Firstpassage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)
Abstract
We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with driftdiffusion dynamics dX_{t} = a (X_{t}) dt + b (X_{t}) dW_{t}. We formulate descriptions of Brownian motion and general driftdiffusion processes on surfaces. We consider statistics of the form u (x) =E^{x} [∫_{0τ}^{g} (X_{t}) dt ] +E^{x} [ f (X_{τ}) ] for a domain Ω and the exit stopping time τ =inf_{t} { t > 0 X_{t} ∉ Ω }, where f , g are general smooth functions. For computing these statistics, we develop highorder Generalized Moving Least Squares (GMLS) solvers for associated surface PDE boundaryvalue problems based on BackwardKolmogorov equations. We focus particularly on the mean First Passage Times (FPTs) given by the case f = 0 , g = 1 where u (x) =E^{x} [ τ ] . We perform studies for a variety of shapes showing our methods converge with highorder 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.
 Publication:

Journal of Computational Physics
 Pub Date:
 March 2022
 DOI:
 10.1016/j.jcp.2021.110932
 arXiv:
 arXiv:2102.02421
 Bibcode:
 2022JCoPh.45310932G
 Keywords:

 Surface PDEs;
 Stochastic processes on surfaces;
 Meshless methods;
 Generalized Moving Least Squares (GMLS);
 First passage time statistics;
 Pathdependent statistics;
 Mathematics  Numerical Analysis;
 Physics  Data Analysis;
 Statistics and Probability;
 Quantitative Biology  Quantitative Methods;
 Statistics  Computation
 EPrint:
 Journal of Comp. Phys.,453, (2022)