When Random Walks Come Back:Understanding Stochastic Recurrence

Abstract

In this lecture, we explore the phenomenon of recurrence in random walks, a fundamental concept in probability theory. Students will first learn the Recurrence Criterion, which precisely characterizes whether a random walk returns to its starting point infinitely often. Using combinatorial reasoning and asymptotic estimates, we demonstrate how to apply this criterion to one-dimensional, two-dimensional, and three-dimensional simple random walks, explaining why lower-dimensional walks are recurrent while higher-dimensional walks are transient. We conclude by highlighting how the Recurrence Criterion provides a powerful tool for understanding the long-term dynamics of random walks.

Biography

Dr Jianyu Hu is currently a Research Fellow in the School of Physical and Mathematical Sciences at Nanyang Technological University. He received his BSc in Mathematics and Applied Mathematics and PhD in Statistics from the School of Mathematics at Huazhong University of Science and Technology. His main research interests include structure-preserving machine learning methods and metastable transitions in stochastic dynamical systems. He has published multiple papers in journals such as Journal of Nonlinear Science, Mathematics of Computation, Chaos, and Physica D.