2023/02/13 by Jaywan Chung, Chung, Jaywan, Yong-Jung Kim +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Applied mathematics #Computer science #Continuous-time random walk #Diffusion #Diffusion equation #Diffusion process #Discrete time and continuous time #FOS: Mathematics #Heterogeneous random walk in one dimension #Innovation diffusion #Lattice (music) #Markov process #Mathematics #Monte Carlo method #Physics #Probability (math.PR) #Random walk #Statistical physics #Statistics #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2302.06275
openalex publication_date 2023/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a discrete-time random walk model on a one-dimensional lattice with a nonconstant sojourn time and prove that the discrete density converges to a solution of a continuum diffusion equation. Our random walk model is not Markovian due to the heterogeneity in the sojourn time, in contrast to a random walk model with a nonconstant walk length. We derive a Markovian process by choosing appropriate subindexes of the time-space grid points, and then show the convergence of its discrete density through the parabolic-scale limit. We also find the Green's function of the continuum diffusion equation and present three Monte Carlo simulations to validate the random walk model and the diffusion equation.