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.
翻译:我们开发了用于计算普通形状表面的随机进程统计数据的数值方法。 我们考虑的是以漂浮的动态来计算普通形状表面的色化进程的数据 $d\mathbf{X ⁇ }t= a(\\mathbf{X}dt+\mathbbf{b}}(mathb{X}d\mathbf{W}}d\mathbf{W}t$。我们为一个域绘制关于棕色运动和一般漂移过程的描述 $(mathb{mab{xf{x}liv) 格式的统计数据 $(mathb}xxxx=maxxxxx dreax freditions) 以美元计, 以美元=xxxxxxxxxxxxxxxxxxxx 的表解。