Explicit representations for multiscale Lévy processes and asymptotics of multifractal conservation laws

Abstract

Nonlinear conservation laws driven by Lévy processes have solutions which, in the case of supercritical nonlinearities, have an asymptotic behavior dictated by the solutions of the linearized equations. Thus, the explicit representation of the latter is of interest in the nonlinear theory. In this paper, we concentrate on the case where the driving Lévy process is a multiscale stable (anomalous) diffusion, which corresponds to the case of multifractal conservation laws considered in Biler et al. [J. Differ. Equations 148, 9 (1998); Stud. Math. 135, 231 (1999); Ann. Inst. Henri Poincare, Anal. Nonlineaire 18, 613 (2001); and Stud. Math. 148, 171 (2001)]. The explicit representations, building on the previous work on single-scale problems (see, e.g., Górska and Penson [Phys. Rev. E 83, 061125 (2011)]), are developed in terms of the special functions (such as Meijer G functions) and are amenable to direct numerical evaluations of relevant probabilities.