Wiener measure

The probability measure $ \mu _ {W} $ on the space $ C[ 0, 1] $ of continuous real-valued functions $ x $ on the interval $ [ 0, 1] $, defined as follows. Let $ 0 < t _ {1} < \dots < t _ {n} \leq 1 $ be an arbitrary sample of points from $ [ 0, 1] $ and let $ A _ {1} \dots A _ {n} $ be Borel sets on the real line. Let $ C( t _ {1} \dots t _ {n} ; A _ {1} \dots A _ {n} ) $ denote the set of functions $ x \in C[ 0, 1] $ for which $ x( t _ {k} ) \in A _ {k} $, $ k = 1 \dots n $. Then

$$ \tag{* } \mu _ {W} ( C ( t _ {1} \dots t _ {n} ; A _ {1} \dots A _ {n} )) = $$

$$ = \ \int\limits _ { A _ 1 } p ( t _ {1} , x _ {1} ) dx _ {1} \int\limits _ { A _ 2 } p ( t _ {2} - t _ {1} , x _ {2} - x _ {1} ) dx _ {2} \dots $$

$$ {} \dots \int\limits _ { A _ n } p ( t _ {n} - t _ {n-} 1 , x _ {n} - x _ {n-} 1 ) dx _ {n} , $$

where

$$ p ( t, x) = { \frac{1}{\sqrt {2 \pi t } } } e ^ {- x ^ {2} / 2 t } . $$

Using the theorem on the extension of a measure it is possible to define the value of the measure $ \mu _ {W} $ on all Borel sets of $ C[ 0, 1] $ on the basis of equation (*).

Comments

The Wiener measure was introduced by N. Wiener [a1] in 1923; it was the first major extension of integration theory beyond a finite-dimensional setting. The construction outlined above extends easily to define Wiener measure $ \mu _ {W} $ on $ C [ 0, \infty ) $. The coordinate process $ x( t) $ is then known as Brownian motion or the Wiener process. Its formal derivative "dxt/dt" is known as Gaussian white noise.

References

[a1]	N. Wiener, "Differential space" J. Math. & Phys. , 2 (1923) pp. 132–174
[a2]	T. Hida, "Brownian motion" , Springer (1980)
[a3]	I. Karatzas, S.E. Shreve, "Brownian motion and stochastic calculus" , Springer (1988)
[a4]	L. Partzsch, "Vorlesungen zum eindimensionalen Wienerschen Prozess" , Teubner (1984)
[a5]	J. Yeh, "Stochastic processes and the Wiener integral" , M. Dekker (1973)
[a6]	S. Albeverio, J.E. Fenstad, R. Høegh-Krohn, T. Lindstrøm, "Nonstandard methods in stochastic analysis and mathematical physics" , Acad. Press (1986)