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Compact symmetry groups and generators for sub-Markovian semigroups
Journal article

Compact symmetry groups and generators for sub-Markovian semigroups

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol.63(1), pp.17-27
03/1983
DOI: 10.1007/BF00534173

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Abstract

We consider closed infinitesimal operators L with dense domain D in the algebra of all continuous functions, vanishing at infinity, on a given locally compact Hausdorff space X. Necessary and sufficient conditions are given for L to be the infinitesimal generator of a strongly continuous sub-Markovian semigroup on C 0(X). The conditions involve: (1) commutativity with a compact group G acting continuously on X (i.e., spatial G-symmetry), (2) a tangential notion defined in terms of the G-action, and finally, (3) a dispersive estimate for L. The dispersive estimate is satisfied on a subalgebra which is a core for L, and contained in the domain D. If G is also a Lie group, we explicitly construct such a core algebra in the domain.

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