L\'evy mixing related to distributed order calculus, subordinators and slow diffusions

The study of distributed order calculus usually concerns about fractional derivatives of the form ${\int }_0^1{\mathrm{\partial }}^{\alpha }um(d\alpha )$ for some measure $m$, eventually a probability measure. In this paper an approach based on L'evy mixing is proposed. Non-decreasing L'evy processes associated to L'evy triplets of the form \l a(y), b(y), \nu(ds, y) \r are considered and the parameter $y$ is randomized by means of a probability measure. The related subordinators are studied from different point of views. Some distributional properties are obtained and the interplay with inverse local times of Markov processes is explored. Distributed order integro-differential operators are introduced and adopted in order to write explicitly the governing equations of such processes. An application to slow diffusions is discussed.