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 general shape with drift-diffusion dynamics dXt=a(Xt)dt+b(Xt)dWt. formulate descriptions Brownian motion and surfaces. consider the form u(x)=Ex[∫0τg(Xt)dt]+Ex[f(Xτ)] a domain Ω exit stopping time τ=inft⁡{t>0|Xt∉Ω}, where f,g are smooth functions. For these statistics, we high-order Generalized Moving Least Squares (GMLS) solvers associated surface PDE boundary-value problems based Backward-Kolmogorov equations. focus particularly mean First Passage Times (FPTs) given by case f=0,g=1 u(x)=Ex[τ]. perform studies variety shapes showing our converge accuracy both in capturing geometry solutions. then how influenced geometry, drift dynamics, spatially dependent diffusivities.