Brownian Motion

A real-valued stochastic process {B(t):t>=0} is a Brownian motion which starts at x in R if the following properties are satisfied:

1. B(0)=x.

2. For all times 0=t_0<=t_1<=t_2<=...<=t_n, the increments B(t_k)-B(t_(k-1)), k=1, ..., n, are independent random variables.

3. For all t>=0, h>0, the increments B(t+h)-B(t) are normally distributed with expectation value zero and variance h.

4. The function t|->B(t) is continuous almost everywhere. The Brownian motion B(t) is said to be standard if B(0)=0.

It is easily shown from the above criteria that a Brownian motion has a number of unique natural invariance properties including scaling invariance and invariance under time inversion. Moreover, any Brownian motion B(t) satisfies a law of large numbers so that


almost everywhere. Moreover, despite looking ill-behaved at first glance, Brownian motions are Hölder continuous almost everywhere for all values alpha<1/2. Contrarily, any Brownian motion is nowhere differentiable almost surely.

The above definition is extended naturally to get higher-dimensional Brownian motions. More precisely, given independent Brownian motions B_1,...,B_d which start at x_1,...,x_d, one can define a stochastic process {beta(t):t>=0} by

 beta(t)=[B_1(t); |; B_d(t)].

Such a beta is called a d-dimensional Brownian motion which starts at (x_1,...,x_d)^(T) in R^d.

See also

Hölder Condition, Independent Statistics, Law of Large Numbers, Normal Distribution, Random Variable, Random Walk, Random Walk-1-Dimensional, Stochastic Process, Wiener Sausage

This entry contributed by Christopher Stover

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Mörters, P. and Peres, Y. "Brownian Motion." 2008.

Cite this as:

Stover, Christopher. "Brownian Motion." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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