  We study the asymptotic behavior of solutions to stochastic evolution
equations with monotone drift and multiplicative Poisson noise in the
variational setting, thus covering a large class of (fully) nonlinear partial
differential equations perturbed by jump noise. In particular, we provide
sufficient conditions for the existence, ergodicity, and uniqueness of
invariant measures. Furthermore, under mild additional assumptions, we prove
that the Kolmogorov equation associated to the stochastic equation with
additive noise is solvable in $L_1$ spaces with respect to an invariant
measure.
