Asymmetric L\'evy walks driven by convex combination of fractional material derivatives

Abstract

We analyze a class of linear partial differential equations that arise as deterministic descriptions of the scaling limits of L\'evy walks, in which transport is driven by a convex combination of fractional material derivatives and a source term. Using techniques of Fourier-Laplace transforms, we first prove the existence of mild solutions for continuous initial data. Using a recently obtained pointwise representation of the fractional material derivative, we then identify a necessary and sufficient condition on the source term that guaranties the solution to remain a probability density for all times (non-negativity and unit mass). Motivated by the need to preserve these probabilistic properties in computations, we construct a finite-volume discretization that is probability conservative by construction. We establish discrete stability and a convergence result for the continuous weak solution as space and time steps tend to zero. Extensive numerical experiments validate the scheme: total mass is conserved, non-negativity is maintained, and the computed solutions reproduce the known analytic representations of the probability density functions associated with the L\'evy walk process. The combined theoretical and numerical framework provides a reliable tool for studying anomalous transport governed by fractional dynamics.