The Branching Diffusion, Stochastic Equations and Travelling Wave Solutions to the Equation of Kolmogorov — Petrovskii — Piskunov

A Brownian particle walks on a line, with negative drift coefficient. After an exponential lifetime it dies giving birth to a certain number of ‘descendants’ each of which then proceeds independently, following the same rule. We first re-derive a necessary and sufficient condition for P(X’ < ∞) = 1 and P(X < ∞) = 1 where X and X’ are the suprema of the deviation of particles in the positive direction taken, respectively, over the infinite time interval ℝ + = [0, ∞) and over all epochs of death (in a slightly weaker form, a similar result may be deduced from an earlier paper by (McKean 1975)). We then check that functions P(X < x) and P(X’ < x) give (unique) maximal solutions to certain stochastic equations and we prove the existence of other solutions to these equations.