Brownian motion with random drift defined on the disk . The SDE is: , where , is a random field, and is 2-D Brownian motion (independent of ). The realizations typically point radially outward, except when for a random .
The infinitesimal generator associated with this SDE is: . If you define as the boundary stopping time, and let , then the function solves the BVP .
The function can be found via Monte Carlo simulation.
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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.