  We introduce two probabilistic models for $N$ interacting Brownian motions
moving in a trap in $\mathbb {R}^d$ under mutually repellent forces. The two
models are defined in terms of transformed path measures on finite time
intervals under a trap Hamiltonian and two respective pair-interaction
Hamiltonians. The first pair interaction exhibits a particle repellency, while
the second one imposes a path repellency. We analyze both models in the limit
of diverging time with fixed number $N$ of Brownian motions. In particular, we
prove large deviations principles for the normalized occupation measures. The
minimizers of the rate functions are related to a certain associated operator,
the Hamilton operator for a system of $N$ interacting trapped particles. More
precisely, in the particle-repellency model, the minimizer is its ground state,
and in the path-repellency model, the minimizers are its ground product-states.
In the case of path-repellency, we also discuss the case of a Dirac-type
interaction, which is rigorously defined in terms of Brownian intersection
local times. We prove a large-deviation result for a discrete variant of the
model. This study is a contribution to the search for a mathematical
formulation of the quantum system of $N$ trapped interacting bosons as a model
for Bose--Einstein condensation, motivated by the success of the famous 1995
experiments. Recently, Lieb et al. described the large-N behavior of the ground
state in terms of the well-known Gross--Pitaevskii formula, involving the
scattering length of the pair potential. We prove that the large-N behavior of
the ground product-states is also described by the Gross--Pitaevskii formula,
however, with the scattering length of the pair potential replaced by its
integral.
