  A generalised stochastic Gross-Pitaevskii equation describing a partially
condensed trapped Bose gas with rotating thermal component is presented. We
elucidate the manner in which the rotation changes the role of the high energy
cutoff and introduces centrifugal effects in the classical field evolution. The
rotation of the cloud means that Bose-enhanced collision processes occur
preferentially into states with the same angular momentum as the thermal cloud,
thus favouring vortex formation. We use the formalism to obtain a first
principles theory of vortex lattice formation caused by thermodynamic
instability.
