# Wiener measure(2)

The probability distribution of a Brownian motion $B ( t, \omega )$, $t \geq 0$, $\omega \in \Omega$, where $( \Omega, {\mathcal B}, {\mathsf P} )$ is a probability space. The Wiener measure is denoted by $m$ or $\mu ^ {W}$. Brownian motion $B$ is a Gaussian process such that

$${\mathsf E} ( B ( t ) ) \equiv 0, \quad {\mathsf E} ( B ( t ) \cdot B ( s ) ) = \min ( t, s ) .$$

Given a Brownian motion $B ( t, \omega )$, one can form a new Brownian motion ${\overline{B}\; } ( t, \omega )$ satisfying:

i) ${\overline{B}\; } ( t, \omega )$ is continuous in $t$ for almost all $\omega$.

ii) ${\mathsf P} ( {\overline{B}\; } ( t, \omega ) = B ( t, \omega ) ) = 1$ for every $t$.

Such a process ${\overline{B}\; } ( t, \omega )$ is called a continuous version of $B ( t, \omega )$.

The Kolmogorov–Prokhorov theorem tells that the probability distribution $m$ of the Brownian motion $B ( t )$ can be introduced in the space $C = C [ 0, \infty )$ of all continuous functions on $[ 0, \infty )$.

Let ${\mathcal B}$ be the topological Borel field (cf. also Borel field of sets) of subsets of $C$. The measure space $( C, {\mathcal B}, m )$ thus obtained is the Wiener measure space.

The integral of a ${\mathcal B}$- measurable functional on $C$ with respect to $m$ is defined in the usual manner. (See also Stochastic integral.)

An elementary and important example of a ${\mathcal B}$- measurable functional of $y \in C$ is a stochastic bilinear form, given by $\langle { {\dot{y} } , f } \rangle$, where $f$ is an $L _ {2} [ 0, \infty )$- function. It is usually denoted by $f ( y )$. It is, in fact, defined by $- \int _ {0} ^ \infty {y ( t ) {\dot{f} } ( t ) } {dt }$ for smooth functions $f$. For a general $f$, $f ( y )$ can be approximated by stochastic bilinear forms defined by smooth functions $f$. An integral of this type is called a Wiener integral. Under certain restrictions, such as non-anticipation, the integral can be extended to the case where the integrand is a functional of $t$ and $y$. And an even more general case has been proposed.

The class of general (non-linear) functionals of $y$ is introduced as follows. Let $H$ be the Hilbert space of all complex-valued, square- $m$- integrable functionals on $C$. Then, $H$ admits a direct sum decomposition (Fock space)

$$H = \oplus _ { n } {\mathcal H} _ {n} .$$

The subspace ${\mathcal H} _ {n}$ is spanned by the Fourier–Hermite polynomials of degree $n$, which are of the form

$$\prod _ { j } H _ {n _ {j} } \left ( { \frac{\left \langle {y,f _ {j} } \right \rangle }{\sqrt 2 } } \right ) ,$$

where $\Sigma n _ {j} = n$ and $\{ f _ {j} \}$ is a complete orthonormal system in the Hilbert space $L _ {2} [ 0, \infty )$. The space $H _ {n}$ can be interpreted as the space of multiple Wiener integrals of degree $n$, due to K. Itô.

#### References

 [a1] R. Cameron, W.T. Martin, "The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals" Ann. of Math. (2) , 48 pp. 385–392 [a2] T. Hida, "Brownian motion" , Applications of Mathematics , 11 , Springer (1980)
How to Cite This Entry:
Wiener measure(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wiener_measure(2)&oldid=49221
This article was adapted from an original article by T. Hida (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article