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Renormalization destroys a finite time bifurcation in the $Φ^4_2$ equation

Alexandra Blessing, Nicolas Perkowski, Chara Zhu

TL;DR

The paper addresses stability of the singular $\\Phi^4_2$ SPDE on the two-torus driven by space-time white noise near a pitchfork bifurcation. It develops a rigorous framework based on paracontrolled calculus and a support theorem for the stationary solution and its renormalized square to study the linearized dynamics and the resulting finite-time Lyapunov exponents. The central finding is that the distribution of the finite-time Lyapunov exponent $\\lambda_T$ has full support on $\\mathbb{R}$ for every bifurcation parameter $\\alpha$, implying no finite-time sign change and showing that renormalization couples to the bifurcation parameter in a way that erases the finite-time bifurcation signature. This advances understanding of renormalization effects in high-dimensional singular SPDEs and provides groundwork for amplitude-equation approaches in the small-noise limit, with potential extensions to other renormalized bifurcations.

Abstract

We study the singular $Φ^4_2$ equation at a pitchfork bifurcation of the underlying deterministic dynamics. To this aim, we linearize the SPDE along its stationary solution and show that the support of its finite-time Lyapunov exponents (FTLEs) is the real line, regardless of the bifurcation parameter and in sharp contrast to the non-singular $Φ^4_1$ equation. The proof relies on a support theorem for the stationary solution and its renormalized square.

Renormalization destroys a finite time bifurcation in the $Φ^4_2$ equation

TL;DR

The paper addresses stability of the singular SPDE on the two-torus driven by space-time white noise near a pitchfork bifurcation. It develops a rigorous framework based on paracontrolled calculus and a support theorem for the stationary solution and its renormalized square to study the linearized dynamics and the resulting finite-time Lyapunov exponents. The central finding is that the distribution of the finite-time Lyapunov exponent has full support on for every bifurcation parameter , implying no finite-time sign change and showing that renormalization couples to the bifurcation parameter in a way that erases the finite-time bifurcation signature. This advances understanding of renormalization effects in high-dimensional singular SPDEs and provides groundwork for amplitude-equation approaches in the small-noise limit, with potential extensions to other renormalized bifurcations.

Abstract

We study the singular equation at a pitchfork bifurcation of the underlying deterministic dynamics. To this aim, we linearize the SPDE along its stationary solution and show that the support of its finite-time Lyapunov exponents (FTLEs) is the real line, regardless of the bifurcation parameter and in sharp contrast to the non-singular equation. The proof relies on a support theorem for the stationary solution and its renormalized square.
Paper Structure (11 sections, 13 theorems, 87 equations)

This paper contains 11 sections, 13 theorems, 87 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. Literature
  3. Plan of the paper
  4. Acknowledgements
  5. Preliminaries
  6. Functional analytic tools
  7. White noise stochastic semi-flows
  8. Main result
  9. Appendix
  10. Proof of Theorem \ref{['thm:continuity']}
  11. Proof of Lemma \ref{['lem:supp']}

Key Result

Theorem 1

The support of the finite time Lyapunov exponent of eq:phi is the real line, that is, for any value of $\alpha \in \mathbb{R}$ and $T>0$:

Theorems & Definitions (32)

  • Theorem : Main result, see Theorem \ref{['thm:main']}
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: Continuity of the Solution Map
  • proof
  • ...and 22 more