Population heterogeneity in the fractional master equation, ensemble self-reinforcement and strong memory effects

TL;DR

This paper develops a fractional master equation modeling population heterogeneity and self-reinforcement in random walks, revealing superdiffusive behavior and connecting ensemble heterogeneity with strong memory effects.

Contribution

It introduces a novel fractional master equation framework linking population heterogeneity, self-reinforcement, and memory effects in random walks.

Findings

Ensemble averaged solution derived via subordination with fractional Poisson process

Exact variance solution shows superdiffusion as fractional exponent approaches 1

Connects heterogeneous populations with strong memory in stochastic processes

Abstract

We formulate a fractional master equation in continuous time with random transition probabilities across the population of random walkers such that the effective underlying random walk exhibits ensemble self-reinforcement. The population heterogeneity generates a random walk with conditional transition probabilities that increase with the number of steps taken previously (self-reinforcement). Through this, we establish the connection between random walks with a heterogeneous ensemble and those with strong memory where the transition probability depends on the entire history of steps. We find the ensemble averaged solution of the fractional master equation through subordination involving the fractional Poisson process counting the number of steps at a given time and the underlying discrete random walk with self-reinforcement. We also find the exact solution for the variance which exhibits superdiffusion even as the fractional exponent tends to 1.