Brownian Motion
A real-valued stochastic process ![<span style=]()
{B(t):t>=0}" src="https://mathworld.wolfram.com/images/equations/BrownianMotion/Inline1.gif" style="height:15px; width:70px" /> 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 ![<span style=]()
{beta(t):t>=0}" src="https://mathworld.wolfram.com/images/equations/BrownianMotion/Inline19.gif" style="height:15px; width:69px" /> by
Such a 
is called a 
-dimensional Brownian motion which starts at 
.
REFERENCES:
Mörters, P. and Peres, Y. "Brownian Motion." 2008. http://www.stat.berkeley.edu/~peres/bmbook.pdf.
