Chapter 1

One-Dimensional

Brownian Motion

1.1. Some motivation

The biologist Robert Brown noticed almost two hundred years ago that bits

of pollen suspended in water undergo chaotic behavior. The bits of pollen

are much more massive than the molecules of water, but of course there are

many more of these molecules than there are bits of pollen. The chaotic

motion of the pollen is the result of many infinitesimal jolts by the water

molecules. By the central limit theorem (CLT), the law of the motion of

the pollen should be closely related to the normal distribution. We now call

this law Brownian motion.

During the past half century or so, Brownian motion has turned out

to be a very versatile tool for both theory and applications. As we will

see in Chapter 6, it provides a very elegant and general treatment of the

Dirichlet problem, which asks for harmonic functions on a domain with

prescribed boundary values. It is also the main building block for the theory

of stochastic calculus, which is the subject of Chapter 5. Via stochastic

calculus, it has played an important role in the development of financial

mathematics.

As we will see later in this chapter, Brownian paths are quite rough —

they are of unbounded variation in every time interval. Therefore, integrals

with respect to them cannot be defined in the Stieltjes sense. A new type

of integral must be defined, which carries the name of K. Itˆ o, and more

recently, of W. Doeblin. This new integral has some unexpected properties.

Here is an example: If B(t) is standard Brownian motion at time t with

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