Large systems of path-repellent Brownian motions in a trap at positive temperature
Stefan Adams, Jean-Bernard Bru, and Wolfgang König
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Submission date: 13. Dec. 2005 (revised version: June 2006)
published in: Electronic journal of probability, 11 (2006) 19, p. 460-485
DOI number (of the published article): 10.1214/EJP.v11-330
MSC-Numbers: 60F10, 60J65, 82B10, 82B26
Keywords and phrases: interacting brownian motions, brownian intersection local times, large deviations, occupation measure, gross-pitaevskii formula
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We study a model of N mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the N paths. In fact, this interaction is an N-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes.
The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross-Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional.
This study is a continuation of the analysis in [ABK04] where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions.