TITLE: Random versions of deterministic problems
QUESTION [23 upvotes]: What are the examples of situations where "randomizing" a problem (or some part of it) and analyzing it using probabilistic techniques yields some insight into its deterministic version?
An example of what I have in mind: it is a well-known conjecture that the Hausdorff dimension of the graph of Weierstrass function (everywhere continuous, nowhere differentiable) is given by a certain simple formula, depending on the amplitudes and phases of the cosines in the series. This is still open; however, in the paper "The Hausdorff Dimension of Graphs of Weierstrass Functions" Hunt proved that if you add a uniformly distributed independent random "noise" to each phase, the conjectured formula holds with probability 1. So while the "randomized" approach does not solve the original problem, it somehow lends credibility to the original conjecture and thus gives us some insight about the problem.

REPLY [5 votes]: 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.