  In this paper, we are interested in the long-time behaviour of stochastic
systems of n interacting vortices: the position in R2 of each vortex evolves
according to a Brownian motion and a drift summing the influences of the other
vortices computed through the Biot and Savart kernel and multiplied by their
respective vorticities. For fixed n, we perform the rescalings of time and
space used successfully by Gallay and Wayne [5] to study the long-time
behaviour of the vorticity formulation of the two dimensional incompressible
Navier-Stokes equation, which is the limit as n $\rightarrow$ $\infty$ of the
weighted empirical measure of the system under mean-field interaction. When all
the vorticities share the same sign, the 2n-dimensional process of the rescaled
positions of the vortices is shown to converge exponentially fast as time goes
to infinity to some invariant measure which turns out to be Gaussian if all the
vorticities are equal. In the particular case n = 2 of two vortices, we prove
exponential convergence in law of the 4-dimensional process to an explicit
random variable, whatever the choice of the two vorticities. We show that this
limit law is not Gaussian when the two vorticities are not equal.
