W dniach 7-16 maja 2013r. gościem Instytutu Matematycznego Uniwersytetu Wrocławskiego będzie prof. Eugene Lytvynov (Swansea University, UK). Wygłosi on cykl wykładów w ramach Środowiskowych Studiów Doktoranckich Nauk Matematycznych pt. Orthogonal decompositions for generalized stochastic processes with independent values. Plan wykładów wtorek 7.05.2013, godz. 12.00-14.00, sala WS czwartek 9.05.2013, godz. 10.00-12.00, sala 603 oraz godz. 14.00-16.00, sala 605 czwartek 16.05.2013, godz. 10.00-12.00, sala 603 oraz godz. 14.00-16.00, sala 605. Streszczenie: A generalized stochastic process is a probability measure on a space of generalized functions on the Euclidean space, that is, a random generalized function. One says that a generalized stochastic processes has independent values if the value of this function at a given point is independent of the values of the function at other points. (The latter is, of course, only a heuristic definition as one can not rigorously speak of a value of a generalized function at a given point.) A generalized stochastic processes with independent values over the real line can also be treated as the derivative of a classical stochastic processes with independent value (in particular, a Levy process). Our aim is the study of the space of square integrable functionals of a generalized stochastic processes with independent values. We will show that there is a natural unitary operator between a certain symmetric Fock space and the L^2-space. This extends the well known results related to the Wiener-Ito-Segal isomorphism for the Gaussian measure and a similar orthogonal decomposition in multiple stochastic integrals for the Poisson measure. We will also study other natural realizations of the L^2-space in terms of Fock-type spaces, in particular, the Nualart-Schoutens decomposition. In our research, we will heavily use not only probability theory, but also functional analysis, in particular, the theory of unbounded self-adjoint operators.
