From semi-Markov random evolutions to scattering transport and superdiffusion

Abstract

We 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.

References in corpus (16)

• Lévy walks
• Fractional Cauchy problems on bounded domains
• Towards deterministic equations for Levy walks: the fractional material derivative
• Convolution-type derivatives, hitting-times of subordinators and time-changed $C_0$-semigroups
• Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics
• One-Dimensional Stochastic Lévy-Lorentz Gas
• Semi-Markov approach to continuous time random walk limit processes
• Relaxation patterns and semi-Markov dynamics
• Generalised Fractional Evolution Equations of Caputo Type
• Single integro-differential wave equation for Lévy walk
• Variable Order Fractional Fokker-Planck Equations derived from Continuous Time Random Walks
• Solvable non-Markovian dynamic network
• Semi-Markov models and motion in heterogeneous media
• Lévy mixing related to distributed order calculus, subordinators and slow diffusions
• Fokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits
• Extinction time of non-Markovian self-similar processes, persistence, annihilation of jumps and the Fréchet distribution