3:00pm Michel Ledoux (Toulouse)
Heat flow, Harnack inequalities, and optimal transportation.
Abstract: The talk will develop heat flow methods towards functional Sobolev-type inequalities and parabolic Harnack inequalities in the context of diffusion operators with curvature bounded from below. Through heat kernel inequalities, connections to optimal transportation and heat flow contraction properties along Wasserstein distances are emphasized.
4:30pm Michael Röckner (Bielefeld)
Stochastic nonlinear Schrödinger equations with linear multiplicative noise: the rescaling approach.
Abstract: Joint work with: Viorel Barbu (Romanian Academy) and Deng Zhang (University of Bielefeld)
We present well-posedness results for stochastic nonlinear Schrödinger equations with linear multiplicative Wiener noise including the non-conservative case. Our approach is different from the standard literature on stochastic nonlinear Schrödinger equations. By a rescaling transformation we reduce the stochastic equation to a random nonlinear Schrödinger equation with lower order terms and treat the resulting equation by a fixed point argument, based on generalizations of Strichartz estimates proved by J. Marzuola, J. Metcalfe and D. Tataru in 2008. This approach allows to improve earlier well-posedness results obtained in the conservative case by a direct approach to the stochastic Schrödinger equation. In contrast to the latter, we obtain well-posedness in the full range (1, 1 + 4/d) of admissible exponents in the non-linear part (where d is the dimension of the underlying Euclidean space), i.e. in exactly the same range as in the deterministic case.