Time-Inhomogeneous Jump Processes and Variable Order Operators

Abstract

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.

Semi-Markov Models and Motion in Heterogeneous Media

“…The time-change by a dependent time process. In order to have an interpretation of (3.1) as a time-changed process, we need the notion of multistable subordinator (see for example [35] and [37]). Strictly speaking, a multistable subordinator is a generalization of a stable subordinator, in the sense that the stability index is a function of time α = α(t) ∈ (0, 1).…”

“…Hence all the finite dimensional distributions are completely determined by the distribution of the increments which can be obtained from (3.3). Therefore (see [37,Section 2] for details on this point)…”

“…Multistable subordinators are particular cases of a larger class of processes, known as non-homogeneous subordinators, which were introduced in [37]. Definition 3.1.…”

“…The construction of Theorem 3.2 is based on the notion of multistable subordinator. In [37] the authors introduced the more general class of inhomogeneous subordinators, i.e., non decreasing processes with independent and non stationary increments. By using these, it is possible to define a new type of CTRWs, which is constructed in the same way as (3.1), except for the distributions of the waiting times, which are no more Mittag-Leffler.…”

“…This leads to the fundamental fact regarding the time-change relation X(t) = M (L(t)): the time process L and the Markov process M are not independent. Such a construction is far from trivial, since L is the right continuous inverse of a non-decreasing additive process also called time-inhomogeneous subordinator (for basic information consult [41] and [37]).…”