[Paper Review] Forward and Backward Governing EQuations for Anomalous Diffusion Models Based on the Continuous Time Random Walk

Peter Straka|arXiv (Cornell University)|Dec 5, 2012

Probabilistic and Robust Engineering Design22 references3 citations

TL;DR

This paper derives forward and backward Kolmogorov equations with non-local time operators for anomalous diffusion processes modeled via coupled continuous time random walks (CTRWs) in space- and time-dependent environments. It establishes convergence conditions for CTRW scaling limits and characterizes the resulting dynamics through four coefficients, enabling modeling of subdiffusion with time-dependent potentials, spatially varying waiting times, and Lévy walks with drift.

ABSTRACT

Continuous Time Random Walks (CTRWs) are jump processes with random waiting times between jumps. We study scaling limits for CTRWs where the distribution of jumps and waiting times is coupled and varies in space and time. Such processes model e.g. anomalous diffusion processes in a space- and time-dependent potential. Conditions for the process-convergence of CTRWs are given, and the limits are characterised by four coefficients. Kolmogorov forwards and backwards equations with non-local time operators are derived, and three models for anomalous diffusion are presented: i) Subdiffusion in a time-dependent potential, ii) subdiffusion with spatially varying waiting times and iii) Lévy walks with space- and time-dependent drift.

Motivation & Objective

• To establish sufficient conditions for the weak convergence of coupled CTRWs with space- and time-dependent jump and waiting time distributions.
• To characterize the scaling limits of such CTRWs through four macroscopic coefficients representing drift, diffusion, waiting time, and potential effects.
• To derive forward and backward Kolmogorov equations with non-local time operators that govern the evolution of probability densities in anomalous diffusion processes.
• To apply the framework to three concrete models: subdiffusion in time-dependent potentials, subdiffusion with spatially varying waiting times, and Lévy walks with space- and time-dependent drift.
• To provide a unified mathematical foundation for modeling complex anomalous diffusion in heterogeneous and non-stationary environments.

Proposed method

• Formal derivation of scaling limits for CTRWs with coupled, space- and time-dependent jump and waiting time distributions using weak convergence techniques.
• Identification of four limiting coefficients—drift, diffusion, waiting time, and potential—through asymptotic analysis of jump and waiting time distributions.
• Derivation of the forward Kolmogorov equation with a non-local time operator to describe the evolution of the probability density function.
• Derivation of the backward Kolmogorov equation with a non-local time operator to describe first-passage and first-exit time statistics.
• Application of the framework to three distinct anomalous diffusion models: subdiffusion in time-dependent potentials, subdiffusion with spatially varying waiting times, and Lévy walks with space- and time-dependent drift.
• Use of non-local operators to capture memory effects and non-Markovian dynamics inherent in anomalous diffusion.

Experimental results

Research questions

• RQ1 Under what conditions does a coupled CTRW with space- and time-dependent jump and waiting time distributions converge to a scaling limit?
• RQ2 How can the limiting dynamics be characterized by a set of four macroscopic coefficients in the context of anomalous diffusion?
• RQ3 What form do the forward and backward Kolmogorov equations take when the time evolution involves non-local time operators?
• RQ4 How can the derived equations model subdiffusion in time-dependent potentials?
• RQ5 How do spatially varying waiting times and space- and time-dependent drifts affect the governing equations in Lévy walk models?

Key findings

• The paper establishes sufficient conditions for the weak convergence of CTRWs with coupled, space- and time-dependent jump and waiting time distributions to a scaling limit.
• The limiting process is fully characterized by four coefficients: drift, diffusion, waiting time, and potential, which emerge from the asymptotic behavior of the jump and waiting time distributions.
• The forward and backward Kolmogorov equations derived feature non-local time operators, reflecting the non-Markovian nature of the anomalous diffusion processes.
• The framework successfully models subdiffusion in time-dependent potentials by incorporating time-varying external forces into the governing equations.
• The model accounts for spatially varying waiting times by allowing the waiting time distribution to depend on position, leading to a space-dependent subdiffusion exponent.
• The Lévy walk model with space- and time-dependent drift is captured through a non-local time operator that incorporates both ballistic motion and memory effects.