General Fractional Calculus, Evolution Equations, and Renewal Processes

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.