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Chaos de Wiener et integrale de Feynman. (Wiener chaos and Feynman integral). (French) Zbl 0644.60081
Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 51-71 (1988).
[For the entire collection see Zbl 0635.00013.]
One of the current interpretations of the Feynman integral of a function f consists in taking an analytic continuation to the value \(\sigma^ 2=- i\) of the function \(E[f;\sigma^ 2]\) representing the expectation of f for a Wiener measure with variance parameter \(\sigma^ 2\). In this note we are not at all concerned with analytic continuation, but rather with the meaning of the above sentence: what does it mean to take the expectation of “the same r.v. f” under two Wiener measures which are mutually singular. This requires some regularity on f, and we have chosen to express this regularity in terms of the Wiener chaos expansion of f, which is a whole sequence of symmetric functions on the product spaces \(({\mathbb{R}}_+)^ n\). It turns out that the kind of regularity we need concerns the possibility of defining the traces of these symmetric functions on the diagonals.
The first part of the paper is essentially expository. In the second part we show that the coefficients of the expansion of solutions of one- dimensional smooth stochastic differential equations do possess traces of all orders.
Reviewer: Y.Z.Hu

60J65 Brownian motion
60H05 Stochastic integrals
Zbl 0635.00013
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