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∞ C ∞ -category, while its deterministic counterpart is only well-posed in the Gevrey s classes with 1 ≤ s <2 1 ≤ s < 2 .