Theta Functions and Brownian Motion

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Theta Functions and Brownian Motion Tyrone E. Duncan1 Received: 16 June 2018 / Revised: 30 May 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with su(2). Keywords Macdonald identity · Theta functions · Product–sum formulae for theta functions Mathematics Subject Classification (2010) Primary 58J65 · 22E65; Secondary 60J90 · 22E67

1 Introduction Every student of multivariable calculus is introduced to some mathematical work of C. G. J. Jacobi by the definitions of the Jacobian matrix and the Jacobian determinant. This determinant describes the change of the volume element for integration induced by a smooth bijection of the space. However, probably Jacobi’s most well-known major mathematical work is his investigation of theta functions and more generally elliptic functions [1]. In this paper, it is shown that a theta function for a connected and simply connected compact simple Lie group can be described by a Jacobian determinant. The transformation for this determinant arises from the transformation of the formal

Research supported by NSF Grant DMS 1411412, AFOSR Grant FA9550-12-1-0384, and ARO Grant W911NF-14-10390.

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Tyrone E. Duncan [email protected] Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA

123

Journal of Theoretical Probability

measure of a sequence of independent standard normal random variables (often called the canonical normal distribution) to a pinned Brownian motion conditioned on the initial value and the orbit for the endpoint determined by a special drift term that occurs from the radial part of the Laplacian on the connected and simply connected compact Lie group and its relation to the Laplacian on a maximal torus. This approach to the definition of a theta function allows for the determination of the Jacobi–Macdonald product–sum identity for theta functions as a change of variables from the so-called canonical normal distribution, e.g., [2]. This product expression for a theta function has combinatorial and number-theoretic applications e.g., [3]. Specifically for many dimensions, clearly different direct sums of simple Lie algebras can be defined on the space. Since similar powers in two of these associated infinite product representations have to be equal, various number-theoretic identities are implied. With this stochastic approach to theta functions, an interpretation is also provided for the radial part of the

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