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Brownian motion with random drift defined on the disk $\mathbb{D}$.  The SDE is:  $dX_t = b(X_t, \omega)dt + \sigma dW_t$, where $\sigma>0$, $b:\mathbb{R}^2\times\Omega\to\mathbb{R}^2$ is a random field, and $W_t$ is 2-D Brownian motion (independent of $b$).  The realizations $b(\cdot,\omega)$ typically point radially outward, except when $|x|\approx r_0$ for a random $r_0>0$.

The infinitesimal generator associated with this SDE is: $A := b_i\frac{\partial}{\partial x_i} + \frac{1}{2}\sigma^2 \Delta$.  If you define $\tau$ as the boundary stopping time, and let $f:\partial\mathbb{D}\to\mathbb{R}$, then the function $u(x) := \mathbb{E}\left\{f(X_\tau)|\, X_0=x \right\}$ solves the BVP $Au=0, \,\,\, u\vert_{\partial\mathbb{D}}=f$.

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.