First-Passage 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 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 $\tau ={\inf }_t\{t>0|{\mathbf{X}}_t\notin \Omega \}$, 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.

References in corpus (15)

• Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control
• An RBF-FD closest point method for solving PDEs on surfaces
• Meshfree Methods on Manifolds for Hydrodynamic Flows on Curved Surfaces: A Generalized Moving Least-Squares (GMLS) Approach
• First Passage Problems in Biology
• Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes
• Mesh-free Semi-Lagrangian Methods for Transport on a Sphere Using Radial Basis Functions
• Hydrodynamic Coupling of Particle Inclusions Embedded in Curved Lipid Bilayer Membranes
• A Fully Lagrangian Meshfree Framework for PDEs on Evolving Surfaces
• RBF-LOI: Augmenting Radial Basis Functions (RBFs) with Least Orthogonal Interpolation (LOI) for Solving PDEs on Surfaces
• A phase transition in the first passage of a Brownian process through a fluctuating boundary: implications for neural coding
• Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
• A First-Passage Kinetic Monte Carlo Method for Reaction-Drift-Diffusion Processes
• Generalized moving least squares and moving kriging least squares approximations for solving the transport equation on the sphere
• Reachable Space Characterization of Markov Decision Processes with Time Variability
• First Passage Value

Cited by in corpus (2)

• Drift-Diffusion Dynamics and Phase Separation in Curved Cell Membranes and Dendritic Spines: Hybrid Discrete-Continuum Methods
• Kinetic Monte Carlo methods for three-dimensional diffusive capture problems in exterior domains