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Probabilistic representation of ODE solutions with quantitative estimates

Qiao Huang, Nicolas Privault

TL;DR

The paper develops a probabilistic representation of solutions to $d$-dimensional autonomous ODEs $\dot x = f(x)$ using marked branching trees, connecting branching-process explosion to the existence and uniqueness of solutions. It introduces an infinite system indexed by marks $\mathcal{C}$ and a representation $x_g(t)=\mathbb{E}[\mathcal{H}_t(\mathcal{B}^{0,g})]$, with conditions ensuring integrability and uniform integrability of the tree functionals so that $x_{\mathrm{Id}}(t)$ solves the ODE in a probabilistic sense. The main contributions include explicit integrability bounds under bounded and unbounded derivative growth, stochastic-dominance techniques to control the branching process, and constructive existence intervals, enabling Monte Carlo-style estimation of ODE solutions without assuming prior solvability. These results bridge Butcher-series representations and stochastic branching methods, providing quantitative explosion-time bounds and a robust framework for probabilistic ODE solvers with potential numerical applications.

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.

Probabilistic representation of ODE solutions with quantitative estimates

TL;DR

The paper develops a probabilistic representation of solutions to -dimensional autonomous ODEs using marked branching trees, connecting branching-process explosion to the existence and uniqueness of solutions. It introduces an infinite system indexed by marks and a representation , with conditions ensuring integrability and uniform integrability of the tree functionals so that solves the ODE in a probabilistic sense. The main contributions include explicit integrability bounds under bounded and unbounded derivative growth, stochastic-dominance techniques to control the branching process, and constructive existence intervals, enabling Monte Carlo-style estimation of ODE solutions without assuming prior solvability. These results bridge Butcher-series representations and stochastic branching methods, providing quantitative explosion-time bounds and a robust framework for probabilistic ODE solvers with potential numerical applications.

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.

Paper Structure

This paper contains 10 sections, 20 theorems, 118 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Marked branching trees
  3. ODE systems
  4. Probabilistic solution of ODEs
  5. Stochastic dominance of random binary trees
  6. Integrability - bounded marks
  7. Integrability - unbounded marks
  8. Examples
  9. Branching trees vs. Butcher trees
  10. Acknowledgements.

Key Result

Lemma 3.1

Let $T>0$ such that the ODE arxiv:ODE admits a solution $x\in {\cal C}^1([0,T],\mathbb{R}^d)$. Then, the family $(x_c)_{c\in \{\textbf{\rm Id}\}\cup \left\{ \nabla^k f \right\}_{k\geq 0}} : = (c(x))_{c\in \{\textbf{\rm Id}\}\cup \left\{ \nabla^k f \right\}_{k\geq 0}}$, solves the ODE system with the initial conditions $x_c(0)=c(x_0)$, for $c \in \{\textbf{\rm Id}\}\cup \left\{ \nabla^k f \right\}

Figures (3)

  • Figure 1: Sample of $\mathcal{K}_t$ for the random tree ${\mathcal{B}}^{0,c}$ started from $c=\textbf{\rm Id}$.
  • Figure 2: Sample of $\mathcal{K}_t$ for the random tree ${\mathcal{B}}^{0,c}$ started from $c\not=\textbf{\rm Id}$.
  • Figure 3: Sample of $\widetilde{\mathcal{K}}_t$ for the random tree $\widetilde{\mathcal{B}}^{0,j}$ started from $j\in \mathbb{N}$.

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • proof
  • Theorem 4.1: Existence
  • ...and 35 more