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Well-posedness of stochastic reacting particle systems with non-local and Lennard-Jones interactions

Daniela Morale, Giulia Rui, Stefania Ugolini

TL;DR

The paper addresses well-posedness for a finite stochastic particle system with a strongly singular Lennard–Jones drift, a nonlocal environmental drift coupled through a regularized empirical density, and a reaction mechanism with field-dependent hazard. The authors extend a contraction-based, regularization approach to handle the singular drift and environmental coupling, and then employ an interlacing technique to construct a global strong solution in the presence of particle removal. They prove existence and pathwise uniqueness for the no-kill system, show non-attainment of the singular set, and establish a global, pathwise unique solution for the full system with reactions, with the interlacing construction terminating after finitely many switches. The results provide a rigorous microscopic stochastic framework for processes like cultural heritage sulphation, where randomness, strong short-range repulsion, environmental feedback, and irreversible particle removal interact. All results are cast with precise regularity and growth controls, ensuring robust well-posedness under the proposed modeling assumptions.

Abstract

We establish well-posedness results for systems of a finite number of stochastic particles driven by independent Brownian motions and subject to a strongly singular drift induced by a Lennard-Jones interaction. In addition to the pairwise force, the dynamics includes a nonlocal drift mediated by an environmental field, whose evolution is coupled to the particle configuration through a regularized empirical density. We then extend the analysis to a reaction model in which the switching (or killing) rate also depends on the field. An interlacing technique is considered for establishing the well-posedness of the full system. The model is motivated by the challenge to provide a stochastic microscopic description of the sulphation phenomenon in cultural heritage materials.

Well-posedness of stochastic reacting particle systems with non-local and Lennard-Jones interactions

TL;DR

The paper addresses well-posedness for a finite stochastic particle system with a strongly singular Lennard–Jones drift, a nonlocal environmental drift coupled through a regularized empirical density, and a reaction mechanism with field-dependent hazard. The authors extend a contraction-based, regularization approach to handle the singular drift and environmental coupling, and then employ an interlacing technique to construct a global strong solution in the presence of particle removal. They prove existence and pathwise uniqueness for the no-kill system, show non-attainment of the singular set, and establish a global, pathwise unique solution for the full system with reactions, with the interlacing construction terminating after finitely many switches. The results provide a rigorous microscopic stochastic framework for processes like cultural heritage sulphation, where randomness, strong short-range repulsion, environmental feedback, and irreversible particle removal interact. All results are cast with precise regularity and growth controls, ensuring robust well-posedness under the proposed modeling assumptions.

Abstract

We establish well-posedness results for systems of a finite number of stochastic particles driven by independent Brownian motions and subject to a strongly singular drift induced by a Lennard-Jones interaction. In addition to the pairwise force, the dynamics includes a nonlocal drift mediated by an environmental field, whose evolution is coupled to the particle configuration through a regularized empirical density. We then extend the analysis to a reaction model in which the switching (or killing) rate also depends on the field. An interlacing technique is considered for establishing the well-posedness of the full system. The model is motivated by the challenge to provide a stochastic microscopic description of the sulphation phenomenon in cultural heritage materials.
Paper Structure (3 sections, 4 theorems, 71 equations)

This paper contains 3 sections, 4 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. No reacting particles: an existence result
  3. Well-posedness of the full system with particle removal

Key Result

Theorem 1.1

MoraleRuiUgolini2025 Let $\left(\Omega,\mathcal{F},\mathbb F=(\mathcal{F}_{t})_{t\in [0,T]},\mathbb{P}\right)$be a filtered probability space with a fixed time horizon $T>0$ and let $W$ be an $m$--dimensional $\mathbb F$-adapted Wiener process. Let us consider the following SDE, for any $t\in [0,T]$ Let $U\subset\mathbb R^d$ be an open set. Suppose that the following conditions on the functions in

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1.1
  • Remark 2: $H_3$ as a Lyapunov condition.
  • Proposition 2.1: Environmental regularity
  • proof
  • Remark 3
  • Theorem 2.2: Well-posedness of the $\widetilde{N}$-particle system
  • proof
  • Theorem 3.1: Global well-posedness with particle removal
  • proof