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
Tutti gli interessati sono invitati a partecipare.
m.
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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.
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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.