CTRW approximations for fractional equations with variable order

TL;DR

This paper develops and rigorously proves the convergence of CTRW-based approximations to variable-order fractional diffusion equations, extending classical diffusion limits to position-dependent fractional derivatives in multidimensional settings.

Contribution

It introduces a novel approach to approximate variable-order fractional equations using CTRWs with fixed jump sizes and position-dependent intensities, providing rigorous convergence proofs.

Findings

Established convergence of CTRW approximations to variable-order fractional equations.

Extended results to multidimensional diffusions and general Feller processes.

Provided a mathematical framework for modeling variable fractional derivatives.

Abstract

The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The "rough" diffusion limit does not feel the difference. The situation changes, if we model jump-type approximations via CTRW with non-exponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an $\alpha$-stable law with a constant intensity $\al$, then the standard scaling would lead in the limit of small jumps and large intensities to the most standard fractional diffusion equation. However, if we choose the CTRW approximations with fixed jump sizes and use the diffusion coefficient to distinguish intensities at different points, then we obtain in the limit the equations with variable position-dependent fractional derivatives. In this paper we build rigorously these approximations and prove their convergence to the corresponding fractional equations for the cases of multidimensional diffusions and more general Feller processes.