PROBABILJTY AND MATHEMATICAL STATISTICS A NOTE ON INVARIANT SETS

Abstracb. A measurable set A is invariant with respect to a not necessarily symmetric sub-Markovian operator T on LP(X, m) if TIA < lA, and strongly invariant if TIA = 1,. 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 stochas-tic processes. In finite measure spaces or iI T * 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 determincd by a single operator, TI. Mathematics Snbject Classification: 31C15,47DO7,60J35,60545. Key words and phrases: Invariant set, sub-Markov operator, re