  We develop a two-dimensional stochastic dissipative theory for the
description of the transport of exciton polaritons accounting for their
interaction with the environment of acoustic phonons. Our approach is based on
the explicit modeling of the corresponding microscopic processes using a Monte
Carlo framework rather than modeling from phenomenological principles. We show
the dynamic formation of a condensate and investigate its characteristics,
including threshold-like behavior in populations and the formation of spatial
and temporal coherence at different temperatures of the environment and
accounting for the stimulated nonlinear scattering, caused by
system-environment interaction. The spatial coherence reveals a transition from
an exponential to polynomial decay which can be attributed to the
Berezinskii-Kosterlitzh-Thouless-like phase.
