Editing Brownian motion (section)

===Lévy characterisation===

The French mathematician [[Paul Lévy (mathematician)|Paul Lévy]] proved the following theorem, which gives a necessary and sufficient condition for a continuous {{math|'''R'''<sup>''n''</sup>}}-valued stochastic process {{math|''X''}} to actually be {{mvar|n}}-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.

Let {{math|1=''X'' = (''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>)}} be a continuous stochastic process on a [[probability space]] {{math|(Ω, Σ, '''P''')}} taking values in {{math|'''R'''<sup>''n''</sup>}}. Then the following are equivalent:

# {{math|''X''}} is a Brownian motion with respect to {{math|'''P'''}}, i.e., the law of {{math|''X''}} with respect to {{math|'''P'''}} is the same as the law of an {{mvar|n}}-dimensional Brownian motion, i.e., the [[push-forward measure]] {{math|''X''<sub>∗</sub>('''P''')}} is [[classical Wiener measure]] on {{math|''C''<sub>0</sub>({{closed-open|0, ∞}}; '''R'''<sup>''n''</sup>)}}.
# both
## {{math|''X''}} is a [[martingale (probability theory)|martingale]] with respect to {{math|'''P'''}} (and its own [[natural filtration]]); and
## for all {{math|1 ≤ ''i'', ''j'' ≤ ''n''}}, {{math|''X''<sub>''i''</sub>(''t'') ''X''<sub>''j''</sub>(''t'') − ''δ''<sub>''ij''</sub> ''t''}} is a martingale with respect to {{math|'''P'''}} (and its own [[natural filtration]]), where {{math|''δ''<sub>''ij''</sub>}} denotes the [[Kronecker delta]].