Solution for the Poisson problem

Question

Consider a continuous function $f:\mathbb{R}^q \to \mathbb{R}$.

I woukd like to check if there are any analytical or probabilistic way to find a function $h:\mathbb{R}^q \to \mathbb{R}$ such that $\Delta h=f$ on $B(0,1).$

Reference are also appreciated.

Answer 1

Let $B$ be a $q$-dimensional standard Brownian motion, and let $T:=\inf\{t>0: |B_t|=2\}$. Then $$ h(x):=\Bbb E^x\left[\int_0^T f(B_t)\,dt\right],\qquad x\in B(0,1), $$ should do the job for you. The superscript $x$ on $\Bbb E^x$ is to indicate that the Brownian motion starts at $x$.