Representations of the canonical commutation relations--algebra and the operators of stochastic calculus

TL;DR

This paper explores a family of CCR-algebra representations, linking them to Gaussian stochastic calculus and deriving operators of Malliavin calculus through an algebraic approach, simplifying the handling of unbounded operators.

Contribution

It introduces admissible CCR-representations that connect to stochastic calculus, providing explicit formulas and a duality framework that simplifies operator domain issues.

Findings

Established a correspondence between admissible CCR-representations and Gaussian stochastic calculus.

Derived explicit formulas for stochastic calculus operators from an algebraic perspective.

Produced new results in multi-variable operator theory using CCR representation theory.

Abstract

We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. And we thus get the operators of Malliavin's calculus of variation from a more algebraic approach than is common. And we obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.