Bochner-Minlos theorem

The Bochner-Minlos theorem , named after Salomon Bochner and Robert Adolfowitsch Minlos , makes a statement about the connection between probability measures and characteristic functions in nuclear spaces . According to the sentence, there is a one-to-one connection between the two concepts; H. a characteristic function can be calculated for each probability measure, and conversely, a uniquely determined probability measure is obtained from every characteristic function. Both objects are linked to one another by a Fourier transformation .

The theorem is a generalization of Bochner's theorem about characteristic functions . ${\ displaystyle \ mathbb {R} ^ {d}}$

Statement of the sentence

For every characteristic function on real nuclear space there is a probability measure such that ${\ displaystyle \ varphi: E \ rightarrow \ mathbb {C}}$ ${\ displaystyle E}$ ${\ displaystyle \ nu}$

{\ displaystyle \ varphi (\ xi) = \ int _ {E ^ {\ prime}} \ exp (i \ langle \ xi, x \ rangle) d \ nu (x), \ quad \ xi \ in E}

is. Conversely, the Fourier transform of a probability measure is always a characteristic function . ${\ displaystyle \ nu}$ ${\ displaystyle E ^ {\ prime}}$ ${\ displaystyle E}$

Here are the strong dual space of and the dual pairing . ${\ displaystyle E ^ {\ prime}}$ ${\ displaystyle E}$ ${\ displaystyle \ langle \ xi, x \ rangle}$

example

If one considers the Gaussian function in the one-dimensional case

{\ displaystyle x \ mapsto \ exp \ left (- {\ frac {1} {2}} x ^ {2} \ right)}

as a characteristic function, the associated probability measure is the measure with Gaussian density ${\ displaystyle \ nu}$

{\ displaystyle d \ nu (y) = \ exp \ left (- {\ frac {1} {2}} y ^ {2} \ right) dy}

.

This result can be generalized for the infinite-dimensional case. The black space is an example of an (infinite-dimensional) nuclear space. There you can see the characteristic function ${\ displaystyle S (\ mathbb {R})}$

{\ displaystyle \ varphi: S (\ mathbb {R}) \ rightarrow \ mathbb {R}, \ varphi (\ xi) = \ exp \ left (- {\ frac {1} {2}} \ langle \ xi, \ xi \ rangle \ right)}

define. According to the theorem, there is then a probability measure on the space of the tempered distributions with the properties mentioned above. This measure is in the White Noise Analysis as white noise measure referred. ${\ displaystyle \ mu}$

Individual proof

↑ Obata, Nobuaki: White Noise Calculus and Fock Space , Springer, 1994, ISBN 978-3-540-57985-4 , section 1.5.

Web links

Jordan Bell: The Bochner-Minlos theorem

[1] Obata, Nobuaki: White Noise Calculus and Fock Space , Springer, 1994, ISBN 978-3-540-57985-4 , section 1.5.