Large systems of path-repellent Brownian motions in a trap at positive temperature

Stefan Adams, Jean-Bernard Bru, Wolfgang Koenig
2006 Electronic Journal of Probability  
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
more » ... ochastic 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 [ABK06] 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. MSC 2000. 60F10; 60J65; 82B10; 82B26.
doi:10.1214/ejp.v11-330 fatcat:t6n4dnqgk5f3xdbwdvjp42v6wi