Asymmetric Lévy walks driven by convex combination of fractional material derivatives

We analyze a class of linear partial differential equations that arise as deterministic descriptions of the scaling limits of Lévy 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 Lévy walk process. The combined theoretical and numerical framework provides a reliable tool for studying anomalous transport governed by fractional dynamics.

💡 Research Summary

This paper investigates a class of linear partial differential equations that serve as deterministic macroscopic descriptions of the scaling limits of Lévy walks, focusing on the case where transport is driven by a convex combination of two fractional material derivatives and an external source term. The fractional material derivative, introduced in earlier work, couples time‑fractional differentiation with spatial advection and is defined in the Fourier‑Laplace domain by the multipliers (s ∓ iξ)^α. The governing equation reads

 p (∂_t − ∂_x)^α u(x,t) + (1 − p) (∂_t + ∂_x)^α u(x,t) = f(x,t),

with the initial condition expressed via a Riemann–Liouville integral I_{1−α} t u(x,t) = g(x). Here 0 ≤ p ≤ 1 measures the asymmetry of the walk, while 0 < α < 2 is the fractional order.

Analytical results.
Using a pointwise representation of the fractional material derivative, the authors rewrite the operator as a convolution in time with a kernel (t−s)^{−α} and a spatial shift x ∓ (t−s). This representation relaxes regularity requirements and makes the non‑local nature explicit. By applying Fourier and Laplace transforms, the equation reduces to an algebraic relation

 U(ξ,s) =