  We establish the existence (and in an appropriate sense uniqueness) of
Markovian solutions for ergodic BSDEs under a novel monotonicity condition. Our
monotonicity condition allows us to prove existence even when the driver f has
arbitrary (in particular super-quadratic) growth in z, which reveals an
interesting trade-off between monotonicity and growth for ergodic BSDEs. The
technique of proof is to establish a probabilistic representation of the
derivative of the Markovian solution, and then use this representation to
obtain a-priori estimates. Our study is motivated by applications to ergodic
control, and we use our existence result to prove the existence of optimal
controls for a class of ergodic control problems with potentially
super-quadratic Hamiltonians. We also treat a class of drivers coming from the
construction of forward performance processes, and interpret our monotonicity
condition in this setting.
