# Wiener measure

The probability measure on the space of continuous real-valued functions on the interval , defined as follows. Let be an arbitrary sample of points from and let be Borel sets on the real line. Let denote the set of functions for which , . Then

(*) |

where

Using the theorem on the extension of a measure it is possible to define the value of the measure on all Borel sets of 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 on . The coordinate process 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. A.V. Skorokhod (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Wiener_measure&oldid=17302