Lectures:
Tuesday 23.07.2019
Michael Röckner, University of Bielefeld, Germany and and Academy for Mathematics and Systems Science, CAS, Beijing
Title: A natural extension of Markov processes and applications to singular SDEs
Time: 10:15 - 11:00
Room: Anf. 2
joint work with:
Lucian Beznea (Romanian Academy, Bucharest, Romania)
Iulian Cîmpean (Romanian Academy, Bucharest, Romania)
Abstract: We develop a general method for extending Markov processes to a larger state space such that the added points form a polar set.
The so obtained extension is an improvement on the standard trivial extension in which case the process is made stuck in the added points, and it renders a new technique of constructing extended solutions to S(P)DEs from all starting points, in such a way that they are solutions at least after any strictly positive time. Concretely, we adopt this strategy to study SDEs with singular coefficients on an infinite dimensional state space (e.g.~SPDEs of evolutionary type), for which one often encounters the situation where not every point in the space is allowed as an initial condition.
The same can happen when constructing solutions of martingale problems or Markov processes from (generalized) Dirichlet forms, to which our new technique also applies.
Nora Müller, University of Bielefeld, Germany and
Title: Stochastic Transport Equations: Method of Characteristics versus Scaling Transform Approach
Time: 11:15 - 12:00
Room: Anf. 2
Wednesday 24.07.2019
Wolfgang Bock, University of Kaiserslautern, Germany
Title: On the Stochastic Quantization of the Fractional Edwards Measure
Time: 10:15 11:00
Room: Anf. 2
Cresente O. Cabahug, University of Kaiserslautern, Germany
Title: Recent Numerical Results for Self-rrepelling fBm
Time: 11:15 12:00
Room: Anf. 2
Thursday 25.07.2019
Michael Röckner, University of Bielefeld, Germany and Academy for Mathematics and Systems Science, CAS, Beijing
Title: The evolution to equilibrium of solutions to nonlinear Fokker-Planck equations
Time: 10:15 - 11:00
Room: Anf. 2
joint work with Viorel Barbu (Romanian Academy, Iasi)
Abstract: The talk is about the so-called H -Theorem for a class of nonlinear Fokker-Planck
equations which are of porous media type on the whole Euclidean space perturbed
by a transport term. We first construct a solution in the sense of mild solutions on
L^1 through a nonlinear semigroup of contractions. Then we study the asymptotic
behavior of the solutions when time tends to infinity. For a large class M of initial
conditions we show their relative compactness with respect to local L^1 convergence,
while all limit points belong to L^1. Under an additional assumption we obtain that we
in fact have convergence in L^1, if the initial condition is a probability density. The limit
is then identified as the unique stationary solution in M to the nonlinear Fokker-Planck
equation. This solution is thus an invariant measure of the solution to the corresponding
distribution dependent SDE whose time marginals converge to it in L^1. It turns out that
under our conditions the underlying nonlinear Kolmogorov operator is a (both in the
second and first order part) nonlinear analog of the generator of a distorted Brownian
motion. The solution of the above mentioned distribution dependent SDE can thus be
interpreted as a “nonlinear distorted Brownian motion“. Our main technique for the proofs
is to construct a suitable Lyapunov function acting nonlinearly on the path in L^1, which is
given by the nonlinear contraction semigroup applied to the initial condition, and then adapt
a classical technique of Pazy to our situation. This Lyapunov function is given by a
generalized entropy function (which in the linear case specializes to the usual
Boltzmann-Gibbs entropy) plus a mean energy part.