One problem that I am aware of is the problem of finding the Poisson boundary of a manifold, that is, the bounded harmonic functions. There is a probabilistic way of approaching this problem using Brownian motion (c.f. here) which has evolved out of the probabilistic proof of Liouville's theorem.
If we let $B=(B_t:t \geq 0)$ be a Brownian motion and $Inv(B)$ be the sigma-algebra of events of the form $\{B_t \in A\}$ iff $\{B_{t+s}\in A\}$ for all $s \geq 0$, then there is a one-to-one correspondence between $Inv(B)$ measurable random variables and bounded harmonic functions.
Liouville's theorem holds on $\mathbb{R}^d$ because if we can start two Brownian motions at $x,y \in \mathbb{R}^d$ and couple them in a finite time, so the invariant sigma algebra is trivial. On manifolds where Liouville's theorem does not hold, then by looking at the invariant algebra, one can work out all of the bounded harmonic functions.