Correlated random walks, hyperbolic systems and Fokker-Planck equations

Abstract

One-dimensional position dependent discrete unrestricted standard and correlated random walks together with their continuum limits are studied. The standard random walks yield Fokker-Planck equations in the Ito form in the continuum limit, while the correlated random walks give rise to hyperbolic equations and systems. As certain parameters in the hyperbolic equations tend to infinity, with the consequence of removing the correlation effects, these equations reduce to Fokker-Planck equations which can be in the Ito or Stratonovich form. The reason for the differences in these limiting equations is given.