Regularization by noise for Gevrey well-posedeness of a weakly hyperbolic operator

TL;DR

This paper demonstrates that adding a specific stochastic noise perturbation to a weakly hyperbolic PDE can transform its ill-posed deterministic problem into a well-posed stochastic one within the smooth function space, extending well-posedness results.

Contribution

It provides a novel example showing how noise regularization can improve well-posedness of weakly hyperbolic operators beyond classical Gevrey classes.

Findings

Noise perturbation induces well-posedness in the $C^{ abla}$-category.

Deterministic problem is only well-posed in Gevrey classes with $1 \,< s < 2$.

Stochastic perturbation achieves well-posedness in smooth functions.

Abstract

We present an example of a linear partial differential equation whose Cauchy problem becomes well-posed when perturbed by noise. Specifically, we make clear how a suitable multiplicative Stratonovich perturbation of Brownian type renders a weakly hyperbolic operator with double involutive characteristics well-posed in the $C^{\infty}$-category, while its deterministic counterpart is only well-posed in the Gevrey $ s $ classes with $ 1 \leq s <2 $ .