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

TL;DR

This paper extends non-linear Young equations to two dimensions and demonstrates how noise can regularize stochastic equations on the plane, including wave equations with noisy boundaries, through local time regularity.

Contribution

It introduces a 2D non-linear Young framework and applies it to prove regularization by noise for stochastic PDEs on the plane, including fractional Brownian fields.

Findings

Existence and uniqueness of 2D non-linear Young equations.

Regularization by noise for stochastic equations on the plane.

Well-posedness of a 1D non-linear wave equation with noisy boundary.

Abstract

We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stochastic Process. Appl., 2016). 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 the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.