Sazonov's theorem

In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov ({{lang|ru|Вячесла́в Васи́льевич Сазо́нов}}), is a theorem in functional analysis.

It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if it is a Hilbert–Schmidt operator. The result is also important in the study of stochastic processes and the Malliavin calculus, since results concerning probability measures on infinite-dimensional spaces are of central importance in these fields. Sazonov's theorem also has a converse: if the map is not Hilbert–Schmidt, then it is not γ-radonifying.

Statement of the theorem

Let G and H be two Hilbert spaces and let T : GH be a bounded operator from G to H. Recall that T is said to be γ-radonifying if the push forward of the canonical Gaussian cylinder set measure on G is a bona fide measure on H. Recall also that T is said to be a Hilbert–Schmidt operator if there is an orthonormal basis {{nowrap|{ ei : iI }}} of G such that

: \sum_{i \in I} \| T(e_i) \|_H^2 < + \infty.

Then Sazonov's theorem is that T is γ-radonifying if it is a Hilbert–Schmidt operator.

The proof uses Prokhorov's theorem.

Remarks

The canonical Gaussian cylinder set measure on an infinite-dimensional Hilbert space can never be a bona fide measure; equivalently, the identity function on such a space cannot be γ-radonifying.

See also

  • {{annotated link|Cameron–Martin theorem}}
  • {{annotated link|Girsanov theorem}}
  • {{annotated link|Radonifying function}}
  • {{annotated link|Minlos–Sazonov theorem}}

References

{{reflist}}

  • {{citation|mr=0426084

|last=Schwartz|first= Laurent

|title=Radon measures on arbitrary topological spaces and cylindrical measures.

|series=Tata Institute of Fundamental Research Studies in Mathematics|issue= 6|publisher= Oxford University Press |publication-place=London|year= 1973|pages= xii+393}}

{{Measure theory}}

{{Analysis in topological vector spaces}}

{{Functional analysis}}

Category:Stochastic processes

Category:Theorems in functional analysis

Category:Theorems in measure theory