Path Laplacian operators and superdiffusive processes on graphs. II. Two-dimensional lattice

TL;DR

This paper investigates superdiffusive behavior on a 2D lattice using generalized diffusion equations derived from Mellin transforms of the k-path Laplacian, identifying parameter ranges for superdiffusion and normal diffusion.

Contribution

It extends the analysis of path Laplacian operators to two-dimensional lattices and characterizes diffusion regimes based on Mellin transform parameters.

Findings

Superdiffusion occurs for s in (2,4)

Normal diffusion occurs for s > 4

Provides mathematical proof of diffusion regimes

Abstract

In this paper we consider a generalized diffusion equation on a square lattice corresponding to Mellin transforms of the $k$-path Laplacian. In particular, we prove that superdiffusion occurs when the parameter $s$ in the Mellin transform is in the interval $(2,4)$ and that normal diffusion prevails when $s>4$.