Voting models and semilinear parabolic equations
It is well known from the pioneering work by H. McKean that the Fisher-KPP equation can be described in terms of the binary branching Brownian motion. It turns out that this elegant interpretation can be extended to a large class of parabolic equations, and systems of parabolic equations, via voting schemes on the branching tree of BBM. Sadly, this beautiful connection has so far produced essentially no new information about the solutions to PDEs. On the other hand, the PDE results indicate that some voting schemes should have interesting algebraic properties for reasons that we do not understand. A totally open question is whether such voting schemes on trees generated by other log-correlated systems have any special properties or if BBM is an outlier in this class.