Voting Models and Semilinear Parabolic Equations

The talk presents a probabilistic interpretation of solutions to semilinear parabolic equations with polynomial nonlinearities via voting models constructed on genealogical trees of branching Brownian motion (BBM).
These results extend the connection between the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation and BBM, originally discovered by McKean.
We consider voting models with a random outcome" and a random threshold". It is shown that any model of this type can be associated with a Cauchy problem for a parabolic equation with nonlinearity $f$ satisfying $f(0)=f(1)=0$.
Several examples of such correspondences are discussed, based on the paper: A. Jing, C. Henderson, L. Ryzhik, \textit{Voting models and semilinear parabolic equations.