Fractional Cauchy problem on random snowflakes

TL;DR

This paper investigates fractional diffusion processes on random Koch fractal domains, analyzing the effects of different boundary conditions and providing asymptotic results for time-changed Brownian motions with fractional derivatives.

Contribution

It introduces a novel analysis of fractional Cauchy problems on random fractal domains with various boundary conditions, including Robin, Neumann, and Dirichlet, using inverse subordinators.

Findings

Asymptotic behavior characterized for fractional diffusions on fractals.

Extension of fractional Cauchy problem solutions to complex fractal boundaries.

Comparison of boundary condition impacts on diffusion processes.

Abstract

We consider time-changed Brownian motions on random Koch (pre-fractal and fractal) domains where the time change is given by the inverse to a subordinator. In particular, we study the fractional Cauchy problem with Robin condition on the pre-fractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.