Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions

Abstract

Stochastic partial differential equations for the one-dimensional telegraph equation and the two-dimensional linear transport equation are derived from basic principles. The telegraph equation and the linear transport equation are well-known correlated random walk (CRW) models, that is, transport models characterized by correlated successive-step orientations. In the present investigation, these deterministic CRW equations are generalized to stochastic CRW equations. To derive the stochastic CRW equations, the possible changes in direction and particle movement for a small time interval are carefully determined. As the time interval decreases, the discrete stochastic models lead to systems of It stochastic differential equations. As the position intervals decrease, stochastic partial differential equations are derived for the telegraph and transport equations. Comparisons between numerical solutions of the stochastic partial differential equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.