Abstracts - Facultatea de Matematică

Viability with probabilistic knowledge of initial condition,
application to differential games
Marc QUINCAMPOIX
Département de Mathématiques, Université de Bretagne Occidentale
6 Avenue Victor Le Gorgeu F-29200 Brest, France
marc.quincampoix@univ-brest.fr
We consider a deterministic control system with state constraints. The main specificity
of the question we address now concerns with the case where the state-space is
only unperfectly known by the controller: Instead of knowing the initial condition , he
know only the probability measure of the initial condition. The problem is then reduced
to a new problem where the state variable appears to be be a measure. We provide a
result characterizing the compatibility of the constraints and the control systems with
probabilistic knowledge of the state space.
Then we use our approach to characterize the value function of an differential game
with probabilistic knowledge of initial condition in term of the unique solution of a suitably
defined Hamilton Jacobi Isaacs equation (written on the set of measures). This
creates some difficulties mainly because the set of measure on R N is not finite dimensional
and it is not a normed space: we will introduce and use the so-called Wasserstein
distance between probability measures.Nevertheless, we give a new result of existence
of a value for a differential game with unperfect information.
Elliptic and parabolic PDE for measures
Michael G. ROECKNER
Fakultät für Mathematik, Universität Bielefeld
Postfach 100131, D-33501 Bielefeld, Germany
roeckner@math.uni-bielefeld.de
Invariant measures and transition probabilities of continuous stochastic processes
satisfy second order PDE of elliptic and parabolic type respectively, however, with coefficients
of possibly low regularity. This motivates the study of such equations for
measures from a purely analytic point of view. In the first part of the talk we shall
start with reviewing existence, uniqueness and local regularity results in the elliptic
case. In particular, the measures solving the PDE (essentially) always have densities
with respect to Lebesgue measure. Subsequently, we shall present some recent global
regularity results for these densities as well as conditions implying that they decay polynomially
or exponentially at infinity. In the second part of the talk we shall pass to the
parabolic case. Here the solutions will be measures on space time. Results on existence
and local regularity are quite similar to the elliptic case. Results on uniqueness and
global regularity have been established only very recently and are quite different from
those in the elliptic case. Finally, it should be mentioned that though in this talk only
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