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)