In dimension $d = 2$ the function $g(x) = - (2 \pi)^{-1} \log |x|$ is the Green's kernel associated to Laplace's equation. For example, if $f \in C_c^\infty(\mathbb{R}^2)$ then the solution of $-\Delta u = f$ can be given by $$ u(x) = \int g(x-y) f(y) dy $$
One also expects that the solution can be given by $$ u(x) = \int_0^\infty \mathbb{E}_x[f(B_t)] dt $$ where under $\mathbb{P}_x$, the process $(B_t)_{t \ge 0}$ is a Brownian motion starting from $x \in \mathbb{R}^2$. But this doesn't work, and the integral can be seen to be diverging, thanks to the recurrence of the process. For example, consider $f \ge 0$.
Is there a way of fixing the argument so that we can represent the solution $u$ still in terms of a Brownian motion? I would expect the following might work, but I can't prove it: $$ u(x) = \int_0^\infty \big( \mathbb{E}_x[f(B_t)] - \mathbb{E}_0[f(B_t)] \big) dt $$
