Il giorno 11 febbraio 2014, presso l'aula seminari del Dipartimento di Matematica dell'Università di Pisa, si terranno i seguenti seminari:
10:30 Olga Aryasova (Institute of Geophysics, National Academy of Sciences of Ukraine): A representation for the derivative with respect to the initial data of the solution of an SDE with non-regular drift 11:30 E. Orsingher (Roma Sapienza): Random flights
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_______________________________________________ abstract Aryasova
We consider a multidimensional SDE with an identity diffusion matrix and a drift vector being a bounded measurable vector field. According to [Ver81] there exists a unique strong solution to such an equation. Recently the Sobolev differentiability of the solution with respect to the initial data was proved under rather weak assumptions on the drift (c.f. [Fed13,Moh12]). If the drift coefficient is smooth the derivative can be represent as a solution of an integral equation. For non-regular drift in dimension one such a representation was obtained using the local time of the initial process (see [Ary12,Att10]). It is well known that the solution does not have a local time at a point in multidimensional situation. We obtained the representation of the derivative using the theory of continuous additive functionals developed by Dynkin [Dyn63]. This method can be considered as a generalization of the local time approach to the multidimensional case.
[Ary12] O. V. Aryasova and A. Yu. Pilipenko, On properties of a flow generated by an SDE with discontinuous drift, Electron. J. Probab. 17:no. 106, 1--20, 2012. [Att10] S. Attanasio, Stochastic flows of diffeomorphisms for one-dimensional {SDE} with discontinuous drift, Electron. Commun. Probab. 15:no. 20, 213--226, 2010. [Dyn63] E. B. Dynkin, Markov Processes, Fizmatlit, Moscow, 1963. [Translated from the Russian to the English by J. Fabius, V. Greenberg, A. Maitra, and G. Majone. Academic Press, New York; Springer, Berlin, 1965. vol. 1, xii + 365 pp.; vol. 2, viii + 274 pp.]. [Fed13] E. Fedrizzi and F. Flandoli, Hölder flow and differentiability for {SDE}s with nonregular drift, Stochastic Analysis and Applications, 31(4):708--736, 2013. [Moh12] S. E. A. Mohammed, T. Nilssen, and F. Proske, Sobolev differentiable stochastic flows of SDE's with measurable drift and applications, arXiv:1204.3867. [Ver81] A. Y. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR Sborn, 39(3):387--403, 1981.
_______________________________________________ abstract Orsingher
In this talk we present different types of random flights and examine their dynamics, probability laws and governing equations. We first consider the telegraph process (a random flight on the line), discuss its distribution, the connections with the telegraph equations, the first-passage time and the limiting case. We consider also the asymmetric telegraph process and its reduction to the symmetric one by means of relativistic transformations. Planar motions with a finite number of directions (in particular, four orthogonal directions) and an infinite number of directions, chosen at Poisson times with uniform law, are examined and several explicit distributions derived. A particular attention is devoted to the second model, where conditional and unconditional distributions are presented and the related equation of damped planar vibrations probabilistically derived. Random flights in R^d are subsequently considered and characteristic functions of the position of moving particles performing the random flights, obtained. The cases d = 2, d = 4, are investigated in detail. We present random flights in R^d with Dirichlet joint distribution for displacements, hyper-spherical uniform law for the orientation of steps and with a fractional Poisson number of changes of direction. Two types of fractional extensions of the above material are presented. The first one is obtained by considering Dzerbayshan-Caputo types of time- fractional derivatives. The second fractional extension is obtained by considering fractionalisation of Klein-Gordon equations and by applying the McBride approach to fractional powers of Bessel-type operators.