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 \cite{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.
\bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~K\"onig},
\newblock Large deviations for trapped interacting Brownian particles and
paths, \newblock {\it Ann. Probab.}, to appear (2004).