Lévy mixing related to distributed order calculus, subordinators and slow diffusions

The study of distributed order calculus usually concerns fractional derivatives of the form ∫01∂αum(dα) for some measure m  , eventually a probability measure. In this paper an approach based on Lévy mixing is proposed. Non-decreasing Lévy processes associated with Lévy triplets of the form (a(y),b(y),ν(ds,y))(a(y),b(y),ν(ds,y)) are considered and the parameter y is randomized by means of a probability measure. The related subordinators are studied from different points of view. 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 (delayed Brownian motion) is discussed.