Directionally Reinforced Random Walks

This paper introduces and analyzes a class of directionally reinforced random walks The work is motivated by an elementary model for time and space correlations in ocean surface wave elds We develop some basic properties of these walks For instance we investigate recurrence properties and give conditions under which the limiting continuous versions of the walks are Gaussian di usion processes MAULDIN MONTICINO V WEIZS ACKER Introduction This paper introduces a class of directionally reinforced random walks The motivation for our work was to develop and analyze an elementary model which simulates some of the time and space correlations observed in ocean surface wave elds In particular as discussed by West the direction of motion of a eld tends to be reinforced so that at any time a eld is more likely to continue moving in its current direction than it is to change direction We give an elementary mathematical model of this reinforcement and develop some basic properties of the resulting stochastic processes Directionaly reinforced random walks in Z are de ned in the next section In section we investigate the recurrence properties of these walks For instance it is shown that a walk in Z is recurrent if and only if it changes direction at some time with probability one An interesting example is given for which a walk in Z is recurrent yet changes direction only a nite number of times within any bounded spatial interval Thus eventually the walk visits only during fantastically long runs in a particular direction Under moment conditions on the time until the walk changes direction we show that the walk is recurrent when d and is transient for d The analysis is facilitated by de ning a related stopping time process which is a random walk with i i d increments and then applying classic results In section we consider the limiting continuous time version of a directionally reinforced walk Conditions on the reinforcement are presented under which the limiting version is a Gaussian di usion process We calculate the di usion coe cient for a speci c example The greater the directional reinforcement in this example the larger the di usion coe cient On the other hand an example shows that the limiting process is not in general necessarily Brownian motion In the last section we interpret some of the above results in terms of wave elds and raise some further questions Note that the random walks considered here have a di erent type of reinforce ment than that considered in other works on reinforced random walks such as Davis Pemantle and Mauldin and Williams There broadly speaking the path crossed by the walk is reinforced in some permanent way Here the reinforcement is on the current direction that the walk is moving Once the walk changes direction the previous reinforcement is forgotten and the new direc tion of motion is reinforced This lack of memory approximates the relaxation of reinforcement in a previous direction of motion Directionally Reinforced Walks in Z For k let g k The g k s will characterize the directional reinforcement of the walks de ned below Denote the set of unit vectors in the d dimensional lattice by U And let DIRECTIONALLY REINFORCED WALKS u u d be an arbitrary enumeration of the d vectors in U Now de ne a sequence of U valued random vectors X X such that for each ui U P X ui d For any ui U and for all n and k P Xn k uijXn k ui Xn ui Xn ui Xn ujn X uj P Xn k uijXn k ui Xn ui Xn ui g k