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A probabilistic interpretation of a non-conservative and path-dependent nonlinear reaction-advection-diffusion system

Daniela Morale, Leonardo Tarquini, Stefania Ugolini

TL;DR

This work provides a rigorous probabilistic interpretation for a non-conservative, path-dependent nonlinear reaction–diffusion system modeling marble sulphation by deriving a regularised Feynman–Kac framework (MKFK-SDE) with drift depending on past history through time-marginal laws discounted by the reaction term. It establishes well-posedness of the regularised Feynman–Kac equation and the associated MKFK-SDE, and links the stochastic representation to a measure-valued PDE via a mollified density $u^m=K*\gamma^m$, together with a demonstration of propagation of chaos for an interacting particle system that approximates the dynamics. The results provide a solid mathematical bridge between nonlinear non-conservative PDEs in cultural heritage modelling and stochastic mean-field representations, enabling particle-based numerical schemes and deeper insights into the microscale sulphation mechanism. Overall, the paper advances the theory of non-Markovian McKean–Vlasov–Feynman–Kac equations and their use for complex, non-conservative reaction–diffusion processes.

Abstract

Given a reaction-advection-diffusion system modelling the sulphation phenomenon, we derive a single regularised non-conservative and path-dependent nonlinear partial differential equation and propose a probabilistic interpretation via a non-Markovian McKean-Vlasov stochastic differential equation coupled with a Feynman-Kac-type equation. We discuss the well-posedness of such a stochastic model, and establish the propagation of chaos property for the associated interacting particle system.

A probabilistic interpretation of a non-conservative and path-dependent nonlinear reaction-advection-diffusion system

TL;DR

This work provides a rigorous probabilistic interpretation for a non-conservative, path-dependent nonlinear reaction–diffusion system modeling marble sulphation by deriving a regularised Feynman–Kac framework (MKFK-SDE) with drift depending on past history through time-marginal laws discounted by the reaction term. It establishes well-posedness of the regularised Feynman–Kac equation and the associated MKFK-SDE, and links the stochastic representation to a measure-valued PDE via a mollified density , together with a demonstration of propagation of chaos for an interacting particle system that approximates the dynamics. The results provide a solid mathematical bridge between nonlinear non-conservative PDEs in cultural heritage modelling and stochastic mean-field representations, enabling particle-based numerical schemes and deeper insights into the microscale sulphation mechanism. Overall, the paper advances the theory of non-Markovian McKean–Vlasov–Feynman–Kac equations and their use for complex, non-conservative reaction–diffusion processes.

Abstract

Given a reaction-advection-diffusion system modelling the sulphation phenomenon, we derive a single regularised non-conservative and path-dependent nonlinear partial differential equation and propose a probabilistic interpretation via a non-Markovian McKean-Vlasov stochastic differential equation coupled with a Feynman-Kac-type equation. We discuss the well-posedness of such a stochastic model, and establish the propagation of chaos property for the associated interacting particle system.
Paper Structure (14 sections, 22 theorems, 143 equations)

This paper contains 14 sections, 22 theorems, 143 equations.

Table of Contents

  1. Motivation and background
  2. Notation.
  3. The regularised Feynman-Kac equation: study of the well-posedness
  4. Existence and uniqueness
  5. Regularity, stability and non-anticipating properties
  6. The MKFK-SDE: well-posedness and its relation with the PDE model
  7. Existence and uniqueness
  8. The probabilistic interpretation of the PDE model
  9. Particle system and propagation of chaos
  10. Appendix
  11. Proof of Proposition \ref{['prop:stability_Feynman_Kac']}
  12. Proof of Proposition \ref{['eq:stability_2']}
  13. Proof of Proposition \ref{['prop:stability_3']}
  14. Proof of Proposition \ref{['prop:continuity_b']}

Key Result

Lemma 2.2

For any $t\in[0,T]$ and $z,z^\prime \in C^+$, the function $\Lambda$ defined in eq:Lambda_Feynman-Kac is such that

Theorems & Definitions (52)

  • Definition 2.1
  • Remark 1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 42 more