Evolutionary semigroups on path spaces

Abstract: We introduce the concept evolutionary semigroups on path spaces, generalizing the notion of transition semigroups to possibly non-Markovian stochastic processes. We study the basic properties of evolutionary semigroups and, in particular, prove that they always arise as the composition of the shift semigroup and a single operator called the expectation operator of the semigroup. We also prove that the transition semigroup of a Markov process can always be extended to an evolutionary semigroup on the path space whenever the Markov process can be realized with the appropriate path regularity. As first examples of evolutionary semigroups associated to non-Markovian processes, we discuss deterministic evolution equations and stochastic flows driven by L\'evy processes. The latter in particular include certain stochastic delay equations.