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General Fractional Calculus, Evolution Equations, and Renewal Processes

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We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form \({(\mathbb D_{(k)} u)(t)=\frac{d}{dt} \int \nolimits_0^tk(t-\tau )u(\tau )\,d\tau-k(t)u(0)}\) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation \({\mathbb D_{(k)} u=-\lambda u}\), λ > 0, proved to be (under some conditions upon k) continuous on [0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.

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Correspondence to Anatoly N. Kochubei.

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This work was partially supported by the Scientific Program of the National Academy of Sciences of Ukraine, Project No. 0107U002029, and by DFG under the project “Neue Klassen von Evolutionsgleichungen und verwandte Probleme der Spektraltheorie”.

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Kochubei, A.N. General Fractional Calculus, Evolution Equations, and Renewal Processes. Integr. Equ. Oper. Theory 71, 583–600 (2011). https://doi.org/10.1007/s00020-011-1918-8

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