Probabilistic representation of ODE solutions with quantitative estimates

Abstract: This paper considers the probabilistic representation of the solutions of ordinary differential equations (ODEs) by the generation of random trees. We present sufficient conditions on equation coefficients that ensure the integrability and uniform integrability of the functionals of random trees used in this representation, and yield quantitative estimates of its explosion time. Those conditions rely on the analysis of a marked branching process that controls the growth of random trees, in which marks can be interpreted as mutant types in population genetics models. We also show how branching process explosion is connected to existence and uniqueness of ODE solutions.