Brownian Motion: The Limit of a Random Walk
A Brief Intuitive Explanation
Introduction
In the 19th century, a Scottish botanist named Robert Brown noticed that pollen grains which were suspended in water displayed random movements. His work revealed this random movement is in fact a general property of matter in that state, and this phenomena was termed Brownian Motion.
Brownian Motion has numerous applications like Physics, Engineerging, Finance, Economics, etc. Most notably, Albert Einstein used Brownian Motion in his work to prove the existence of atoms. To build an intuition for what exactly Brownian motion is, think of it as some kind of random movement in space.
Random Movements in Space
We want to generate some kind of random movement in space. A simple way to do this is to generate a random walk.
Suppose that we toss a coin infintely many times, and record the result of each of the i tosses. We label the outcome of the i-th toss X ᵢ, where X ᵢ takes one of the values H or T. Now for each i, define the random variable ζ ᵢ by setting
So ζ = (ζ₁, ζ₂, ζ₃, … ) is a sequence of -1’s and 1’s representing the outcome of our coin tosses. Now since each coin toss is independent of the previous coin tosses, and each coin coin toss has the same probability distribution, we say that the random variables ζ ᵢ are independently and identically distributed (abbreviated as iid).
Generating a Random Walk from ζ
Let’s take a Random Walk. We’ll denote our position at time n by Sₙ. Assume that the point at which we begin our walk is 0, i.e. S₀=0. On the first step, we will either go up, or down, depending on the value of the ζ₁. So our position at the first step is S₁=ζ₁.
For each step, say step n, the value of ζ ₙ tells us whether to walk one step in the positive direction, or one step in the negative direction. So for the second step, S₂ will either be S₁+1 or S₁ -1, depending on ζ₂. So S₂=S₁+ζ₂.
In general, the position (or state) of the random walk at the n-th step is given by the equation:
Thus, we define a random walk by the sequence S=(S₀, S₁, S₂,…). Note that the position of a random walk at some time is determined by the previous position.
The Limit of a Random Walk
Although Random Walks do a good job of representing random movement space, those movements are in discrete steps. Movements in nature (such as that of a particle dispensed in a liquid, or the stock market) are continuous.
So the next logical step towards defining Brownian motion is to make random walks continuous. There is a simple way to do this using a version of the Central Limit Theorem called Donsker’s invariance principle (sometimes referred to as the functional central limit theorem).
The exact statement of the Theorem is given below, and is somewhat technical, but it is the basis for which one constructs Brownian motion from a random walk.
Donsker’s Theorem.
Let X₁,…,Xₙ be independent identically distributed random variables with mean 0 and variance 1. Let S be defined as above. Define the diffusively rescaled random walk by the equation
where t is in the interval [0,1]. Then for each t, there exists a limit W(t), such that
Explanation of the Theorem
The limit W(t) is a called a Wiener Process, and is in fact, Brownian motion, so we are done! By rescaling our Random walk, we “squeeze” it into the interval [0,1], which in a sense, makes our random walk continuous.
We can visualize what Wₙ(t) looks like as n gets large. The interactive plot below is available for free on my Github.
Summary
We constructed Simple Brownian motion as the limit of a one dimensional random walk. Wiener Processes, and Rescaled Random Walks were briefly introduced. Although we constrained ourselves to one dimension, we can simulate Brownian motion in any number of dimensions, and even on surfaces. Also, this construction is based on Random Walks, but there are other ways to construct Brownian Motion with martingales (the Levy Characterization). Feel free to contact me if you have any questions, or comments!
