Correction to "Well-posedness of semilinear stochastic wave equations with H\"{o}lder continuous coefficients''

TL;DR

This paper demonstrates that adding white noise to semilinear wave equations with Hölder continuous coefficients can restore well-posedness and uniqueness, highlighting a regularization effect of noise in stochastic PDEs.

Contribution

It introduces a novel approach using backward stochastic differential equations to establish well-posedness for wave equations with Hölder continuous drifts, even when deterministic counterparts lack uniqueness.

Findings

Well-posedness achieved for α-Hölder continuous coefficients with α in (2/3,1).

Regularization by noise restores well-posedness and uniqueness.

Method applies despite the associated linear semigroup not being strong Feller.

Abstract

We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an $\alpha$-H\"{o}lder continuous drift coefficient, if $\alpha \in (2/3,1)$. The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations, differentiability along subspaces and control theoretic results. We stress that the well-posedness holds despite the Markov semigroup associated to the linear stochastic wave equation is not strong Feller.