Abstract: A measurable set is invariant with respect to a not necessarily symmetric
sub-Markovian operator on if and strongly invariant if
We show that these definitions accommodate many of the usual definitions of
invariance, e.g., those used in Dirichlet form theory, ergodic theory or for stochastic
processes. In finite measure spaces or if is sub-Markovian and recurrent, the notions of
invariance and strong invariance coincide. We also show that for certain analytic semigroups
of sub-Markovian operators, (strongly) invariant sets are already determined by a single
operator,
2000 AMS Mathematics Subject Classification: 31C15,47D07, 60J35, 60J45.
Key words and phrases: Invariant set, sub-Markov operator, recurrence, transience,
fractional power, analytic semigroup.