Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups
TLDR
In this paper, the authors take under consideration subordinators and their inverse processes (hitting-times) and present the governing equations of such processes by means of convolution-type integro-differential operators similar to the fractional derivatives.Abstract:
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.read more
Citations
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Spectral projections correlation structure for short-to-long range dependent processes
Pierre Patie,Anna Srapionyan +1 more
TL;DR: In this paper, the spectral projections correlation functions have been introduced for stochastic Markov processes, which can be used to make inferences about the path properties of the process (presence of jumps), distance from symmetry (self-adjoint or non-selfadjoint) and short-to-long-range dependence.
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On the Construction of Some Deterministic and Stochastic Non-Local SIR Models
TL;DR: In this paper, the authors considered the generalization of a simple SIR model in the context of generalized fractional calculus and studied the main features of such model, and constructed semi-Markov stochastic epidemic models by using time changed continuous time Markov chains.
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Spectral Analysis of Fractional Hyperbolic Diffusion Equations with Random Data
Nikolai Leonenko,Jayme Vaz +1 more
TL;DR: In this paper, the fundamental solutions to fractional in time hyperbolic diffusion equation or telegraph equations and their properties were studied and derived in terms of series expansions of isotropic in space spherical random fields on the unit sphere.
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Heat Kernel Estimates for Fractional Heat Equation
TL;DR: In this paper, the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes is studied.
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Fractional SIS epidemic models and their solutions
TL;DR: The explicit representation and the numerical schemes for the limit case of the fractional order $\alpha\uparrow1$, corresponding to the well-known ordinary SIS model, are examined.
References
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Book
One-Parameter Semigroups for Linear Evolution Equations
Klaus-Jochen Engel,Rainer Nagel +1 more
TL;DR: In this paper, Spectral Theory for Semigroups and Generators is used to describe the exponential function of a semigroup and its relation to generators and resolvents.
Journal ArticleDOI
Random Walks on Lattices. II
TL;DR: In this paper, the mean first passage times and their dispersion in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions have been derived by the method of Green's functions.
Book
Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models
TL;DR: The Eulerian Functions The Bessel Functions The Error Functions The Exponential Integral Functions The Mittag-Leffler Functions The Wright Functions as mentioned in this paper The Eulerians Functions