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

TL;DR

This paper investigates how renormalization affects the bifurcation structure of the singular $\

Contribution

It demonstrates that renormalization eliminates a finite-time bifurcation in the $\

Findings

Support of FTLEs is the entire real line regardless of bifurcation parameter.

Renormalization alters the bifurcation behavior compared to non-singular cases.

Support theorem established for stationary solution and its renormalized square.

Abstract

We study the singular $\Phi^4_2$ 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 $\Phi^4_1$ equation. The proof relies on a support theorem for the stationary solution and its renormalized square.