On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations

Abstract

From the point of view of stochastic analysis the Caputo and Riemann-Liouville derivatives of order $\al \in (0,2)$ can be viewed as (regularized) generators of stable Lévy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. These extensions are introduced and some well-posedness results are obtained that generalize, simplify and unify lots of known facts. This probabilistic analysis leads one to study a class of Markov processes that can be constructed from any given Markov process in ${\mathbb{R}}^d$ by blocking (or interrupting) the jumps that attempt to cross certain closed set of 'check-points'.

Submitted to Fract. Calc. Appl. Anal

References in corpus (3)

• Fractional Cauchy problems on bounded domains
• Leapover lengths and first passage time statistics for Lévy flights
• Time-fractional derivatives in relaxation processes: a tutorial survey

Cited by in corpus (14)

• Generalised Fractional Evolution Equations of Caputo Type
• First passage time moments of asymmetric Lévy flights
• First passage properties of asymmetric Lévy flights
• Green's function estimates for time fractional evolution equations
• On the probabilistic approach to the solution of generalized fractional differential equations of Caputo and Riemann-Liouville type
• Maximum principles for time-fractional Cauchy problems with spatially non-local components
• Censored Stable Subordinators and Fractional Derivatives
• Stochastic representation of solution to nonlocal-in-time diffusion
• Self-similar Cauchy problems and generalized Mittag-Leffler functions
• Abstract McKean-Vlasov and HJB equations, their fractional versions and related forward-backward systems on Riemannian manifolds
• Time Fractional Poisson Equations: Representations and Estimates
• An averaging principle for fractional stochastic differential equations with Lévy noise
• Multi-agent interaction and nonlinear Markov games
• A Meyer-Itô Formula for Stable Processes via Fractional Calculus