Abstract. This paper takes under consideration subordinators and their inverse processes (hitting-times). The governing equations of such processes are presented by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore the concept of time-changed C 0 -semigroup is discussed in case the time-change is performed by means of the hitting-time of a subordinator. Such time-change gives rise to bounded linear operators governed by integro-differential time-operators. Because these operators are non-local the presence of long-range dependence is investigated.

In this paper we consider point processes N f (r), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure v. We obtain the general expression of the probability generating functions Gf of N f , the equations governing the state probabilities p{ of N f , and their corresponding explicit forms. We also give the distribution of the first-passage times T! of N f , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times T? of jumps with height 1j (LJ=1 1j = k) under the condition N (t) = k for all these special processes is investigated in detail.

Abstract. In this work we construct compositions of vector processes of the form S 2β n c 2 L ν (t) , t > 0, ν ∈ 0, 1 2 , β ∈ (0, 1], n ∈ N, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S 2β n whose random time is represented by the inverse L ν (t), t > 0, of the superposition of independent positively-skewed stable processes,, independent stable subordinators). As special cases for n = 1, ν = ) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stabledistributed time.

Exponential relaxation to equilibrium is a typical property of physical systems, but inhomogeneities are known to distort the exponential relaxation curve, leading to a wide variety of relaxation patterns. Power law relaxation is related to fractional derivatives in the time variable. More general relaxation patterns are considered here, and the corresponding semi-Markov processes are studied. Our method, based on Bernstein functions, unifies three different approaches in the literature.

Abstract. In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not Lévy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bernštein 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.