Brownian motion
Definition.
Onedimensional Brownian motion^{} is a stochastic process^{} $W(t)$, defined for $t\in [0,\mathrm{\infty})$ such that

1.
$W(0)=0$ almost surely

2.
The sample paths $t\mapsto W(t)$ are almost surely continuous.

3.
For any finite sequence^{} of times $$, the increments
$$W({t}_{1})W({t}_{0}),W({t}_{2})W({t}_{1}),\mathrm{\dots},W({t}_{n})W({t}_{n1})$$ are independent.

4.
For any times $$, $W(t)W(s)$ is normally distributed with mean zero and variance $ts$.
Definition.
A $d$dimensional Brownian motion is a stochastic process $W(t)=({W}_{1}(t),\mathrm{\dots},{W}_{d}(t))$ in ${\mathbb{R}}^{d}$ whose coordinate processes ${W}_{i}(t)$ are independent onedimensional Brownian motions.
Title  Brownian motion 

Canonical name  BrownianMotion 
Date of creation  20130322 15:12:46 
Last modified on  20130322 15:12:46 
Owner  skubeedooo (5401) 
Last modified by  skubeedooo (5401) 
Numerical id  16 
Author  skubeedooo (5401) 
Entry type  Definition 
Classification  msc 60J65 
Synonym  Wiener process 
Related topic  WienerMeasure 
Related topic  StochasticCalculusAndSDE 