Random Walks on Finite Discrete Velocity Fields

This work presents a discrete-time random velocity field on a class of one-dimensional finite lattices with periodic boundary conditions. We define the Eulerian and Lagrangian location processes, analyzing their relationship through circulant and permutation transition matrices. By examining the second-largest eigenvalue modulus, we characterize the convergence rate of the Lagrangian location process to its invariant distribution. We explore how spatial domain size and parity influence convergence behavior, providing insights into stochastic transport dynamics in discrete settings.