First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)

We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics dX(t) = alpha(X-t)dt + b(X-t)dW(t). We formulate descriptions of Brownian motion and general drift-diffusion processes on surfaces. We consider statistics of the form u(x) = E-x [integral(tau)(0) g(X-t)dt] + E-x [ f(X-tau)] for a domain Omega and the exit stopping time tau = inf(t) {t > 0 vertical bar X-t is not an element of 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(x) = E-x [tau]. 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. (C) 2021 Published by Elsevier Inc.

Automated summary

In this paper, we develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics. We focus on the mean First Passage Times (FPTs) and develop high-order Generalized Moving Least Squares (GMLS) solvers for associated surface PDE boundary-value problems. Our studies show that our methods converge with high-order accuracy and can capture the influence of surface geometry, drift dynamics, and spatially dependent diffusivities on the statistics.