Chapter 4 - Dynamic random walks in a random scenery

Abstract

This chapter illustrates dynamic random walks in a random scenery. Some of the valued random variables with zero mean and finite positive variance can play the role of random scenery. Lewis established a self-normalized law of the iterated logarithm for random walk in random scenery. Lewis obtained a strong law for the range and the number of self-intersection of the random walk with the help of deterministic normalizers. This result covers the transient random walks and the recurrent random walk in two-dimensions. Lewis provides solution for stable processes in random scenery to get simple, symmetric random walk in random scenery with a stronger normalization than the natural one. Some random electrical charges can correspond to the random variables. Some of the physical modes considered proteins as very long charged molecules. Their stereochemical shape is the one minimizing the total electrical energy. The theoretical impacts about classical random walks in random sceneries can be extended to the case of dynamic random walks.