Consider the Laplace equation in $\mathcal{R}^3$
\begin{equation}
\Delta u = f, ~~~\lim\_{x\to \infty} u(x) = 0.
\end{equation}
Here we assume $f$ is a smooth, compactly supported function. Of course, $u$ can be explicitly solved with the Green function. I am considering if we can use a stochastic process (like Brownian motion) to express its solution? I know the solution can be expressed as an expectation of a stochastic process (with a stopping time) if we consider a bounded domain with a certain boundary condition. In this case, the boundary seems to play a quite important role. Can we have a similar result for the whole space case?



Use stochastic process to express solution to Laplace equation in the whole space
If $f(x) / (1 + |x|)$ is integrable, then the solution $u$ is equal to the Newtonian potential of $f$:
$$ -u(x) = \frac{1}{4\pi} \int\_{\mathbb R^3} \frac{f(y)}{|x - y|} \, dy . $$
And the Newtonian potential kernel is the occupation density of the Brownian motion:
$$ \frac{1}{4\pi} \, \frac{1}{|x|} = \int\_0^\infty \frac{1}{(4 \pi t)^{3/2}} \, e^{-|x|^2 / (4 t)} dt . $$
Thus,
$$ -u(x) = \mathbb E^x \int\_0^\infty f(B\_t) dt , $$
where $B\_t$ is the 3-D Brownian motion (with covariance matrix $2 \operatorname{Id}$).


