Large deviations for many Brownian bridges with symmetrised initial-terminal condition
Stefan Adams and Wolfgang König
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Submission date: 30. Mar. 2006
published in: Probability theory and related fields, 142 (2008) 1-2, p. 79-124
DOI number (of the published article): 10.1007/s00440-007-0099-5
MSC-Numbers: 60F10, 60J65, 82B10, 81S40
Keywords and phrases: Brownian motions, symmetrised distribution, large deviations, systems of Bosons
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Consider a large system of N Brownian motions in with some non-degenerate initial measure on some fixed time interval with symmetrised initial-terminal condition. That is, for any i, the terminal location of the i-th motion is affixed to the initial point of the -th motion, where is a uniformly distributed random permutation of . Such systems play an important role in quantum physics in the description of Boson systems at positive temperature .
In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Dirac measures in the N paths) and of the mean of the normalised occupation measures of the N motions in terms of large deviations principles. The rate functions are given as variational formulas involving certain entropies and Fenchel-Legendre transforms. Consequences are drawn for asymptotic independence statements and laws of large numbers.
In the special case related to quantum physics, our rate function for the occupation measures turns out to be equal to the well-known Donsker-Varadhan rate function for the occupation measures of one motion in the limit of diverging time. This enables us to prove a simple formula for the large-N asymptotic of the symmetrised trace of , where is an N-particle Hamilton operator in a trap.