Asymmetric Lévy walks driven by convex combination of fractional material derivatives

TL;DR

The paper addresses deterministic modeling of scaling limits for asymmetric Lévy walks using a linear PDE driven by a convex combination of fractional material derivatives, $p(\partial_t - \partial_x)^{\alpha} u + (1-p)(\partial_t + \partial_x)^{\alpha} u = f(x,t)$. It first proves the existence and uniqueness of mild solutions via Fourier-Laplace analysis and derives a pointwise representation that leads to a Duhamel-type formula $u(x,t) = G_t * g + \int_0^t (G_{t-\tau} * f)(x,\tau) \, d\tau$, with the kernel $G_t$ built from $\zeta(\xi,s) = p(s - i\xi)^{\alpha} + (1-p)(s + i\xi)^{\alpha}$. A sharp necessary-and-sufficient condition on the source term $f$ is established for the solution to remain a probability density for all times, enabling mass conservation and nonnegativity. The authors then design a mass-conserving finite-volume discretization, prove discrete stability and convergence to the continuous solution, and validate the method with extensive numerical experiments showing conservation of total mass, positivity, and accurate reproduction of known Lévy-walk PDFs. This framework provides a robust tool for simulating anomalous transport governed by fractional material dynamics and opens avenues for nonlinear, heterogeneous, and higher-dimensional generalizations.

Abstract

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.

Figures (11)

• Figure 1: Comparison of analytical solution and numerical scheme for WF Lévy walk for $\alpha=0.5$ and different values of $p$.
• Figure 2: Comparison of analytical solution and numerical scheme for JF Lévy walk for $\alpha=0.5$ and different values of $p$.
• Figure 3: Comparison of analytical solution and numerical scheme for standard Lévy walk for $\alpha=0.5$ and different values of $p$.
• Figure 4: Comparison of analytical solution and numerical scheme for WF Lévy walk for $p=0.25$ and different values of $\alpha$.
• Figure 5: Comparison of analytical solution and numerical scheme for JF Lévy walk for $p=0.25$ and different values of $\alpha$.

Theorems & Definitions (20)

• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 2
• proof
• Proposition 3
• proof
• Proposition 4