Journal of the Mathematical Society of Japan

Holomorphic functions and the Itô chaos

Bruce K. DRIVER

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Abstract

This paper is concerned with the characterization of spaces of square integrable holomorphic functions on a complex manifold, $G$, in terms of the derivatives of the function at a fixed point $o\in G$. The reproducing kernel properties of square integrable holomorphic functions are reviewed and a number of examples are given. These examples include square integrable holomorphic functions relative to Gaussian measures on complex Euclidean spaces along with their generalizations to heat kernel measures on complex Lie groups. These results are intimately related to the Itô's chaos expansion in stochastic analysis and to the Fock space description of free quantum fields in physics.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1449-1484.

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951156

Digital Object Identifier
doi:10.2969/jmsj/06741449

Mathematical Reviews number (MathSciNet)
MR3417503

Zentralblatt MATH identifier
1336.32019

Subjects
Primary: 32W30: Heat kernels in several complex variables 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 35H20: Subelliptic equations 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

Keywords
Itô chaos Fock space Taylor map complex groups hypo-elliptic heat Kerenels

Citation

DRIVER, Bruce K. Holomorphic functions and the Itô chaos. J. Math. Soc. Japan 67 (2015), no. 4, 1449--1484. doi:10.2969/jmsj/06741449. https://projecteuclid.org/euclid.jmsj/1445951156


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