Brownian motion

Brownian motion with random drift defined on the disk D.  The SDE is:  dXt=b(Xt,ω)dt+σdWt, where σ>0, b:R2×Ω→R2 is a random field, and Wt is 2-D Brownian motion (independent of b).  The realizations b(⋅,ω) typically point radially outward, except when |x|≈r0 for a random r0>0.

The infinitesimal generator associated with this SDE is: A:=bi∂∂xi+12σ2Δ.  If you define τ as the boundary stopping time, and let f:∂D→R, then the function u(x):=E{f(Xτ)|X0=x} solves the BVP Au=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.