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)