  Describing partially-condensed Bose gases poses a long-standing theoretical
challenge. We present exact stochastic Ehrenfest relations for the stochastic
projected Gross-Pitaevskii equation, including both number and energy damping
mechanisms, and all projector terms that arise from the energy cutoff
separating system from reservoir. We test the theory by applying it to the
centre of mass fluctuations of a harmonically trapped prolate system, finding
close agreement between c-field simulations and analytical results. The
formalism lays the foundation to analytically explore experimentally accessible
hot Bose-Einstein condensates.
