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

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 [G., Souganidis; Comm. Pure Appl. Math. (2017)]. We show that $(\rho,\gamma)$-regularity is a sufficient path-by-path condition implying such regularizing effects. In addition, we introduce a new path-by-path scaling property which is also shown to be sufficient to imply regularizing effects.