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We develop operator renewal theory for flows and apply this to obtain resultson mixing and rates of mixing for a large class of finite and infinite measuresemiflows. Examples of systems covered by our results include suspensions overparabolic rational maps of the complex plane, and nonuniformly expandingsemiflows with indifferent periodic orbits. In the finite measure case, theemphasis is on obtaining sharp rates of decorrelations, extending results ofGou\"ezel and Sarig from the discrete time setting to continuous time. In theinfinite measure case, the primary question is to prove results on mixingitself, extending our results in the discrete time setting. In some cases, weobtain also higher order asymptotics and rates of mixing.

Operator renewal theory for continuous time dynamical systems with finite and infinite measure