On bounded pseudodifferential operators in Wiener spaces

TL;DR

This paper extends the Weyl calculus to infinite-dimensional Wiener spaces, defining bounded pseudodifferential operators using stochastic extensions and hybrid operators, bridging classical and infinite-dimensional analysis.

Contribution

It introduces a novel framework for defining Weyl operators in Wiener spaces, combining stochastic extensions and hybrid operators, expanding pseudodifferential calculus to infinite dimensions.

Findings

Defined Weyl operators via stochastic extensions on Wiener spaces.

Established continuity of the operators on L^2 spaces.

Linked the construction to classical pseudodifferential operators through examples.

Abstract

We aim at extending the definition of the Weyl calculus to an infinite dimensional setting, by replacing the phase space $ \mathbb{R}^{2n}$ by $B^2$, where $(i,H,B)$ is an abstract Wiener space. A first approach is to generalize the integral definition using the Wigner function. The symbol is then a function defined on $B^2$ and belonging to a $L^1$ space for a gaussian measure, the Weyl operator is defined as a quadratic form on a dense subspace of $L^2(B)$. For example, the symbol can be the stochastic extension on $B^2$, in the sense of L. Gross, of a function $F$ which is continuous and bounded on $H^2$. In the second approach, this function $F$ defined on $H^2$ satisfies differentiability conditions analogous to the finite dimensional ones. One needs to introduce hybrid operators acting as Weyl operators on the variables of finite dimensional subset of $H$ and as Anti-Wick operators on the rest of the variables. The final Weyl operator is then defined as a limit and it is continuous on a $L^2$ space. Under rather weak conditions, it is an extension of the operator defined by the first approach. We give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject.