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A note on invariant sets

Schilling R. L.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 1

47--66

A measurable set A is invariant with respect to a not necessarily symmetric sub-Markovian operator T on Lp (X, m) if T1A ≤ 1A, and strongly invariant if T1A = 1A. 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 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 determined by a single operator, T1.