1,720,986 research outputs found
Pseudoprocesses Related to Space-Fractional Higher-Order Heat-Type Equations
No full textIn 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 gamma > 2. The classical pseudoprocesses are very special cases of those investigated here
Shooting randomly against a line in Euclidean and non-Euclidean spaces
No full textIn this paper we study a class of distributions related to the r.v. C-n (t) = ttan(1/n)Theta, for different distributions of Theta. 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 C-n (t) solves the Laplace equation of order 2n, possesses even moments of order 2k < 2n - 1, and has bimodal structure when Theta is uniform. We study also a number of distributional properties of functionals of C-n (t), 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 Pr{h is an element of dw} = dw/pi cosh w of which the main properties are explored. In particular we study the distribution of hyperbolic functions of eta, the law of sums of i.i.d. r.v.'s eta(j) and the distribution of the area of random hyperbolic right triangles
Lévy mixing related to distributed order calculus, subordinators and slow diffusions
No full textAbstract. The study of distributed order calculus usually concerns about fractional derivatives of the form ∫ 1 0 ∂ αum(dα) for some measure m, eventu-ally a probability measure. In this paper an approach based on Lévy mixing is proposed. Non-decreasing Lévy processes associated to Lévy triplets of the form (a(y), b(y), ν(ds, y)) 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 (delayed Brownian motion) is discussed. Content
Convolution-type derivatives, hitting-times of subordinators and time-changed C_0-semigroups
No full textThis 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. Content
Fractional telegraph-type equations and hyperbolic Brownian motion
No full textThis paper is devoted to the interplay between time-fractional telegraph-type equations and processes defined on the n-dimensional Poincare half-space W. 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. (c) 2014 Elsevier B.V. All rights reserved
From semi-Markov random evolutions to scattering transport and superdiffusion
Get PDFWe here study random evolutions on Banach spaces, driven by a class of
semi-Markov processes. The expectation (in the sense of Bochner) of such
evolutions is shown to solve some abstract Cauchy problems. Further, the
abstract telegraph (damped wave) equation is generalized to the case of
semi-Markov perturbations. A special attention is devoted to semi-Markov models
of scattering transport processes which can be represented through these
evolutions. In particular, we consider random flights with infinite mean flight
times which turn out to be governed by a semi-Markov generalization of a linear
Boltzmann equation; their scaling limit is proved to converge to superdiffusive
transport processes
Time-Inhomogeneous Jump Processes and Variable Order Operators
No full textIn 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
Bernˇstein 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
- …
