Fractional governing equations for coupled random walks

Keywords: Fractional calculus Anomalous diffusion Continuous time random walk Central limit theory Operator stable law a b s t r a c t In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space–time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.