Renormalization destroys a finite time bifurcation in the $\hat{I}{¦}_2^4$ equation

Abstract

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

18 pages