Stability analysis of a branching diffusion solver for nonlinear PDEs

TL;DR

This work provides a stability analysis for a stochastic branching diffusion solver of the nonlinear PDE $\partial_t u + \mathcal L u + f(u) = 0$ with terminal data $u(T)=\phi$ by ensuring the integrability of random branching functionals. It develops a coding framework and a dominating binary branching construction to bound the multiplicative progeny via a Hamilton–Jacobi equation, yielding factorial and exponential growth conditions on $\phi$ and derivatives of $f$ that guarantee integrability on a nontrivial time interval. The authors show that, under uniform integrability, probabilistic representations for classical and viscosity solutions hold, and unique mild solutions exist in weighted spaces, linking branching representations to Feynman–Kac–type formulas. The results enable stable, high-dimensional Monte Carlo approaches for fully nonlinear PDEs, specifying precise data-growth and time-horizon requirements for stability and reliability.

Abstract

Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the integrability of random functionals of branching processes in order to ensure the non-explosion of solutions. In this paper, we study the stability of a functional branching representation of PDE solutions by deriving sufficient criteria for the integrability of the multiplicative weighted progeny of stochastic branching processes. We also prove the uniqueness of mild solutions under uniform integrability assumptions on random functionals.

Figures (5)

• Figure 1: Sample of random tree started from $c=\alpha!^{-1}\partial^\alpha$.
• Figure 2: Sample of random tree started from $c=\alpha!^{-1} \partial^\alpha \circ f^{(j)}$.
• Figure 3: Original vs. dominating branching chain.
• Figure 4: Merging of pair branches into a single branch.
• Figure :

Theorems & Definitions (47)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 4.1
• proof
• Proposition 4.2
• proof
• Theorem 4.3
• proof