Consider a standard Wiener process (in 3 dimensions) $(W_t)_{t>0}$, such that $W_0 = x_0 \neq 0$. I am trying to determine the transition density of $(W_t)$ reflected on a sphere of radius $a < \|x_0\|$ centered at $0$.
I considered that the laplacian is a self-adjoint operator, so I can write the Fokker Planck equation to obtain the density function:
$$ \begin{cases} \partial_t p = \Delta p\\ \lim_{t\rightarrow 0} p(t,x) = \delta(x-x_0)\\ \frac{\partial p}{\partial r}\Big|_{r=a} = 0 \end{cases} $$
Is there any way I can obtain a closed form for $p$?
Instead of starting from $x_0$, I could start from a distribution that has spherical symmetry and transform it to 1D equation, but I'd prefer to have a closed form when starting from a point.