Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups

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.

Final version, Potential analysis, 2014

References in corpus (2)

• Fractional Cauchy problems on bounded domains
• Fractional diffusion equations and processes with randomly varying time

Cited by in corpus (19)

• Time fractional equations and probabilistic representation
• Relaxation patterns and semi-Markov dynamics
• Fractional Herglotz Variational Principles with Generalized Caputo Derivatives
• Solutions of fractional logistic equations by Euler's numbers
• Semi-Markov models and motion in heterogeneous media
• Heat Kernel Estimates for Fractional Heat Equation
• Lévy mixing related to distributed order calculus, subordinators and slow diffusions
• Fractional equations via convergence of forms
• Time-changed fractional Ornstein-Uhlenbeck process
• Non-Local Pearson diffusions
• Parametric General Fractional Calculus: Nonlocal Operators Acting on Function with Respect to Another Function
• Tempered relaxation equation and related generalized stable processes
• Random Time Change and Related Evolution Equations: Time Asymptotic Behavior
• Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fréchet distribution
• From semi-Markov random evolutions to scattering transport and superdiffusion
• Analyses of the contour integral method for time fractional subdiffusion-normal transport equation
• On self-similar Bernstein functions and corresponding generalized fractional derivatives
• Asymptotics of fundamental solutions for time fractional equations with convolution kernels
• Moments of generalized fractional polynomial processes