Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorovs backward equations for
a large class of homogeneous semi-Markov processes, having the form of an abstract Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Fractional equations in the time variable are a particular case of our analysis. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorovs equations are identified.
In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Levy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly differen
t Bernv{s}tein functions for each time $t$. By means of these processes, a generalization of subordinate semigroups in the sense of Bochner is proposed. Because of time-inhomogeneity, two-parameter semigroups (propagators) arise and we provide a Phillips formula which leads to time dependent generators. The inverse processes are also investigated and the corresponding governing equations obtained in the form of generalized variable order fractional equations. An application to a generalized subordinate Brownian motion is also examined.
In this article, we consider time-changed models of population evolution $mathcal{X}^f(t)=mathcal{X}(H^f(t))$, where $mathcal{X}$ is a counting process and $H^f$ is a subordinator with Laplace exponent $f$. In the case $mathcal{X}$ is a pure birth pr
ocess, we study the form of the distribution, the intertimes between successive jumps and the condition of explosion (also in the case of killed subordinators). We also investigate the case where $mathcal{X}$ represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size $n_0$. Finally, the subordinated linear birth-death process is considered. A special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.
