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 $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-\tau)u(\tau)\,d\tau -k(t)u(0)$ 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 $Du=-\lambda u$, $\lambda >0$, proved to be (under some conditions upon $k$) continuous on $[(0,\infty)$ 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 $\lambda$, $E(t)$ is an inverse subordinator.