 TOPICS # Brownian Motion

A real-valued stochastic process is a Brownian motion which starts at if the following properties are satisfied:

1. .

2. For all times , the increments , , ..., , are independent random variables.

3. For all , , the increments are normally distributed with expectation value zero and variance .

4. The function is continuous almost everywhere. The Brownian motion is said to be standard if .

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 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 . 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 which start at , one can define a stochastic process by Such a is called a -dimensional Brownian motion which starts at .

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

## Explore with Wolfram|Alpha ## References

Mörters, P. and Peres, Y. "Brownian Motion." 2008. http://www.stat.berkeley.edu/~peres/bmbook.pdf.

## Cite this as:

Stover, Christopher. "Brownian Motion." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/BrownianMotion.html