# Wiener measure

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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)
How to Cite This Entry:
Wiener measure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_measure&oldid=49220
This article was adapted from an original article by A.V. Skorokhod (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article