Probabilistic representation of ODE solutions with quantitative estimates

TL;DR

This paper develops a probabilistic framework using marked random trees to represent ODE solutions, providing conditions for integrability and linking branching process explosion to solution existence and uniqueness.

Contribution

It introduces a novel probabilistic representation of ODE solutions via marked random trees and establishes conditions connecting branching process explosion with solution properties.

Findings

Provided sufficient conditions for integrability of the probabilistic representation.

Linked explosion times of branching processes to ODE solution existence.

Established implicit bounds on explosion times related to solution uniqueness.

Abstract

This paper considers the probabilistic representation of the solutions of ordinary differential equations (ODEs) by the generation of marked random trees in which marks can be interpreted as mutant types in population genetics models. We present sufficient conditions on equation coefficients that ensure the integrability and uniform integrability of the functionals of random trees used in this representation. Those conditions rely on the analysis of a marked branching process that controls the growth of random trees and provide implicit lower bounds on the explosion time of the underlying ODE, thus providing a connection between branching process explosion and the existence and uniqueness of ODE solutions.