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Pathwise Solvability and Bubbling in 2D Stochastic Landau-Lifshitz-Gilbert Equations

Ben Goldys, Chunxi Jiao, Christof Melcher

TL;DR

This paper develops a pathwise theory for the stochastic Landau-Lifshitz-Gilbert equation in 2D by employing a Doss–Sussmann-type transformation to a gauged LLG with random gauge $A$, enabling parabolic regularity methods in the transformed setting. It proves strong local well-posedness in the energy space, derives energy and higher-order estimates, and constructs pathwise global weak solutions through an iterative bubbling-accounts framework, where blow-up is governed by energy concentration at finitely many points with a $4\pi$ quantization. The results establish a stochastic Struwe-type solution framework and show uniqueness within the class of cadlag energy processes via a Yamada–Watanabe-type argument, connecting probabilistic and analytic concepts in stochastic geometric flows. The analysis highlights how random gauge fields interact with energy concentration and bubbling, providing a robust mechanism to study singularities and their propagation under infinite-dimensional noise in two dimensions. The work lays groundwork for understanding stochastic topological defects in ferromagnetism and offers a rigorous pathwise foundation for noisy spin dynamics with potential applications to stochastic optimal control and statistical physics.

Abstract

We investigate the stochastic Landau-Lifshitz-Gilbert (LLG) equation on a periodic 2D domain, driven by infinite-dimensional Gaussian noise in a Sobolev class. We establish strong local well-posedness in the energy space and characterize blow-up at random times in terms of energy concentration at small scales (bubbling). By iteration, we construct pathwise global weak solutions, with energy evolving as a c{à}dl{à}g process, and prove uniqueness within this class. These results offer a stochastic counterpart to the deterministic concept of Struwe solutions. The approach relies on a transformation that leads to a magnetic Landau-Lifshitz-Gilbert equation with random gauge coefficients.

Pathwise Solvability and Bubbling in 2D Stochastic Landau-Lifshitz-Gilbert Equations

TL;DR

This paper develops a pathwise theory for the stochastic Landau-Lifshitz-Gilbert equation in 2D by employing a Doss–Sussmann-type transformation to a gauged LLG with random gauge , enabling parabolic regularity methods in the transformed setting. It proves strong local well-posedness in the energy space, derives energy and higher-order estimates, and constructs pathwise global weak solutions through an iterative bubbling-accounts framework, where blow-up is governed by energy concentration at finitely many points with a quantization. The results establish a stochastic Struwe-type solution framework and show uniqueness within the class of cadlag energy processes via a Yamada–Watanabe-type argument, connecting probabilistic and analytic concepts in stochastic geometric flows. The analysis highlights how random gauge fields interact with energy concentration and bubbling, providing a robust mechanism to study singularities and their propagation under infinite-dimensional noise in two dimensions. The work lays groundwork for understanding stochastic topological defects in ferromagnetism and offers a rigorous pathwise foundation for noisy spin dynamics with potential applications to stochastic optimal control and statistical physics.

Abstract

We investigate the stochastic Landau-Lifshitz-Gilbert (LLG) equation on a periodic 2D domain, driven by infinite-dimensional Gaussian noise in a Sobolev class. We establish strong local well-posedness in the energy space and characterize blow-up at random times in terms of energy concentration at small scales (bubbling). By iteration, we construct pathwise global weak solutions, with energy evolving as a c{à}dl{à}g process, and prove uniqueness within this class. These results offer a stochastic counterpart to the deterministic concept of Struwe solutions. The approach relies on a transformation that leads to a magnetic Landau-Lifshitz-Gilbert equation with random gauge coefficients.

Paper Structure

This paper contains 22 sections, 20 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Wiener process
  3. Main result
  4. Transformation
  5. Transformation operator $\mathscr Y$ and $A$
  6. Transformed equation
  7. Solution of transformed equation
  8. Maximal solution in $\mathbb H^2$
  9. Energy estimates
  10. Regularity under small local energy
  11. $L^4$-estimates
  12. Higher-order estimates
  13. Well-posedness in energy space
  14. Proof of existence
  15. Approximations in $C([0,T];\mathbb H^2)$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Assume that eq_q3 holds for $\sigma = 4$. For every $\boldsymbol{m}_0 \in H^1(\mathbb T^2; \mathbb{S}^2)$, there exists a weak martingale solution $(\Omega, \mathscr F, (\mathscr F_t)_{t \in [0,T]}, \mathbb P, W, \boldsymbol{m})$ of eq: sLLG with $\boldsymbol{m}(0) = \boldsymbol{m}_0$ in the sense o

Theorems & Definitions (44)

  • Definition 1.1: Weak martingale solution
  • Definition 1.2: Local strong solution
  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Example 2.4
  • ...and 34 more