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A Smoluchowski-Kramers approximation for the stochastic variational wave equation

Billel Guelmame, Julien Vovelle

TL;DR

The paper analyzes the Smoluchowski–Kramers approximation for a one-dimensional stochastic variational wave equation with state-dependent damping and additive noise on the torus. It first establishes global weak dissipative solutions via a regularized approximated system and energy/dissipation inequalities, then derives uniform-in-$\mu$ estimates to control the small-mass limit. By employing stochastic compactness (Prokhorov–Skorokhod) and the Gyöngy–Krylov approach, the authors prove that as $\mu\to 0$ the solutions converge in probability to a solution of a stochastic quasi-linear parabolic equation for $\Gamma(u)$ (or its related variables), with an Itô correction term arising from the non-constant friction. The limiting system is shown to be unique, and the defect measure vanishes, ensuring a rigorous SK-type reduction for this nonlinear SPDE setting. The results extend the SK approximation to stochastic variational wave dynamics with state-dependent coefficients, providing a rigorous passage from a stochastic damped wave model to a stochastic parabolic-like limit with clear physical and mathematical structure.

Abstract

We investigate the Smoluchowski-Kramers approximation for the one-dimensional periodic variational wave equation with state-dependent damping and additive noise. We show that weak ``dissipative'' solutions converge to solutions of a stochastic quasilinear parabolic equation.

A Smoluchowski-Kramers approximation for the stochastic variational wave equation

TL;DR

The paper analyzes the Smoluchowski–Kramers approximation for a one-dimensional stochastic variational wave equation with state-dependent damping and additive noise on the torus. It first establishes global weak dissipative solutions via a regularized approximated system and energy/dissipation inequalities, then derives uniform-in- estimates to control the small-mass limit. By employing stochastic compactness (Prokhorov–Skorokhod) and the Gyöngy–Krylov approach, the authors prove that as the solutions converge in probability to a solution of a stochastic quasi-linear parabolic equation for (or its related variables), with an Itô correction term arising from the non-constant friction. The limiting system is shown to be unique, and the defect measure vanishes, ensuring a rigorous SK-type reduction for this nonlinear SPDE setting. The results extend the SK approximation to stochastic variational wave dynamics with state-dependent coefficients, providing a rigorous passage from a stochastic damped wave model to a stochastic parabolic-like limit with clear physical and mathematical structure.

Abstract

We investigate the Smoluchowski-Kramers approximation for the one-dimensional periodic variational wave equation with state-dependent damping and additive noise. We show that weak ``dissipative'' solutions converge to solutions of a stochastic quasilinear parabolic equation.

Paper Structure

This paper contains 23 sections, 22 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. The equations and main results
  3. The stochastic variational wave equation
  4. An approximated system
  5. Global solutions
  6. The small-mass limit
  7. The energy estimates
  8. Uniform estimates
  9. Parabolic estimates
  10. Improved estimate on the wave energy
  11. Existence of martingale solutions with good controls
  12. The small-mass limit: the Smoluchowski--Kramers approximation
  13. Compactness results
  14. The convergence result
  15. Some elements of convergence
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let $c,f$ and $\gamma$ satisfying coeff-c and coeff-gammaf. Let $\Phi$ satisfy defq. Then SVWEep admits a unique global smooth solution $(R^{\mu, \varepsilon}, S^{\mu, \varepsilon})$, in the following sense:

Theorems & Definitions (43)

  • Remark 2.1: Notations
  • Theorem 2.1: Global existence of regular solutions
  • Definition 2.2: Weak martingale solution
  • Remark 2.2
  • Theorem 2.3: Global existence of weak martingale solutions
  • Definition 2.4: Weak solution to the limit stochastic quasi-linear equation on $\theta$
  • Definition 2.5: Weak solution to the limit stochastic quasi-linear equation on $u$
  • Theorem 2.6: Smoluchowski--Kramers approximation
  • Proposition 3.1: Conservation law for the energy
  • proof : Proof of Proposition \ref{['prop:energy identity']}
  • ...and 33 more