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

Abstract

The study of distributed order calculus usually concerns about fractional derivatives of the form ${\int }_0^1{\partial }^{\alpha }um(d\alpha )$ for some measure $m$, eventually 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 $ła(y), b(y), ν(ds, y) \r$ 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.

arXiv commentin Journal of Mathematical Analysis and Applications (2015)

References in corpus (10)

• Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
• Distributed Order Calculus and Equations of Ultraslow Diffusion
• Fractional Cauchy problems on bounded domains
• Distributed-order fractional Cauchy problems on bounded domains
• Distributed-Order Fractional Kinetics
• Fractional diffusion equations and processes with randomly varying time
• Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups
• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Distributed Order Derivatives and Relaxation Patterns
• Random-time processes governed by differential equations of fractional distributed order

Cited by in corpus (3)

• Relaxation patterns and semi-Markov dynamics
• Semi-Markov models and motion in heterogeneous media
• From semi-Markov random evolutions to scattering transport and superdiffusion