Introduction to the continuous time random walk

Igor Sokolov

In this lecture we will discuss the basic concepts and some tools of the theory of continuous time random walk. The lecture serves as the introduction to several other ones. The plan is as follows:

1) Historical introduction. Simple random walks: Bachelier, Einstein, Pearson, and Smoluchowski. Postulates and models. Mean squared displacement (MSD). Lattice random walks and their properties.

2) Sums of independent random variables, and the central limit theorem. Scaling limit of random walks. The diffusion equation.

3) Waiting times, and continuous time random walks: Models and physical background. Mean number of steps and MSD in a decoupled CTRW. Related models: Levy walks.

4) Renewal processes: ordinary, aged and equilibrium. Forward waiting time, and inspection paradox. Aging in CTRW.

5) CTRW as a subordinated process. Scaling limits of CTRW.

6) Another approach to CTRW: the generalized Master equation, and corresponding diffusion limits.

7) Simplest first passage problems for CTRW.