On heterogeneous diffusion processes and the formation of spatial-temporal nonlocality

Area of Science:

• Statistical Physics
• Stochastic Processes
• Mathematical Modeling

Background:

• Heterogeneous diffusion processes are solutions to the overdamped Langevin equation with multiplicative noise.
• The noise amplitude exhibits a power-law dependence on space.

Purpose of the Study:

• To analyze discrete analogs of heterogeneous diffusion processes.
• To investigate a class of processes formed by deforming discrete fractional Brownian motion.
• To model nonlocality in time and space considering spatial heterogeneity.

Main Methods:

• Asymptotic estimation of variance behavior in time for discrete analogs.
• Deformation of discrete fractional Brownian motion using Cantor ladder and inverse transformation.
• Construction of random processes based on discrete heterogeneous processes and memory flow phenomenology.

Main Results:

• An asymptotic estimate for the variance behavior in time of discrete heterogeneous processes was obtained.
• A class of processes structurally similar to discrete heterogeneous processes was identified.
• Sub- and superdiffusion transport regimes were geometrically illustrated.
• A class of random processes modeling spatio-temporal nonlocality and spatial heterogeneity was constructed.

Conclusions:

• Discrete analogs of heterogeneous diffusion processes exhibit complex temporal variance behavior.
• Deformed fractional Brownian motion provides insights into heterogeneous process structures.
• The developed models effectively capture nonlocality and spatial heterogeneity in diffusion.