Past lectures

Abstract

The lectures will follow mainly [1]. As prerequisites a course in probability theory and some basic knowledge about Hilbert spaces would be helpful, though e.g. even the notion of a martingale will be recalled in the lectures. The first part will be a self-contained introduction to stochastic integration on Hilbert spaces, followed by a part on stochastic differential equations (SDEs) on finite dimensional state spaces. Then as the core of the lectures, the variational approach to SDEs on Hilbert spaces will be presented, first under global monotonicity conditions on the coefficients and subsequently under merely local monotonicity as well as generalized coercivity conditions. Applications to standard stochastic partial differential equations, including the stochastic versions of the parabolic porous media, p-Laplace, Cahn-Hillard, Burgers and 2D as well as 3D Navier-Stokes equations will be presented. Finally, the connection to the Fokker-Planck Kolmogorov equations will be discussed and, time permitting, some recent results on the latter explained. This last part will be based on [2].

To detailedness in which the respective parts of the lectures will be presented, will depend on the background knowledge and the interest of the audience. One way to realize this in a sort of individualized manner is to inform the audience about the beginning/end of the next/previous part via e-mail so everyone may decide to skip a part she or he knows about, or join another more advanced part she or he is particularly interested in, respectively.

[1] Wei Liu und Michael Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, 2015, pp. 266

[2] Vladimir I. Bogachev, Nicolai V. Krylov, Michael Röckner und Stanislav V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, Russian version: Izhewsk Institute of Computer Science, 2013, English version: AMS-Monographs to appear, pp. 488.