# Wiener integral

An abstract integral of Lebesgue type over sets in an infinite-dimensional function space of functionals defined on these sets. Introduced by N. Wiener in the nineteentwenties in connection with problems on Brownian motion , .

Let $C _ {0}$ be the vector space of continuous real-valued functions $x$ defined on $[ 0, 1]$ such that $x( 0) = 0$, with norm

$$\| x \| = \max _ {t \in [ 0, 1] } | x ( t) |.$$

The set

$$Q = \{ {x \in C _ {0} } : { a _ {i} < x ( t _ {i} ) \leq b _ {i} ,\ 0 = t _ {0} < t _ {1} < \dots < t _ {n} = 1 } \}$$

is called a quasi-interval of this space. Here, $a _ {i}$ and $b _ {i}$ may be equal to $- \infty$ and $+ \infty$, respectively, but then the symbol $<$ must replace $\leq$. The whole space $C _ {0} = \{ {x ( t) } : {- \infty < x ( 1) < + \infty } \}$ is an example of a quasi-interval.

The Wiener measure of a quasi-interval $Q$ is the number

$$\mu _ {W} ( Q) = \ { \frac{1}{\sqrt {\pi ^ {n} \prod _ { i = 1 } ^ { n } ( t _ {i} - t _ {i-} 1 ) }} } \int\limits _ { a _ 1 } ^ {b _ 1 } \dots \int\limits _ { a _ n } ^ {b _ n} e ^ {- L _ {n} } dx _ {n} \dots dx _ {1} ,$$

where

$$L _ {n} = \sum _ {j = 1 } ^ { n } \frac{( x _ {j} - x _ {j-} 1 ) ^ {2} }{t _ {j} - t _ {j-} 1 }$$

and $x _ {j} = x ( t _ {j} )$. This measure extends to a $\sigma$- additive measure on the Borel field of sets generated by the quasi-intervals, known also as Wiener measure.

Let $F$ be a functional defined on $C _ {0}$ that is measurable with respect to the measure $\mu _ {W}$. The Lebesgue-type integral

$$\int\limits _ { C _ 0 } F ( x) d \mu _ {W} ( x)$$

is known as the Wiener integral, or as the integral with respect to the Wiener measure, of the functional $F$. If $E \subset C _ {0}$ is measurable, then

$$\int\limits _ { E } F ( x) d \mu _ {W} ( x) = \ \int\limits _ { C _ 0 } F ( x) \chi _ {E} ( x) d \mu _ {W} ( x) ,$$

where $\chi _ {E}$ is the characteristic function of the set $E$.

Wiener's integral displays several properties of the ordinary Lebesgue integral. In particular, a functional which is bounded and measurable on a set $E$ is integrable with respect to the Wiener measure on this set and if, in addition, the functional $F$ is continuous and non-negative, then

$$\int\limits _ { C _ 0 } F ( x) d \mu _ {W} ( x) =$$

$$= \ \lim\limits _ {n \rightarrow \infty } \frac{1}{\sqrt { {\pi ^ {n} \prod _ {i = 1 } ^ { n } ( t _ {i} - t _ {i-} 1 ) } }} \int\limits _ { \mathbf R } ^ {n} \frac{F _ {n} ( x _ {1} \dots x _ {n} ) }{e ^ {L} _ {n} } dx _ {1} \dots dx _ {n} ,$$

where $F _ {n} ( x _ {1} \dots x _ {n} )$ is the value of $F$ at linear interpolation of $x( t)$ between points $( t _ {i} , x _ {i} \equiv x( t _ {i} ))$.

The computation of a Wiener integral presents considerable difficulties, even for the simplest functionals. The task may sometimes be reduced to solving a single differential equation .

There is a method by which Wiener's integral may be approximately computed through approximating it by finite-dimensional Stieltjes integrals of a high multiplicity (cf. Stieltjes integral).

How to Cite This Entry:
Wiener integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_integral&oldid=49219
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article