Brownian motion with random drift defined on the disk D. The SDE is: *d**X**t*=*b*(*X**t*,*ω*)*d**t*+*σ**d**W**t*, where *σ*>0, *b*:R2×Ω→R2 is a random field, and *W**t* is 2-D Brownian motion (independent of *b*). The realizations *b*(⋅,*ω*) typically point radially outward, except when |*x*|≈*r*0 for a random *r*0>0.

The infinitesimal generator associated with this SDE is: *A*:=*b**i*∂∂*x**i*+12*σ*2Δ. If you define *τ* as the boundary stopping time, and let *f*:∂D→R, then the function *u*(*x*):=E{*f*(*X**τ*)|*X*0=*x*} solves the BVP *A**u*=0,*u*|∂D=*f*.

The function *u*(*x*) can be found via Monte Carlo simulation.

This image was generated using a Python script.

I like this simple math since it combines four major themes that I’ve been working on: SDE, PDE with random coefficients, Monte Carlo simulation, and scientific computing using Python.