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Fractional diffusion of variable order and anomalous aggregation phenomenon

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FD2W01 - Deterministic and stochastic fractional differential equations and jump processes

In this talk we consider a fractional diffusion of variable order both in time and space which may be thought of as a particle moving in diverse porous milieu. Linking them to general semi-Markov processes we discuss the long-term behaviour of these diffusions when the derivative in time is fractional of variable order and the spatial behaviour is Brownian motion. In particular depending on the variable order of the diffusion, we prove rigorously an anticipated anomalous behaviour, that is the domination of the time the fractional diffusion spends in regions where the order of the diffusion is minimal compared to the time spent elsewhere. Under some conditions and depending on the variable order of the diffusion we also demonstrate that the probability to find the particle in regions where the order of the diffusion is minimal converges to one as time increases. The main techniques do not stem from the integro-differential equation solved by the semigroup (in a mild sense) of these processes, but depend on classical laws of the iterated logarithm for general Levy processes. This is a clear indication of interplay between the purely analytical and the fluctuation approach.   This is joint work with Bruno Toaldo (Torino, Italy)

This talk is part of the Isaac Newton Institute Seminar Series series.

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