Journal of the Mathematical Society of Japan

Holomorphic functions and the Itô chaos


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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.

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

First available in Project Euclid: 27 October 2015

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Zentralblatt MATH identifier

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.

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


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

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