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 process
x (t) =
1
x(t).
log log
As , x (t) will go to 0 in
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 the process develop the subject.
The rst example is the
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] = with cfw_ (x) : x Zd
being independent. Let us sup
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 =1
maps X n M(X ) and the product measure n will genera
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 the initial conguration of the L particles is such that t
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 there is a type 1 particle at x and 0 otherwise
2 (x)
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 assumed to have a stationary transition probability given b
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 with probability 21 where cfw_er are the unit vectors i
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 natural parameter in the problem varies. It is best to think
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 classical particles in a large periodic cube of side
3 .
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
lim
n
1
log Ex exp[V (X1 ) + V (X2 ) + + V (Xn )] = (V
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 chosen randomly and the probability of a given site bein