Wiener measure(2)
The probability distribution of a Brownian motion 
, 
, 
, where 
is a probability space. The Wiener measure is denoted by 
or 
. Brownian motion 
is a Gaussian process such that
 |
Given a Brownian motion 
, one can form a new Brownian motion 
satisfying:
i) 
is continuous in 
for almost all 
.
ii) 
for every 
.
Such a process 
is called a continuous version of 
.
The Kolmogorov–Prokhorov theorem tells that the probability distribution 
of the Brownian motion 
can be introduced in the space 
of all continuous functions on 
.
Let 
be the topological Borel field (cf. also Borel field of sets) of subsets of 
. The measure space 
thus obtained is the Wiener measure space.
The integral of a 
-measurable functional on 
with respect to 
is defined in the usual manner. (See also Stochastic integral.)
An elementary and important example of a 
-measurable functional of 
is a stochastic bilinear form, given by 
, where 
is an 
-function. It is usually denoted by 
. It is, in fact, defined by 
for smooth functions 
. For a general 
, 
can be approximated by stochastic bilinear forms defined by smooth functions 
. 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 
and 
. And an even more general case has been proposed.
The class of general (non-linear) functionals of 
is introduced as follows. Let 
be the Hilbert space of all complex-valued, square-
-integrable functionals on 
. Then, 
admits a direct sum decomposition (Fock space)
 |
The subspace 
is spanned by the Fourier–Hermite polynomials of degree 
, which are of the form
 |
where 
and 
is a complete orthonormal system in the Hilbert space 
. The space 
can be interpreted as the space of multiple Wiener integrals of degree 
, 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) |
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