Theta Functions and Brownian Motion

Tyrone E. Duncan ()
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Tyrone E. Duncan: University of Kansas

Journal of Theoretical Probability, 2021, vol. 34, issue 1, 81-89

Abstract: 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; Primary 58J65; 22E65; Secondary 60J90; 22E67 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-019-00977-y

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