View source for Wiener measure(2)

The [[Probability distribution|probability distribution]] of a [[Brownian motion|Brownian motion]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100602.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100603.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100604.png" /> is a [[Probability space|probability space]]. The Wiener measure is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100605.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100606.png" />. Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100607.png" /> is a [[Gaussian process|Gaussian process]] such that

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Given a Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w1100609.png" />, one can form a new Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006010.png" /> satisfying:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006011.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006012.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006013.png" />.

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006014.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006015.png" />.

Such a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006016.png" /> is called a continuous version of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006017.png" />.

The Kolmogorov–Prokhorov theorem tells that the probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006018.png" /> of the Brownian motion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006019.png" /> can be introduced in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006020.png" /> of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006021.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006022.png" /> be the topological Borel field (cf. also [[Borel field of sets|Borel field of sets]]) of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006023.png" />. The [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006024.png" /> thus obtained is the Wiener measure space.

The integral of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006025.png" />-measurable functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006026.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006027.png" /> is defined in the usual manner. (See also [[Stochastic integral|Stochastic integral]].)

An elementary and important example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006028.png" />-measurable functional of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006029.png" /> is a stochastic bilinear form, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006031.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006032.png" />-function. It is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006033.png" />. It is, in fact, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006034.png" /> for smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006035.png" />. For a general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006037.png" /> can be approximated by stochastic bilinear forms defined by smooth functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006038.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006040.png" />. And an even more general case has been proposed.

The class of general (non-linear) functionals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006041.png" /> is introduced as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006042.png" /> be the [[Hilbert space|Hilbert space]] of all complex-valued, square-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006043.png" />-integrable functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006044.png" />. Then, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006045.png" /> admits a direct sum decomposition ([[Fock space|Fock space]])

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The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006047.png" /> is spanned by the Fourier–Hermite polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006048.png" />, which are of the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006051.png" /> is a complete [[Orthonormal system|orthonormal system]] in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006052.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006053.png" /> can be interpreted as the space of multiple Wiener integrals of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w110/w110060/w11006054.png" />, due to K. Itô.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Hida,   "Brownian motion" , ''Applications of Mathematics'' , '''11''' , Springer  (1980)</TD></TR></table>