Path-by-path regularization by noise for scalar conservation laws

Abstract: We prove a path-by-path regularization by noise result for scalar conservation laws. In particular, this proves regularizing properties for scalar conservation laws driven by fractional Brownian motion and generalizes the respective results obtained in [Gess, Souganidis; Comm. Pure Appl. Math. (2017)]. In addition, we introduce a new path-by-path scaling property which is shown to be sufficient to imply regularizing effects.