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An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

Andrea Di Primio, Marvin Fritz, Luca Scarpa, Margherita Zanella

TL;DR

This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair and proves well-posedness of the model in a strong probabilistic sense, and analyzes its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties.

Abstract

This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties. Eventually, we present an Euler--Maruyama scheme to simulate the model and illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics.

An Allen-Cahn equation with jump-diffusion noise for biological damage and repair processes

TL;DR

This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair and proves well-posedness of the model in a strong probabilistic sense, and analyzes its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties.

Abstract

This paper analyzes a stochastic Allen--Cahn equation for the dynamics of biomolecular damage and repair. The system is driven by two distinct noise processes: a multiplicative cylindrical Wiener process, modeling continuous background stochastic fluctuations, and a jump-type noise, modeling the abrupt, localized damage induced by external shocks. The drift of the equation is singular and covers the typical logarithmic Flory-Huggins potential required in phase-separation dynamics. We prove well-posedness of the model in a strong probabilistic sense, and analyze its long-time behavior in terms of existence and uniqueness of invariant measures, ergodicity, and mixing properties. Eventually, we present an Euler--Maruyama scheme to simulate the model and illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics.
Paper Structure (27 sections, 5 theorems, 191 equations, 5 figures)

This paper contains 27 sections, 5 theorems, 191 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Mathematical preliminaries and notation
  4. Stochastic framework.
  5. Analytical framework.
  6. Assumptions
  7. Main results
  8. Proof of Theorem \ref{['thm:wellposed']}
  9. An approximated problem.
  10. Uniform estimates
  11. First estimate.
  12. Second estimate.
  13. Third estimate.
  14. Passage to the limit $\lambda \to 0^+$.
  15. Uniqueness of solutions.
  16. ...and 12 more sections

Key Result

Theorem 2.4

Let Assumption hyp:structural-hyp:J hold. Fix $p \geq 2$ and let $r := \min\{4,p\}$. Let further $u_0 \in L^p(\Omega, \mathscr{F}_0; H)$ be such that Then, there exists a unique progressively measurable $H$-valued càdlàg process $u$ such that Moreover, given two initial conditions $u_{01}$ and $u_{02}$ satisfying the same assumptions listed above, letting $u_1$ and $u_2$ denote the corresponding

Figures (5)

  • Figure 1: Case 1 (random initial condition, compensated jumps): snapshots of $u(t,\cdot)$ at $t=0,0.25,1,25$ for increasing jump intensities (None/Few/Some/Many). The last column visualizes the jump locations up to final time.
  • Figure 2: Case 1 (compensated jumps): left: evolution of the total damage $\int_\mathcal{O} u(t)\,\mathrm{d}x$ for different jump intensities (ensemble mean with uncertainty bands); right: evolution of $u_{\min}(t)$ and $u_{\max}(t)$ for different jump intensities. The values remain strictly within $(0,1)$, indicating effective enforcement of the logarithmic barrier.
  • Figure 3: Case 2 (circular initial condition, uncompensated jumps): snapshots of $u(t,\cdot)$ at $t=0,0.25,1,4$ for increasing jump intensities (None/Few/Some/Many), $c_\mathrm{noise}=1/2$. The last column visualizes the jump locations up to final time.
  • Figure 4: Case 2 (uncompensated jumps): left: evolution of the total damage $\int_\mathcal{O} u(t)\,\mathrm{d}x$ for different jump intensities (ensemble mean with uncertainty bands). Increasing jump intensity accelerates the growth of total damage; right: evolution of $u_{\min}(t)$ and $u_{\max}(t)$ for different jump intensities. The solution remains strictly within $(0,1)$ for all regimes.
  • Figure 5: Case 2 (circular initial condition, uncompensated jumps): snapshots of $u(t,\cdot)$ at $t=0,0.25,1,4$ for increasing jump intensities (None/Few/Some/Many), with $c_\mathrm{noise}=5$. The last column visualizes the jump locations up to final time.

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Definition 4.1
  • Definition 4.2
  • ...and 5 more