  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.
