Exponential integrability and application to stochastic quantization. (English) Zbl 0903.60046

This paper studies the problem of exponential integrability of certain Wiener functionals, namely multiple stochastic integrals in a Wiener chaos. Elements of the first Wiener chaos always have exponential moments of all orders, while for elements in the second chaos, this is only true if they are negative definite. It is proved that non-zero elements of the third chaos never have any exponential moment. The case of chaos of higher order is much more difficult in geneal. Some results are proved, for a special class of Wiener functionals, constructed from the free field in a bounded connected domain in \(\mathbb{R}^d\). These results are applied to stochastic quantization, as described for example in the paper by G. Jona-Lasinio and P. K. Mitter [Commun. Math. Phys. 101, 409-436 (1985; Zbl 0588.60054)]. The idea of stochastic quantization is to realize a measure \(\mu\) as the invariant measure of some Markov process. In the case under investigation, the measure is of the form \(d\mu= \exp(-I_n(f_n))d\mu_0\) where \(\mu_0\) is a Gaussian measure and \(I_n(f_n)\) is an element in the \(n\)th chaos of this measure. The Markov process with invariant measure \(\mu\) is realized as the solution to an infinite-dimensional stochastic differential equation, whose existence is proved.
Reviewer: Ph.Biane (Paris)


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
81S20 Stochastic quantization


Zbl 0588.60054
Full Text: DOI