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Workshop on Fractional Models for Distributed Systems


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Wolfgang Bock (University of Kaiserlautern):
Recent Results on Grey Brownian Motion
Fractional differential equations are a very intensively studied object in many fields of applications. The fractionality introduces a long-range coupling along the trajectories which gives the possibility to model memory effects. On the other side, the well known connecion of partial differential equations and stochastic processes, which is in the heat equation case given via the famous Feynman–Kac formaula usually breaks down in the fractional case. For a certain class of space-time fractional heat equations however, it is shown by Schneider, that a Feynman–Kac like formula exists for a process which he names grey Brownian motion. In fact the characteristic function of the process is in a similar fashion given as that of a standard Wiener process, where the exponential function is replaced by a Mittag-Leffler function. In this talk we derive solutions of the Ornstein–Uhlenbeck driven by Grey Brownian motion and of a linear Wick-type SDE driven by Grey Brownian motion. The solutions are characterized to be in a suitable distribution space of Mittag-Leffler Analysis introduced by Grothaus, Jahnert, Riemann and Silva. Moreover we give an overview of process properties related to Physics. And discuss recent results of grey Brownian motion. Joint work with: José Luís Silva