Bruno Toaldo

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

In this work, we construct compositions of vector processes of the form , t > 0, , β ∈ (0, 1], , 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 S2βn whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H2ν1, Hν2, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2]) 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.

Association of reversed Robin Hood syndrome with risk of stroke recurrence.

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.

Convolution-Type Derivatives, Hitting-Times of Subordinators and Time-Changed C0-semigroups

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 C0-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.

Pseudoprocesses Related to Space-Fractional Higher-Order Heat-Type Equations

In this article, we construct pseudo random walks (symmetric and asymmetric) that 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 article is the derivation of pseudoprocesses whose law is governed by heat-type equations of real-valued order γ > 2. The classical pseudoprocesses are very special cases of those investigated here.

Lévy mixing related to distributed order calculus, subordinators and slow diffusions

Abstract The study of distributed order calculus usually concerns fractional derivatives of the form ∫ 0 1 ∂ α u m ( d α ) for some measure m, eventually a probability measure. In this paper an approach based on Levy mixing is proposed. Non-decreasing Levy processes associated with Levy triplets of the form ( a ( y ) , b ( y ) , ν ( d s , y ) ) are considered and the parameter y is randomized by means of a probability measure. The related subordinators are studied from different points of view. 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 (delayed Brownian motion) is discussed.

Semi-Markov Models and Motion in Heterogeneous Media

In this paper we study continuous time random walks 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.

Population models at stochastic times

Abstract 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.

Pseudoprocesses on a circle and related Poisson kernels

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.

Shooting randomly against a line in Euclidean and non-Euclidean spaces

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 Poincaré 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.