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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.

In this paper, a Leaky Integrate-and-Fire (LIF) model for the membrane potential of a neuron is considered, in case the potential process is a semi-Markov process. Semi-Markov property is obtained here by means of the time-change of a Gauss-Markov process. This model has some merits, including heavy-tailed distribution of the waiting times between spikes. This and other properties of the process, such as the mean, variance and autocovariance, are discussed.

Dans cet article, les equations de Volterra integro-differentielles dont le noyau de convolution depend de la variable vectorielle sont considerees et une relation entre ces equations et une classe de processus semi-Markoviens est etablie. L’equation de diffusion fractionnelle d’ordre variable $\alpha(x)$ est un cas particulier de notre analyse et elle se revele etre associee a un changement de temps approprie (non independant) du mouvement Brownien. Le processus resultant est semi-markovien et ses trajectoires ont des intervalles de constance, comme cela arrive pour le mouvement Brownien retarde, adapte pour modeliser les effets de piegeage induits par le milieu. Cependant, dans notre scenario, l’intervalle de constance peut dependre de la position et cela signifie des pieges de profondeur variant dans l’espace comme cela se produit dans un milieu desordonne. La force du piegeage est etudiee au moyen du comportement asymptotique du processus: il est demontre que, sous certaines hypotheses techniques sur $\alpha(x)$, les pieges rendent le processus non diffusif en ce sens qu’il passe un temps negligeable hors d’un voisinage de la region $\operatorname{argmin}(\alpha(x))$ vers laquelle il converge en probabilite sous quelques hypotheses plus restrictives sur $\alpha(x)$.

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 Kolmogorov's 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 Kolmogorov's equations are identified.

In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large $t$) and of the distribution function (for small $t$) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leaky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, for example, it makes the intertimes between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.

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.

In this paper, we consider the general space–time fractional equation of the form $$\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)$$ , for $$\nu _j \in \left( 0,1 \right] $$ and $$\beta \in \left( 0,1 \right] $$ with initial condition $$w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)$$ . We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) $$ , $$t>0$$ , where $$\varvec{S}_n^{2\beta }$$ is an isotropic stable process independent from $$\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)$$ , which is the inverse of $$\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)$$ , $$t>0$$ , with $$H^{\nu _j}(t)$$ independent, positively skewed stable random variables of order $$\nu _j$$ . The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition $$\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) $$ , $$t>0$$ , supplies a probabilistic representation for the solutions of the fractional equations above and coincides for $$\beta = 1$$ with the n-dimensional Brownian motion at the random time $$\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)$$ , $$t>0$$ . The iterated process $$\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)$$ , $$t>0$$ , inverse to $$\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) $$ , $$t>0$$ , permits us to construct the process $$\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) $$ , $$t>0$$ , the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For $$r \rightarrow \infty $$ and $$\beta = 1$$ , we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation $$\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)$$ . Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.

In this paper we will take under consideration subordinators and their inverse processes (hitting-times). We will present in general the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives. Furthermore we will discuss the concept of time-changed $C_0$-semigroup in case the time-change is performed by means of the hitting-time of a subordinator. We will show that such time-change give rise to bounded linear operators not preserving the semigroup property and we will present their governing equations by using again integro-differential operators. Such operators are non-local and therefore we will investigate the presence of long-range dependence., Comment: Final version, Potential analysis, 2014

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 pages

Pseudoprocesses, constructed by means of the solutions of higher-order heat-type equations, have been developed by several authors and many related functionals have been analysed by applying the Feynman–Kac functional or by means of the Spitzer identity. We here examine pseudoprocesses wrapped up on circles and derive their explicit signed density measures. By composing the circular pseudoprocesses with positively skewed stable processes, we arrive at genuine circular processes whose distribution is obtained in the form of Poisson kernels. The distribution of circular even-order pseudoprocesses is similar to the Von Mises (or Fisher) circular normal law and to the wrapped up law of Brownian motion. Time-fractional and space-fractional equations related to processes and pseudoprocesses on the unit radius circumference are introduced and analysed.

The study of distributed order calculus usually concerns about fractional derivatives of the form $\int_0^1 \partial^\alpha u \, m(d\alpha)$ for some measure $m$, eventually a probability measure. In this paper an approach based on L\'evy mixing is proposed. Non-decreasing L\'evy processes associated to L\'evy triplets of the form $\l a(y), b(y), \nu(ds, y) \r$ are considered and the parameter $y$ is randomized by means of a probability measure. The related subordinators are studied from different point of views. Some distributional properties are obtained and the interplay with inverse local times of Markov processes is explored. Distributed order integro-differential operators are introduced and adopted in order to write explicitly the governing equations of such processes. An application to slow diffusions is discussed., Comment: in Journal of Mathematical Analysis and Applications (2015)

In this paper we consider time-changed models of population evolution Xf(t) = X(Hf(t)), where X is a counting process and Hf is a subordinator with Laplace exponent f. In the case where 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 X represents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size n0. Finally, the subordinated linear birth–death process is considered. Special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed.

In this paper we study a class of distributions related to the r.v. , for different distributions of . The problem is related to the hitting point of a randomly oriented ray and generalizes the Cauchy distribution in different directions. We show that the distribution of solves the Laplace equation of order , possesses even moments of order , and has bimodal structure when is uniform. We study also a number of distributional properties of functionals of , including those related to the arcsine law. Finally we study the same problem in the Poincare half-plane and this leads to the hyperbolic distribution of which the main properties are explored. In particular we study the distribution of hyperbolic functions of , the law of sums of i.i.d. r.v.'s and the distribution of the area of random hyperbolic right triangles.

This paper is devoted to the interplay between time-fractional telegraph-type equations and processes defined on the n -dimensional Poincare half-space H n . We solve such equations and show that the solutions coincide with the law of the composition of a hyperbolic Brownian motion with the inverse of the sum of two independent stable subordinators. In the case n = 3 , we obtain the explicit form of the solution of the above equation.

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.

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 pages

Background: Reversed Robin Hood syndrome (RRHS) has recently been identified as one of the mechanisms of early neurologic deterioration in acute ischemic stroke (AIS) patients related to arterial blood flow steal from ischemic to nonaffected brain. We sought to investigate the association of RRHS with risk of stroke recurrence in a single-center cohort study. Methods: Consecutive patients with AIS or TIA affecting the anterior circulation were prospectively evaluated with serial NIH Stroke Scale assessments and bilateral transcranial Doppler monitoring with breath-holding test. RRHS was defined according to previously validated criteria. Results: A total of 360 patients (51% women, mean age 62 ± 15 years) had an ischemic stroke (81%) or TIA (19%) in the anterior circulation, and 30 (8%) of them had RRHS. During a mean follow-up period of 6 months (range 1–24), a total of 16 (4%) recurrent strokes (15 ischemic and 1 hemorrhagic) were documented. The cumulative recurrence rate was higher in patients with RRHS (19%; 95% confidence interval [CI] 1–37) compared to the rest (15%; 95% CI 0–30; p = 0.022 by log-rank test). All recurrent strokes in patients with RRHS were cerebral infarcts that occurred in the ipsilateral to the index event anterior circulation vascular territory. After adjusting for demographic characteristics, vascular risk factors, and secondary prevention therapies, RRHS was independently associated with a higher stroke recurrence risk (hazard ratio 7.31; 95% CI 2.12–25.22; p = 0.002). Conclusions: Patients with AIS and RRHS appear to have a higher risk of recurrent strokes that are of ischemic origin and occur in the same arterial territory distribution to the index event. Further independent validation of this association is required in a multicenter setting.