Use stochastic process to express solution to Laplace equation in the whole space 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?
 A: 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 $2 \operatorname{Id}$).
A: consider its boundary condition $lim_{x\to{\infty}}u(x)=0$, to express its inhomogeneity as a hyperbolic term, by inverse function theorem. thus not adequate to define a fast variation function. since stochastic process require a nonzero term under variation two, to make this wave be more frequently, even to make its vibration not ever disappearing.
then define an Ito calculus to represent its vibration, $df=\frac{\partial{f}}{\partial{t}}dt+\frac{\partial{f}}{\partial{x}}dB_t+\frac{1}{2}\frac{\partial^2{f}}{\partial{x}^2}d(B_t)^2=(\frac{\partial{f}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{f}}{\partial{x}^2})dt+\frac{\partial{f}}{\partial{x}}dB_t$, action as a centralized source at both positive and negative side, between $y=(u-\epsilon)t$ and $y=(u+\epsilon)t$ , spanned at a distance of $\Delta{s}$, represented by its non-convergent part.
then define a mandatory function by orthogonal condition, such that $\delta(x-x_0)+c\phi_n{(x)}$. to handle its inhomogeneity boundary by generalized green function. satisfied space symmetry and time reciprocity, and the condition of compact support and smoothness in variation theory.
for example, since $\partial{f}$ is both $t$ and $B_t$'s derivation, are equivalent under the scalar derivation $dB_t$ at order two. so this time's green integral can be expressed as $\int[G(x,t;x',0)$$\psi{(x')}-\phi{(x')}$$\frac{\partial{G(x,t;x',t')}}{\partial{t'}}\vert_{t'=0}]dx'$, correspond to the reciprocal of time.
next, to express its space symmetry as a green integral be similar to the form :
$a^2{\int{[v(t')}}$$\frac{\partial{G}}
{\partial{x'}}\vert_{x'=l}-\mu{(t')}$$\frac{\partial{G}}{\partial{x'}}\vert_{x'=0}]dt'$
is it helpful? thank you.
