arXiv subjects
Publications and source records attributed to A. S. Ustunel.
Let $(W,H,μ)$ be an abstract Wiener space and $L$ be a probability density of class LlogL. Using the measure transportation of Monge-Kantorovitch, we prove that the kernel of the projection of L on the second Wiener chaos defines an (Hilbert-Schmidt) operator which is lower bounded by another Hilbert-Schmidt operator.
We show the existence of the strong solutions of the Monge-Ampere equation for the log-concave measures without any regularity assumption. In particular the measures with density of the form $\exp -f$, where $f$ is an $H$-convex function which may take the value $\infty$ on a set of positive measure are included.
In this work we prove that the unique 1-convex solution of the Monge problem contructed from the solution of the Monge-Kantorovitch problem between the Wiener measure and a target measure which has a log-concave density w.r.to the Wiener measure is also the strong solution of the Monge-Ampere equation in the frame of infinite dimensional Frechet spaces. We enhance also the polar factorization results of the mappings which transform a spread measure to another one of finite Wasserstein distance. Finally we calculate the semimartingale decomposition of the transport process with respect to its natural filtration and make the connection between the curved Brownian motion and the polar decomposition of the corresponding shifts.
We give the solution of the Monge-Kantorovitch problem on the Wiener space for the singular Wasserstein metric which is defined with respect to the distance of the underlying Cameron-Martin space. We show, under the hypothesis that this distance is finite, the existence and the uniquness of the solutions, that they are supported by the graphs of the weak derivatives of H-convex Wiener functionals. then we prove the more general situation, where the measures are not even necessarily absolutely continuous w.r.to the Wiener measure. We give sufficient conditions for the hypothesis about the Wassestein distance is finite with the help of the Girsanov theorem. Finally we give the solutions of the Monge-Ampere equation using the classical Jacobi representation and/or the Ito parametrization of the Wiener space.
This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space, (i.e. transformations induced by general measure preserving transformations, called `rotations', and H-valued shifts) and the associated flows on abstract Wiener spaces
In this paper we study ergodicity and mixing property of some measure preserving transformations on the Wiener space (W,H,μ) which are generated by some random unitary operators defined on the Cameron-Martin space H.
Suppose that T is a map of the Wiener space into itself, of the following type: T=I+u where u takes its values in the Cameron-Martin space H. Assume also that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work we prove that if T preserves the Wiener measure, then necessarily u is in the first Wiener chaos and the transformation corresponding to it is a rotation in the sense of [9]. Afterwards the ergodicity and mixing of such transformations, which are second quantizations of the unitary operators on the Cameron-Martin space, are characterized. Finally, the ergocity of the transformation dY_t=gamma(t)dW_t, 0 \le t \le 1 where W is n-dimensional Wiener and gamma is non random is characterized