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

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 .