Explicit representations for multiscale L\'evy processes, and asymptotics of multifractal conservation laws

TL;DR

This paper develops explicit representations for solutions of nonlinear conservation laws driven by multiscale Le9vy processes, using special functions to facilitate numerical evaluation and analyze asymptotic behaviors in multifractal contexts.

Contribution

It introduces explicit formulas for multiscale Le9vy-driven conservation laws using special functions, extending previous single-scale results to complex multiscale scenarios.

Findings

Explicit representations using Meijer G functions.

Enhanced numerical evaluation of probabilities.

Asymptotic analysis of multifractal conservation laws.

Abstract

Nonlinear conservation laws driven by L\'evy 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 L\'evy process is a multiscale stable (anomalous) diffusion, which corresponds to the case of multifractal conservation laws considered in [1-4]. The explicit representations, building on the previous work on single-scale problems (see, e.g.,[5]), are developed in terms of the special functions (such as Meijer G functions), and are amenable to direct numerical evaluations of relevant probabilities.