Random Walk in Random and Non-Random Environments

The following sections are included:

• Preface

• Contents

• Introduction

The following sections are included:

• Notations and abbreviations
  • Notations
    • General notations
    • Notations to the increments
    • Notations to the Strassen-type theorems
    • Notations to the local time
  • Abbreviations

The following sections are included:

• Random walk

• Dyadic expansion

• Rademacher functions

• Coin tossing

• The language of the probabilist

The following sections are included:

• Exact distributions

• Limit distributions

The following sections are included:

• Recurrence

• The Zero-One Law

The following sections are included:

• Borel - Cantelli lemma and Markov inequality

• The strong law of large numbers THEOREM OF BOREL (1909)

• Between the Strong Law of Large Numbers and the Law of Iterated Logarithm

• The LIL of Khinchine

The following sections are included:

• Definitions

• EFKP LIL

• The laws of Chung and Hirsch

• When will S_n be very big?

• A theorem of Csáki

The following sections are included:

• Two lemmas

• Definition of the Wiener process

• Invariance Principle

The following sections are included:

• Long head-runs

• The increments of a Wiener process

• The increments of S_N

The following sections are included:

• The theorem of Strassen

• Strassen theorems for increments

• The rate of convergence in Strassen's theorems

• A theorem of Wichura

The following sections are included:

• Exact distributions

• Limit distributions

• Definition and distribution of the local time of a Wiener process

The following sections are included:

• An Invariance Principle

• A theorem of Lévy

The following sections are included:

• Strong theorems for ξ(x, n) and ξ(n)

• Increments of η(x, t)

• Increments of ξ(x, n)

• Strassen type theorems

• Stability

• Favourite points

• Rarely visited points

The following sections are included:

• On the Wiener sheet

• The theorem

• Applications

The following sections are included:

• On the distribution of the zeros of a random walk

• Local time and the number of long excursions (Mesure du voisinage)

• The local time of high excursions

• How many times can a random walk reach its maximum?

The following sections are included:

• On the location of the maximum of a random walk

• On the location of the last zero

• The Ornstein - Uhlenbeck process and a theorem of Darling and Erdős

• A discrete version of the Itô formula

Please refer to full text

The following sections are included:

• Notations

Please refer to full text

Let W₁(t),W₂(t),…,W_d(t) be a sequence of independent Wiener processes. The R^d valued process W(t) = {W₁(t),W₂(t),…,W_d(t)} is called a d-dimensional Wiener process. We ask how the random walk S_n can be approximated by W(t)…

Please refer to full text

The following sections are included:

• ξ(0, n) in Z²

• ξ(n) in Z^d

• A few further results

The following sections are included:

• The range of S_n

• Wiener sausage

It is easy to see (and Theorem 20.1 also implies) that the path of a random walk crosses itself infinitely many times for any d ≥ 1 with probability 1. We mean that there exists an infinite sequence {U_n, V_n} of positive integer valued r.v.'s such that S(U_n) = S(U_n + V_n), and 0 ≤ U₁ < U₂ < …, (n = 1,2,…). However, we ask the following question: will selfcrossings occur after a long time? For example, we ask whether the crossing S(U_n) = S(U_n + V_n) will occur for every n = 1, 2,… if we assume that V_n converges to infinity with a great speed and U_n converges to infinity much slower. . In fact Erdõs and Taylor (1960/B) proposed the following two problems…

The following sections are included:

• Completely covered discs in Z²

• Discs covered with positive density

• Completely covered balls in Z^d

• Once more on Z²

Please refer to full text

The following sections are included:

• On the Dirichlet problem

• DLA model

• Percolation

The following sections are included:

• Notations

The sequence {S_n} of Part I was considered as a mathematical model of the linear Brownian motion. In fact it is a model of the linear Brownian motion in a homogeneous (non-random) environment…

In this chapter we study what might have happened during the creation of the universe, i.e. the possible properties of the sequence ɛ are investigated…

The following sections are included:

• The recurrence theorem of Solomon

• Guess how far the particle is going away in an RE

• A prediction of the Lord

• A prediction of the physicist

The following sections are included:

• Two further lemmas on the environment

• On the local time ξ(0, n)

Please refer to full text

The following sections are included:

• Two theorems of Golosov

• Non-nearest-neighbour random walk

• RWIRE in Z^d

• Non-independent environments

• Random walk in random scenery

• Reinforced random walk

The following sections are included:

• References

• Author Index

• Subject Index