Feller property and convergence for semigroups of time-changed processes
Ali BenAmor, Kazuhiro Kuwae
TL;DR
The paper develops a substitute to the Feller property for semigroups of time changed processes on a general state space $X$ using $G$-Kato measures and Revuz correspondence. It proves that under mild regularity the time changed process yields a $C_0$-semigroup on a subspace of $C_0(X)$ with a Hille–Yosida generator, and establishes strong continuity of the resolvent on this subspace. It then proves convergence of resolvents and semigroups when the perturbing measures $\mu_n$ converge vaguely to $\mu_\infty$ with uniform convergence of $G^{\mu_n}1$, leading to convergence of evolution equation solutions, finite time distributions, and weak convergence of the time changed processes. The paper also discusses recovered Feller behavior on the fine support $F$ and connects to Dirichlet forms and boundary value problems, providing a versatile framework for stability and approximation in time changed settings.
Abstract
We give a substitute to Feller property for semigroups of time-changed processes; under some conditions this leads to establish sufficient (new) conditions for the semigroups to be Feller. Moreover, given a standard process and a sequence of measures converging vaguely to a final measure, under some assumptions, we establish convergence of the sequence of the semigroups and the resolvents of the corresponding time changed-processes. Some applications are given: convergence of solutions of evolution equations and convergence of finite time distributions, as well as weak convergence of the related processes.