On the influence of reflective boundary conditions on the statistics of Poisson-Kac diffusion processes

TL;DR

This paper investigates how reflective boundary conditions affect Poisson-Kac diffusion processes, revealing modifications in switching-time statistics and emergent anomalous diffusion behaviors in fractal domains.

Contribution

It provides a detailed analysis of boundary effects on Poisson-Kac processes, especially in complex fractal geometries, highlighting new emergent diffusion features.

Findings

Reflective boundaries alter switching-time statistics.

Fractal boundaries induce anomalous diffusion.

Transition from regular to anomalous diffusion observed.

Abstract

We analyze the influence of reflective boundary conditions on the statistics of Poisson-Kac diffusion processes, and specifically how they modify the Poissonian switching-time statistics. After addressing simple cases such as diffusion in a channel, and the switching statistics in the presence of a polarization potential, we thoroughly study Poisson-Kac diffusion in fractal domains. Diffusion in fractal spaces highlights neatly how the modification in the switching-time statistics associated with reflections against a complex and fractal boundary induces new emergent features of Poisson-Kac diffusion leading to a transition from a regular behavior at shorter timescales to emerging anomalous diffusion properties controlled by walk dimensionality of the fractal set.