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Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation

Guillaume Penent, Nicolas Privault

TL;DR

The paper addresses the existence of viscosity solutions for semilinear fractional elliptic PDEs on open balls with gradient nonlinearities by developing a tree-based probabilistic representation using $α$-stable branching processes. It introduces integration-by-parts weights to handle gradient terms and proves existence under small-domain and small-coefficient assumptions without requiring coercivity on the gradient nonlinearity. The main contributions include a rigorous stochastic representation for the solution and its gradient, uniform $L^p$ bounds for the associated functional on branching trees, and a practical Monte Carlo scheme that performs well in high dimensions (e.g., $d=10$). This probabilistic approach offers a scalable alternative to deterministic methods for nonlocal PDEs with gradient nonlinearities, with demonstrated accuracy in numerical experiments.

Abstract

We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $α\in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a tree-based probabilistic representation of solutions and their partial derivatives using $α$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. In comparison with the existing literature on the regularity of solutions, no polynomial order condition is imposed on gradient nonlinearities. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.

Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation

TL;DR

The paper addresses the existence of viscosity solutions for semilinear fractional elliptic PDEs on open balls with gradient nonlinearities by developing a tree-based probabilistic representation using -stable branching processes. It introduces integration-by-parts weights to handle gradient terms and proves existence under small-domain and small-coefficient assumptions without requiring coercivity on the gradient nonlinearity. The main contributions include a rigorous stochastic representation for the solution and its gradient, uniform bounds for the associated functional on branching trees, and a practical Monte Carlo scheme that performs well in high dimensions (e.g., ). This probabilistic approach offers a scalable alternative to deterministic methods for nonlocal PDEs with gradient nonlinearities, with demonstrated accuracy in numerical experiments.

Abstract

We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index with polynomial gradient nonlinearities on -dimensional balls, . Our approach uses a tree-based probabilistic representation of solutions and their partial derivatives using -stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. In comparison with the existing literature on the regularity of solutions, no polynomial order condition is imposed on gradient nonlinearities. Numerical illustrations demonstrate the accuracy of the method in dimension , solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.
Paper Structure (5 sections, 8 theorems, 90 equations, 6 figures)

This paper contains 5 sections, 8 theorems, 90 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Marked branching process
  4. Probabilistic representation of PDE solutions
  5. Numerical examples

Key Result

Theorem 1.1

Let $\alpha \in (1,2)$ and $d\geqslant 2$. Under Assumption (A), the semilinear elliptic PDE eq:1 admits a viscosity solution in ${\cal C}^1(B(0,R)) \cap {\cal C}^0 ( { \macc@depthne bgroupempty egroup\macc@set@skewchar {\mathord{\mathbb Z}}@ \frozen@everymath{\mathgroup\macc@group} \macc@set@skewch

Figures (6)

  • Figure 1: Tree labelling and marking conventions.
  • Figure 2: Tree labelling and marking conventions.
  • Figure 3: Random tree samples for the PDE \ref{['eq:lg']}.
  • Figure 4: Numerical solution of \ref{['eq:lg']} in dimension $d=10$ with $\alpha =1.75$.
  • Figure 5: Random tree sample for the PDE \ref{['eq:nlg']}.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1.1
  • Lemma 2.1
  • Definition 3.1
  • Definition 4.1
  • Theorem 4.2
  • Proposition 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Proposition 4.6
  • Lemma 4.7