Self-similar blowup from arbitrary data for supercritical wave maps with additive noise

TL;DR

This paper demonstrates that in supercritical wave maps with additive noise, random perturbations can induce self-similar blowup with positive probability, highlighting the influence of noise on singularity formation.

Contribution

It proves that Gaussian additive noise causes self-similar blowup in supercritical wave maps for any initial data, a novel insight into stochastic effects on PDE singularities.

Findings

Additive noise induces blowup with positive probability

Self-similar blowup occurs in all energy-supercritical dimensions

Results suggest noise influences singularity formation in PDEs

Abstract

We consider stochastically perturbed wave maps from $\mathbb{R}^{1+d}$ into $\mathbb{S}^d$, in all energy-supercritical dimensions $d \geq 3$. We show that corotational non-degenerate Gaussian additive noise leads to self-similar blowup with positive probability for any corotational initial data. The same result without noise is conjectured, but unknown, for large data.