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

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 $C^{\mathrm{\infty }}$-category, while its deterministic counterpart is only well-posed in the Gevrey $s$ classes with $1\le s<2$ .