  We study long-term behavior and stationary distributions for stochastic heat
equations forced simultaneously by a multiplicative noise and an independent
additive noise with the same distribution. We prove that nontrivial space-time
translation-invariant measures exist for all values of the parameters. We also
show that if the multiplicative noise is sufficiently strong, the invariant
measure has Cauchy-distributed marginals. Using the same techniques, we prove a
similar result on Cauchy-distributed marginals for a logarithmically attenuated
version of the problem in two spatial dimensions. The proofs rely on stochastic
analysis and elementary potential theory.
