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Opérateurs filtrés et chaînes de tribus invariantes sur un espace probabilisé dénombrable. (Filtered operators and chains of invariant \(\sigma\)-fields on a countable probability space). (French) Zbl 0657.60073

Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 197-213 (1988).
[For the entire collection see Zbl 0635.00013.]
The authors solve a conjecture of C. Dellacherie about the characterization of symmetric operators on \(L^ 2(\Omega,{\mathcal F},\mu)\) as stochastic integral operators. They give an algorithmic solution to this problem in the case of a countable space \(\Omega\). The main tool is the use of maximal chains of \(\sigma\)-fields which are invariant under the action of the operator. Finally, they obtain a geometric criterion for these operators: their matrices define a supermetric geometry.
Reviewer: M.Chaleyat-Maurel

MSC:

60H05 Stochastic integrals

Citations:

Zbl 0635.00013
Full Text: Numdam EuDML