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

Vassili N. Kolokoltsov|arXiv (Cornell University)|Jan 16, 2015

Fractional Differential Equations Solutions45 references40 citations

TL;DR

This paper introduces fully mixed and multidimensional extensions of Caputo and Riemann-Liouville fractional derivatives through a probabilistic framework, interpreting them as generators of Lévy processes interrupted at boundaries. It establishes well-posedness for related fractional differential equations using Markov processes with boundary-blocking dynamics, unifying and generalizing prior results via stochastic analysis and providing explicit solutions for β ≤ 1.

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 L\'evy 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 $\R^d$ by blocking (or interrupting) the jumps that attempt to cross certain closed set of 'check-points'.

Motivation & Objective

• To generalize classical fractional derivatives by introducing fully mixed and multidimensional extensions based on probabilistic interpretations.
• To unify and simplify known results on fractional differential equations using stochastic analysis.
• To develop a rigorous framework for well-posedness of fractional equations through Markov processes interrupted at boundaries.
• To establish explicit solutions for equations involving derivatives of order β ≤ 1.
• To initiate a theory of Markov processes constructed by blocking jumps that cross closed sets of check-points in R^d.

Proposed method

• Interprets Caputo and Riemann-Liouville derivatives of order α ∈ (0,2) as regularized generators of stable Lévy motions interrupted upon crossing a boundary.
• Introduces a class of Markov processes in R^d by blocking jumps that attempt to cross a closed set of check-points, generalizing boundary-interruption dynamics.
• Uses Dynkin’s martingale formula and Feller process theory to derive well-posedness results for boundary value problems.
• Applies stochastic calculus to operators A and Â[a,b] representing interrupted jump dynamics, with ν(x,y) as jump density satisfying moment and regularity conditions.
• Derives equivalent integral representations of fractional derivatives via integration by parts, linking analytic and probabilistic forms.
• Establishes generalized and classical solution uniqueness for equations with λ = 0 under smoothness and monotonicity assumptions on ν and boundary exit probabilities.

Experimental results

Research questions

• RQ1 How can the Caputo and Riemann-Liouville derivatives be naturally extended to fully mixed, two-sided, or multidimensional settings using probabilistic principles?
• RQ2 What is the role of boundary interruption in defining generalized fractional derivatives and their associated stochastic processes?
• RQ3 Under what conditions does the operator representing an interrupted Lévy process generate a unique Feller process?
• RQ4 How do the solutions to fractional differential equations with β ≤ 1 relate to the exit distributions and transition kernels of the interrupted processes?
• RQ5 Can explicit solutions be derived for boundary value problems involving these extended derivatives?

Key findings

• For β ∈ (0,1), the Caputo derivative Dβ_{a+}⋆f(x) equals the Riemann-Liouville derivative of f − f(a), differing by a boundary correction term involving f(a)/(Γ(1−β)(x−a)^β).
• For β ∈ (1,2), the Caputo derivative includes a second-order regularization term involving f′(a), with Dβ_{a+}⋆f(x) = Dβ_{a+}[f − f(a) − f′(a)(·−a)](x) minus boundary correction terms.
• When a = −∞ and f is smooth, bounded, and integrable, the right-sided fractional derivative dβ/dxβ f(x) equals the common value of Dβ_{−∞+}f(x) and Dβ_{−∞+}⋆f(x), given by an integral involving f(x−z)−f(x) or f(x−z)−f(x)+f′(x)z.
• The operator Â[a,b] defined on [a,b] generates a Feller process under regularity and monotonicity conditions on ν(x,y), ensuring unique strong solutions to the associated SDEs.
• Under the assumption that exit probabilities P(Xx(τ) ≤ a) and P(Xx(τ) ≥ b) are in C²[a,b] and the kernel H(x,dy) is weakly differentiable, formula (91) supplies the unique classical solution to the boundary value problem with λ = 0.
• Even without classical solution regularity, formula (91) yields a unique generalized solution to the problem with λ = 0, extending the applicability of the method.