Evolutionary semigroups on path spaces
Robert Denk, Markus Kunze, Michael Kupper
TL;DR
The paper develops a general framework of evolutionary semigroups on path spaces to extend transition semigroups to potentially non-Markovian dynamics. Central to the approach is the representation $\mathbb{T}(t)=\mathbb{E}\Theta_t$ where $\mathbb{E}$ is an expectation operator acting on path-function spaces, encoding history through $\mathscr{X}^-$ and linking to the shift on full path space $\mathscr{X}$. It provides a thorough structural theory, including equivalences characterizing evolutionary semigroups, generator relations with the shift, and core- and continuity-results for both continuous and cadlag path spaces. The paper then demonstrates, via four main examples—deterministic evolution, Markov processes, stochastic delay equations, and Lévy-driven stochastic flows—how this framework yields concrete path-dependent dynamics and connects to path-dependent PDEs and control. The resulting framework broadens the applicability of semigroup methods to path-dependent, history-influenced systems with potential implications for stochastic analysis and mathematical finance.
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 Lévy processes. The latter in particular include certain stochastic delay equations.