Semi-Markov models and motion in heterogeneous media

In this paper we study continuous time random walks (CTRWs) such that the holding time in each state has a distribution depending on the state itself. For such processes, we provide integro-differential (backward and forward) equations of Volterra type, exhibiting a position dependent convolution kernel. Particular attention is devoted to the case where the holding times have a power-law decaying density, whose exponent depends on the state itself, which leads to variable order fractional equations. A suitable limit yields a variable order fractional heat equation, which models anomalous diffusions in heterogeneous media

Counting processes with Bern\v{s}tein intertimes and random jumps

We consider here point processes $N^f(t)$, $t>0$, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bern\v{s}tein functions $f$ with L\'evy measure $\nu$. We obtain the general expression of the probability generating functions $G^f$ of $N^f$, the equations governing the state probabilities $p_k^f$ of $N^f$, and their corresponding explicit forms. We also give the distribution of the first-passage times $T_k^f$ 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 $\tau_j^{l_j}$ of jumps with height $l_j$ ($\sum_{j=1}^rl_j = k$) under the condition $N(t) = k$ for all these special processes is investigated in detail

Pseudoprocesses related to space-fractional higher-order heat-type equations

In this paper we construct pseudo random walks (symmetric and asymmetric) which converge in law to compositions of pseudoprocesses stopped at stable subordinators. We find the higher-order space-fractional heat-type equations whose fundamental solutions coincide with the law of the limiting pseudoprocesses. The fractional equations involve either Riesz operators or their Feller asymmetric counterparts. The main result of this paper is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order $\gamma>2$. The classical pseudoprocesses are very special cases of those investigated here

Population models at stochastic times

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 process, 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

Time-changed processes governed by space-time fractional telegraph equations

In this work we construct compositions of processes of the form \bm{S}_n^{2\beta}(c^2 \mathpzc{L}^\nu (t) \r, t>0, \nu \in (0, 1/2], \beta \in (0,1], n \in \mathbb{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 \bm{S}_n^{2\beta} whose random time is represented by the inverse \mathpzc{L}^\nu (t), t>0, of the superposition of independent positively-skewed stable processes, \mathpzc{H}^\nu (t) = H_1^{2\nu} (t) + (2\lambda \r^{\frac{1}{\nu}} H_2^\nu (t), t>0, (H_1^{2\nu}, H_2^\nu, independent stable subordinators). As special cases for n=1, \nu = 1/2 and \beta = 1 we examine the telegraph process T at Brownian time B (Orsingher and Beghin) and establish the equality in distribution B (c^2 \mathpzc{L}^{1/2} (t)) \stackrel{\textrm{law}}{=} T (|B(t)|), t>0. Furthermore the iterated Brownian motion (Allouba and Zheng) 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 stable-distributed time.Comment: 34 page

Time-inhomogeneous jump processes and variable order operators

In this paper we introduce non-decreasing jump processes with independent and time non-homogeneous increments. Although they are not L\'evy processes, they somehow generalize subordinators in the sense that their Laplace exponents are possibly different Bern\v{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.Comment: 26 page