  Driving and dissipation can stabilize Bose-Einstein condensates. Using
Keldysh field theory, we analyze this phenomenon for Markovian systems that can
comprise on-site two-particle driving, on-site single-particle and two-particle
loss, as well as edge-correlated pumping. Above the upper critical dimension,
mean-field theory shows that pumping and two-particle driving induce
condensation right at the boundary between the stable and unstable regions of
the non-interacting theory. With nonzero two-particle driving, the condensate
is gapped. This picture is consistent with the recent observation that, without
symmetry constraints beyond invariance under single-particle basis
transformations, all gapped quadratic bosonic Liouvillians belong to the same
phase. For systems below the upper critical dimension, the edge-correlated
pumping penalizes high-momentum fluctuations, rendering the theory
renormalizable. We perform the one-loop renormalization group analysis, finding
a condensation transition inside the unstable region of the non-interacting
theory. Interestingly, its critical behavior is determined by a
Wilson-Fisher-like fixed point with universal correlation-length exponent
$\nu=0.6$ in three dimensions.
