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

Enrico Bernardi, Alberto Lanconelli

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 $ .

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

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 -category, while its deterministic counterpart is only well-posed in the Gevrey classes with .
Paper Structure (5 sections, 16 theorems, 86 equations)

This paper contains 5 sections, 16 theorems, 86 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. A brief review of the deterministic case
  3. Sufficiency
  4. Sharpness
  5. The stochastic energy estimate

Key Result

Theorem 1.1

The Cauchy problem for the stochastic Stratonovich perturbation (eq:STRA) is well-posed in $H_x^\infty$ in the continuous sense, i.e. it has a unique solution process in

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • proof : Proof of Theorem \ref{['ill1']}
  • Lemma 3.1: Second moments system
  • ...and 18 more