1. Strassens Law of the Iterated Logarithm.
Let P be the Wiener measure on the space = C [0, ) of continuos functions on [0, )
that starts at time 0 from the point 0. For 3 we dene the rescaled proces
Large Deviations
Spring 2012
February 2, 2012
2
Chapter 1
Introduction
1.1
Outline.
We will examine the theory of large deviations through three concrete examples. We will
work them out fully and in t
Chapter 5
Self diusion.
5.1
Motion of a tagged particle.
Let us look at the simple exclusion process in equilibrium on Zd at density . The distribution is the Bernoulli distribution dened by [ (x) = 1
1. Sanovs Theorem
Here we consider a sequence of i.i.d. random variables with values in some complete
separable metric space X with a common distribution . Then the sample distribution
n =
1
n
n
xj
j
The simplest example is a system of noninteracting particles undergoing independent
motions. For instance we could have on T3 , L N 3 particles all behaving like independent
Browninan Particles. If th
Chapter 6
Non-gradient systems
6.1
Two color system
Let us look at the situation where there are two types of particles. Type 1 and type 2.
d
The state space is N = cfw_0, 1, 2N . We dene
1 (x) = 1 if
1. Markov Chains.
Let us switch from independent to dependent random variables. Suppose X1 , , Xn ,
is a Markov Chain on a nite state spacce F consisting of points x F . The Markov Chain
will be assu
Chapter 3
Large Time.
3.1
Introduction.
The goal of this chapter is to prove the following theorem.
Theorem 3.1.1. Let Sn = X1 + X2 + + Xn be the nearest neighbor random walk on
Zd with each Xi = er w
LARGE DEVIATIONS
S. R. S. VARADHAN
Date : Spring, 2010.
1
2
S. R. S. VARADHAN
1. Introduction
The theory of large deviations deals with rates at which probabilities of certain events
decay as a natura
Chapter 4
Hydrodynamic Scaling
4.1
From Classical Mechanics to Euler Equations.
The basic example of hydrodynamical scaling is naturally hydrodynamics itself. Let us
start with a collection of N
3 cl
1. Empirical Processes.
Let us return to the situation of a Markov chain on a nite state space X , having (x, y )
as the probability of transition from the state x to the state y . We saw before that
1. Application.
We now present an application of the methods of large deviations. Let us consider the
usual random walk on Z d . Suppose certain sites in Z d were traps. The sites that are traps
are c