On a Class of Non-linear Differential Equations Arising from Branching Diffusions

TL;DR

This paper investigates a specific non-linear differential equation from branching Brownian motion, providing explicit solutions, proving uniqueness under certain conditions, and extending the approach to a broader class of probabilistic parabolic equations.

Contribution

It introduces explicit solutions and uniqueness results for a non-linear differential equation from branching Brownian motion and generalizes the concept to probabilistic parabolic equations.

Findings

Explicit solution for the non-linear differential equation

Proof of uniqueness under boundedness conditions

Extension to a new class of probabilistic parabolic equations

Abstract

A non-linear differential equation arising from a stochastic process known as branching Brownian motion is considered. We find an explicit solution and show the uniqueness of the solution under some boundedness conditions using probabilistic ideas. We discuss non-negative solutions. We also generalize this idea to a class of non-linear parabolic differential equations which we call probabilistic parabolic equations.