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Stochastic complex Ginzburg-Landau equation on compact surfaces

Tristan Robert, Younes Zine

TL;DR

The work analyzes the stochastic complex Ginzburg-Landau equation on a compact surface with a magnetic Laplacian Δ_A and white-noise forcing, addressing the ill-defined nonlinearity via Wick renormalization and a truncation-then-limit scheme. It develops a full analytic probabilistic framework on manifolds, including Besov spaces tied to Δ_A and sharp Green's-function estimates, to construct and control the stochastic objects Ψ and their Wick powers. Local well-posedness of the renormalized dynamics is established uniformly in the truncation, with convergence to a limiting stochastic flow; in the defocusing regime, a dispersive-dissipative balance yields deterministic global well-posedness. The results extend singular SPDE techniques to curved geometries with magnetic effects, providing a rigorous foundation for SCGL on compact surfaces and highlighting the role of geometric and spectral tools in stochastic PDE analysis.

Abstract

We study a stochastic complex Ginzburg-Landau equation (SCGL) on compact surfaces with magnetic Laplacian and polynomial nonlinearity, forced by a space-time white noise. After renormalizing the equation in a suitable manner, we show that the dynamics is locally well-posed. Moreover, we prove deterministic global well-posedness for the defocusing SCGL in the weakly dispersive regime.

Stochastic complex Ginzburg-Landau equation on compact surfaces

TL;DR

The work analyzes the stochastic complex Ginzburg-Landau equation on a compact surface with a magnetic Laplacian Δ_A and white-noise forcing, addressing the ill-defined nonlinearity via Wick renormalization and a truncation-then-limit scheme. It develops a full analytic probabilistic framework on manifolds, including Besov spaces tied to Δ_A and sharp Green's-function estimates, to construct and control the stochastic objects Ψ and their Wick powers. Local well-posedness of the renormalized dynamics is established uniformly in the truncation, with convergence to a limiting stochastic flow; in the defocusing regime, a dispersive-dissipative balance yields deterministic global well-posedness. The results extend singular SPDE techniques to curved geometries with magnetic effects, providing a rigorous foundation for SCGL on compact surfaces and highlighting the role of geometric and spectral tools in stochastic PDE analysis.

Abstract

We study a stochastic complex Ginzburg-Landau equation (SCGL) on compact surfaces with magnetic Laplacian and polynomial nonlinearity, forced by a space-time white noise. After renormalizing the equation in a suitable manner, we show that the dynamics is locally well-posed. Moreover, we prove deterministic global well-posedness for the defocusing SCGL in the weakly dispersive regime.

Paper Structure

This paper contains 16 sections, 20 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. The stochastic complex Ginzburg-Landau equation
  3. Renormalization of the nonlinearity
  4. Well-posedness of the renormalized dynamics
  5. Analytic toolbox
  6. Notations and first properties
  7. Geometric background
  8. Spectral properties of the magnetic Laplacian
  9. Function spaces, linear and nonlinear estimates
  10. Regularization of the Green's function
  11. Probabilistic tools
  12. Algebraic properties of the renormalization
  13. On the Wick powers
  14. Well-posedness theory
  15. Local well-posedness
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\alpha_1,\gamma>0$ and $\alpha_2,\beta_1,\beta_2\in\mathbb{R}$. Fix $m \geq 2$ an integer. Let $- \frac{2}{2m-1} < s_0 < 0$ and $0 < \varepsilon \ll 1$. Then, for any renormalized truncated SCGL 1 is pathwise uniformly locally well-posed in $\mathcal{C}^{s_0}(\mathcal{M};\mathbb{C})$, uniformly Moreover, $\{u_N\}_{N \in \mathbb{N}}$ converges $\mathbb{P}$-almost surely to a non-trivial stocha

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • ...and 28 more