The seminar takes place on selected Mondays 9-10 am German time (5-6 pm Korean time).
ID: 968 2387 9152
Patrícia Gonçalves (Lisbon): t.b.a.
Jaehun Lee (Seoul): General law of iterated logarithm for Markov processes
Abstract : Law of iterated logarithm for a stochastic process describes the magnitude of the fluctuations of its sample path behaviors. In this talk, we discuss the law of iterated logarithm for continuous-time Markov processes, which cover a large class of subordinated diffusions, jump processes with mixed polynomial local growth, jump processes with singular jumping kernels and random conductance models with long range jumps. We establish the law of iterated logarithms for the distance and occupation time, in terms of two functions concerned with the first exit time from balls, and uniform bounds on the tails of jumping measure. Especially, we concentrate on the sharp sufficient conditions for both small and large time. This is joint work with Soobin Cho and Panki Kim.
This is the joint work with Soobin Cho and Panki Kim.
Tabea Tscherpel (Bielefeld): On the Sobolev stability of the $L^2$-Projection mapping to finite element spaces
Abstract: The $L^2$-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Especially when dealing with evolution problems Sobolev stability of the $L^2$-projection is needed. While the proof of Sobolev stability is straightforward for uniform meshes, it is much more challenging for graded meshes. We tackle this difficulty by investigating the decay properties of the $L^2$-projection analytically. Inspired by the miracle of extrapolation we introduce a maximal operator. This allows us to obtain equivalence of decay and certain weighted estimates, that eventually lead to Sobolev stability. We present recent results in 2D and 3D for a wide range of polynomial degrees and graded meshes. Those include a large class of meshes used in practical applications.
This is joint work with Lars Diening and Johannes Storn from Bielefeld University (Germany).
Minhyun Kim (Bielefeld): Local regularity theory for nonlocal equations
Abstract: We introduce several nonlocal equations which correspond to classical local models, attempting to develop parallel or unified theory. We study local regularity properties such as local boundedness, weak Harnack inequality, and local Hölder regularity of weak solutions to these nonlocal equations.