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Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus

Nils Berglund, Rita Nader

Abstract

We consider slowly time-dependent singular stochastic partial differential equations on the two-dimensional torus, driven by weak space-time white noise, and renormalised in the Wick sense. Our main results are concentration results on sample paths near stable equilibrium branches of the equation without noise, measured in appropriate Besov and Hölder norms. We also discuss a case involving a pitchfork bifurcation. These results extend to the two-dimensional torus those obtained in [Berglund and Gentz, Proability Theory and Related Fields, 2002] for finite-dimensional SDEs, and in [Berglund and Nader, Stochastics and PDEs, 2022] for SPDEs on the one-dimensional torus.

Concentration estimates for slowly time-dependent singular SPDEs on the two-dimensional torus

Abstract

We consider slowly time-dependent singular stochastic partial differential equations on the two-dimensional torus, driven by weak space-time white noise, and renormalised in the Wick sense. Our main results are concentration results on sample paths near stable equilibrium branches of the equation without noise, measured in appropriate Besov and Hölder norms. We also discuss a case involving a pitchfork bifurcation. These results extend to the two-dimensional torus those obtained in [Berglund and Gentz, Proability Theory and Related Fields, 2002] for finite-dimensional SDEs, and in [Berglund and Nader, Stochastics and PDEs, 2022] for SPDEs on the one-dimensional torus.
Paper Structure (25 sections, 28 theorems, 237 equations)

This paper contains 25 sections, 28 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Set-up and main results
  3. A family of Wick-renormalised singular SPDEs
  4. Besov spaces and the '' Da Prato--Debussche trick''
  5. Main results: Wick powers of the stochastic convolution
  6. Main results: concentration around stable equilibrium branches
  7. The case of bifurcations
  8. Proof of Theorem \ref{['thm:stoch_convolution']}
  9. Relation between Hermite polynomials and binomial formula
  10. Martingale and partition
  11. Summing over $q_0$ and $\mathbf{n}$
  12. The case of a general linearisation $a(t)$
  13. Proof of the concentration results
  14. Proof of Proposition \ref{['prop:stable_det']}
  15. Proof of Lemma \ref{['lem:F0']}
  16. ...and 10 more sections

Key Result

Proposition 2.2

Theorems & Definitions (46)

  • Definition 2.1: Besov spaces
  • Proposition 2.2: Embeddings and products
  • Theorem 2.3: daPratoDebussche
  • Theorem 2.4: Tail estimates on Wick powers of the stochastic convolution
  • Remark 1
  • Remark 2
  • Proposition 2.6: Deterministic case
  • Lemma 2.7
  • Remark 3
  • Theorem 2.8: Concentration estimate for $\phi_1$
  • ...and 36 more