(\mathbf{x}) = \mathbb{E}^{\mathbf{x}}\left[\tau\right]$. We perform studies for a variety of shapes showing our methods converge with high-order accuracy both in capturing the geometry and the surface PDE solutions. We then perform studies showing how statistics are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.