= Solution
The two-dimensional <Fokker-Planck equation> is
$$
\boxed{
\partial_t p
=-a_1\partial_xp-\partial_y(a_2p)
+\partial_{xx}p+\frac12\partial_{yy}(\sigma^2p)
=-\nabla\mathbin\cdot\mathbf J,
}
$$
with <Fokker-Planck probability current>
$$
\boxed{
\mathbf J=
\left(
a_1p-\partial_xp,\,
a_2p-\frac12\partial_y(\sigma^2p)
\right).
}
$$
The point initial condition is
$$
p(x,y,0)=\delta(x)\delta(y),
$$
where $\delta$ is the <Dirac delta function>. The <reflecting boundary condition for a diffusion> is
$$
\boxed{\mathbf J\mathbin\cdot\mathbf n=0
\quad\hbox{on }\partial\Omega,}
$$
with $\mathbf n$ the outward unit normal. This zero-flux condition conserves the integral of the <probability density function> over the square.
Back to article page
