Continuous time random walk limit processes

Abstract

Continuous Time Random Walks (CTRWs) provide stochastic models for the random movement of any entity, e.g. a particle, a derivative price, etc. They are given by a sequence of random waiting times, each of which is followed by a random jump. The distribution of a jump possibly depends on the preceding (or succeeding) waiting time. After every jump, the CTRW is renewed. Waiting times with infinite means usually model long trapping times of particles, slow relaxation to equilibrium, or the system’s dependence on its history. Jumps with infinite variance usually model extremely fast dispersion. The rate at which the distribution of the limit process spreads in space can stay constant or can accelerate or decelerate, with transient behaviour from short to long times being possible. CTRW limits are hence versatile models for anomalous sub- or superdiffusion, and have found applications in a wide range of fields including biology, physics, hydrology, finance and ecology.

This thesis is concerned with scaling limits of CTRWs on the level of stochastic processes. Similar to the convergence of random walks to Brownian motion, we identify limit processes when waiting times and jump lengths are rescaled to zero. We then show that these limit processes are jump-diffusions whose time evolution is governed by the inverse of an increasing additive process. Moreover, it turns out that there can be two different limit processes, depending on whether the length of a jump depends on the preceding or the succeeding waiting time. For independent identically distributed waiting times, we show the existence of a governing “generalized Fokker-Planck equation,” which is a partial differential equation whose solutions are given by the densities of CTRW limit process. Finally, we establish a continuous semi-Markov property for CTRW limits by embedding them into Markov processes. This property can then be used to resolve problems which arise from the missing Markov property of CTRW limits.