RANDOM WALKS IN Zd AND THE DIRICHLET PROBLEM
نویسنده
چکیده
Random walks can be used to solve the Dirichlet problem – the boundary value problem for harmonic functions. We begin by constructing the random walk in Z and showing some of its properties. Later, we introduce and examine harmonic functions in Z in order to set up the discrete Dirichlet problem. Finally, we solve the Dirichlet problem using random walks. Throughout the paper, we discuss connections with the continuous analog.
منابع مشابه
Reversed Dirichlet environment and directional transience of random walks in Dirichlet environment
We consider random walks in a random environment given by i.i.d. Dirichlet distributions at each vertex of Zd or, equivalently, oriented edge reinforced random walks on Zd . The parameters of the distribution are a 2d-uplet of positive real numbers indexed by the unit vectors of Zd . We prove that, as soon as these weights are nonsymmetric, the random walk is transient in a direction (i.e., it ...
متن کاملCentral Limit Theorems for Open Quantum Random Walks∗
Open Quantum Random Walks, as developed in [1], are the exact quantum generalization of Markov chains on finite graphs or on nets. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random...
متن کاملAn Introduction to Random Walks
In this paper, we investigate simple random walks in n-dimensional Euclidean Space. We begin by defining simple random walk; we give particular attention to the symmetric random walk on the d-dimensional integer lattice Zd. We proceed to consider returns to the origin, recurrence, the level-crossing phenomenon, and the Gambler’s Ruin.
متن کاملRecurrence Theorems in Various Types of Random Walk
In 1921 George Polya famously resolved the question of recurrence versus transience of the simple random walk on integer lattices. In this talk we will study the analogous question for random basic walks, which are random processes related to the problem of graph exploration by a mobile entity. After discussing recurrence results on other generalizations of simple random walks, we’ll resolve th...
متن کاملA monotonicity property for random walk in a partially random environment
We prove a law of large numbers for random walks in certain kinds of i.i.d. random environments in Zd that is an extension of a result of Bolthausen, Sznitman and Zeitouni [4]. We use this result, along with the lace expansion for self-interacting random walks, to prove a monotonicity result for the first coordinate of the speed of the random walk under some strong assumptions on the distributi...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011