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The study of distributed order calculus usually concerns about fractional derivatives of the form $int_0^1 partial^alpha u , m(dalpha)$ for some measure $m$, eventually a probability measure. In this paper an approach based on Levy mixing is proposed. Non-decreasing Levy processes associated to Levy triplets of the form $l a(y), b(y), u(ds, y) r$ 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.
We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respe
Consider a spectrally positive Stable($1+alpha$) process whose jumps we interpret as lifetimes of individuals. We mark the jumps by continuous excursions assigning sizes varying during the lifetime. As for Crump-Mode-Jagers processes (with characteri
For a difference approximations of multidimensional diffusion, the truncated local limit theorem is proved. Under very mild conditions on the distribution of the difference terms, this theorem provides that the transition probabilities of these appro
In this paper, we construct a Malliavin derivative for functionals of square-integrable Levy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian
Among Markovian processes, the hallmark of Levy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Levy laws, as well as Gaussians, can also be the limit distributions of processes with long range memory that exhibit very