M.Sc Thesis | |

M.Sc Student | Lev-Ari Anat |
---|---|

Subject | Random Walks in Cookie Environment |

Department | Department of Mathematics |

Supervisor | PROF. Ross Pinsky |

In this work we consider two generalizations of a simple random walk on the integers. In the first one, we take a transient random walk with the following transition probabilities: there is a constant K such that at the i-th visit to a point x, for i≤K, the probability for the process to jump right is ω(x,i)≤1/2 and to jump left is 1-ω(x,i). After the K-th visit to x, the probabilities to jump left and right change to 1/2. The sequence

ω={{ω(x,i)
}_{i=1,...,K}}_{x}_{∈Z}
is called a one-sided cookie environment with a uniformly bounded number of
cookies per site. We give a new and simpler proof that if ω(x,i)=p≥1/2
for all integer number x, i=1,...,K, and p>1/2+1/K, then it is also
ballistic, i.e. liminf_{n→∞}X_{n}/n>0 a.s.. The
transient random walk will visit each site finitely many times a.s. and
eventually escape to infinity leaving a trail of leftover cookies. Those
leftovers cookies make a new cookie environment, which we call the leftovers
environment, and we can place a new random walk on it. We show that if ω={{ω(x,i)
}_{i=1,...,K}}_{x}_{∈Z}
is an i.i.d. sequence, the second process is transient if and only if the first
one is ballistic.

In the second generalization, we consider a transient random walk with the following transition probabilities: at each point x the probability to jump right is ω(x)=p≥1/2 as long as the process keeps jumping right from x. The first time it jumps left from x the transition probabilities change for 1/2 to each direction. We show that the leftovers environment is stationary and ergodic, give a necessary and sufficient condition for the second random walk to be transient or to be recurrent, and give a sufficient condition for the random walk to be ballistic.