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Axioms 2016, 5(2), 12; doi:10.3390/axioms5020012

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

1,†
and
2,†,*
1
Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
2
Department of Mathematics, Hampton University, Hampton, VA 23668, USA
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: Humberto Bustince
Received: 8 April 2016 / Revised: 11 May 2016 / Accepted: 11 May 2016 / Published: 17 May 2016
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Abstract

We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. 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. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further 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. View Full-Text
Keywords: canonical commutation relations; representations; unbounded operators; closable operator; unbounded derivations; spectral theory; duality; Gaussian fields; probability space; stochastic processes; discrete time; path-space measure; stochastic calculus canonical commutation relations; representations; unbounded operators; closable operator; unbounded derivations; spectral theory; duality; Gaussian fields; probability space; stochastic processes; discrete time; path-space measure; stochastic calculus
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Jorgensen, P.; Tian, F. Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus. Axioms 2016, 5, 12.

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