  We introduce a stochastic model of two-dimensional Brownian vortices
associated with the canonical ensemble. The point vortices evolve through their
usual mutual advection but they experience in addition a random velocity and a
systematic drift generated by the system as a whole. The statistical
equilibrium state of this stochastic model is the Gibbs canonical distribution.
We consider a single species system and a system made of two types of vortices
with positive and negative circulations. At positive temperatures, like-sign
vortices repel each other (plasma case) and at negative temperatures, like-sign
vortices attract each other (gravity case). We derive the stochastic equation
satisfied by the exact vorticity field and the Fokker-Planck equation satisfied
by the N-body distribution function. We present the BBGKY-like hierarchy of
equations satisfied by the reduced distribution functions and close the
hierarchy by considering an expansion of the solutions in powers of 1/N, where
N is the number of vortices, in a proper thermodynamic limit. For spatially
inhomogeneous systems, we derive the kinetic equations satisfied by the smooth
vorticity field in a mean field approximation valid for $N\to +\infty$. For
spatially homogeneous systems, we study the two-body correlation function, in a
Debye-H\"uckel approximation valid at the order O(1/N). The results of this
paper can also apply to other systems of random walkers with long-range
interactions such as self-gravitating Brownian particles and bacterial
populations experiencing chemotaxis. Furthermore, for positive temperatures,
our study provides a kinetic derivation, from microscopic stochastic processes,
of the Debye-H\"uckel model of electrolytes.
