Markovian and non-Markovian Transport on Lattices

: From Data to Analytics via Random Walks with Internal States

Abstract

The amount of time for a stochastic agent to complete a search process is a fundamental quantity across a range of scientific disciplines. Examples include the duration of an animal’s foraging task or the time a stock price will cross a threshold. Given this ubiquity, the study of stochastic search processes has expanded significantly but has mainly focused on determining the first moment of the distribution of search times only. This is largely because the first moment can be determined without the need to find the time dependent occupation probability for the searcher. However, using the first moment to represent the search not only reduces the entire spatio-temporal process into a single number, hiding nuances in the dynamics, but it also overestimates empirical search times.

This thesis goes beyond standard approaches by deriving closed form expressions for the occupation probability of a searcher in a range of systems. We do so largely by employing random walks with internal states, which allow for the study of both Markovian and non-Markovian systems. We focus primarily on dynamics bounded in finite space deriving exact representations of the spatio-temporal dynamics of a random walker on the hexagonal, honeycomb, kagome, and square-octagonal lattices, as well as on more general network structures. For non-Markovian dynamics, we derive equivalent quantities for the persistent random walk on hypercubic lattices of arbitrary dimensions, as well as on two-dimensional hexagon and honeycomb lattices. We use these findings to study the entire spatio-temporal search dynamics across a range of stochastic process in finite space. We relate our findings to experimental observations by parametrising the analytic representation of the honeycomb persistent random walk using movement data to model the foraging behaviour of the ant species Aphaenogaster
senilis.

Date of Award	30 Sept 2025
Original language	English
Awarding Institution	University of Bristol
Supervisor	Luca Giuggioli (Supervisor)