Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

Abstract

We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli in [6]. To this end we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of regularity of the local time associated to the perturbing stochastic field. As an illustration we prove wellposedness of a 1D non-linear wave equation with a noisy boundary given by two independent fractional Brownian motions. A discussion of several open problems and further investigations is provided.