Tue, 21 May 2024
16:00
L6

TBA

Gernot Akemann
(Bielefeld University)
Abstract

TBA

Mon, 01 Mar 2021

16:00 - 17:00

Nonlinear Fokker=Planck equations with measure as initial data and McKean-Vlasov equations

MICHAEL ROECKNER
(Bielefeld University)
Abstract

Nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations This talk is about joint work with Viorel Barbu. We consider a class of nonlinear Fokker-Planck (- Kolmogorov) equations of type \begin{equation*} \frac{\partial}{\partial t} u(t,x) - \Delta_x\beta(u(t,x)) + \mathrm{div} \big(D(x)b(u(t,x))u(t,x)\big) = 0,\quad u(0,\cdot)=\mu, \end{equation*} where $(t,x) \in [0,\infty) \times \mathbb{R}^d$, $d \geq 3$ and $\mu$ is a signed Borel measure on $\mathbb{R}^d$ of bounded variation. In the first part of the talk we shall explain how to construct a solution to the above PDE based on classical nonlinear operator semigroup theory on $L^1(\mathbb{R}^d)$ and new results on $L^1- L^\infty$ regularization of the solution semigroups in our case. In the second part of the talk we shall present a general result about the correspondence of nonlinear Fokker-Planck equations (FPEs) and McKean-Vlasov type SDEs. In particular, it is shown that if one can solve the nonlinear FPE, then one can always construct a weak solution to the corresponding McKean-Vlasov SDE. We would like to emphasize that this, in particular, applies to the singular case, where the coefficients depend "Nemytski-type" on the time-marginal law of the solution process, hence the coefficients are not continuous in the measure-variable with respect to the weak topology on probability measures. This is in contrast to the literature in which the latter is standardly assumed. Hence we can cover nonlinear FPEs as the ones above, which are PDEs for the marginal law densities, realizing an old vision of McKean.

References V. Barbu, M. Röckner: From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Prob. 48 (2020), no. 4, 1902-1920. V. Barbu, M. Röckner: Solutions for nonlinear Fokker-Planck equations with measures as initial data and McKean-Vlasov equations, J. Funct. Anal. 280 (2021), no. 7, 108926.

Mon, 19 Nov 2018

17:00 - 18:00
L4

Higher Regularity of the p-Poisson Equation in the Plane

Lars Diening
(Bielefeld University)
Abstract

In recent years it has been discovered that also non-linear, degenerate equations like the $p$-Poisson equation $$ -\mathrm{div}(A(\nabla u))= - \mathrm{div} (|\nabla u|^{{p-2}}\nabla u)= -{\rm div} F$$ allow for optimal regularity. This equation has similarities to the one of power-law fluids. In particular, the non-linear mapping $F \mapsto A(\nabla u)$ satisfies surprisingly the linear, optimal estimate $\|A(\nabla u)\|_X \le c\, \|F\|_X$ for several choices of spaces $X$. In particular, this estimate holds for Lebesgue spaces $L^q$ (with $q \geq p'$), spaces of bounded mean oscillations and Holder spaces$C^{0,\alpha}$ (for some $\alpha>0$).

In this talk we show that we can extend this theory to Sobolev and Besov spaces of (almost) one derivative. Our result are restricted to the case of the plane, since we use complex analysis in our proof. Moreover, we are restricted to the super-linear case $p \geq 2$, since the result fails $p < 2$. Joint work with Anna Kh. Balci, Markus Weimar.

Wed, 24 Oct 2018
16:00
C1

Finding fibres for free factors

Benjamin Brück
(Bielefeld University)
Abstract

"Fibre theorems" in the style of Quillen's fibre lemma are versatile tools used to study the topology of partially ordered sets. In this talk, I will formulate two of them and explain how these can be used to determine the homotopy type of the complex of (conjugacy classes of) free factors of a free group.
The latter is joint work with Radhika Gupta (see https://arxiv.org/abs/1810.09380).

Fri, 04 Nov 2016
11:00
C5

Gauge theory and Fueter maps

Andriy Haydys
(Bielefeld University)
Abstract

A Fueter map between two hyperKaehler manifolds is a solution of a Cauchy-Riemann-type equation in the quaternionic context. In this talk I will describe relations between Fueter maps, generalized Seiberg-Witten equations, and Yang-Mills instantons on G2-manifolds (so called G2-instantons).

 
 
Thu, 17 Oct 2013

15:00 - 16:00
L2

The root posets (and the hereditary abelian categories of Dynkin type)

Claus Ringel
(Bielefeld University)
Abstract

Given a root system, the choice of a root basis divides the set of roots into the positive and the negative ones, it also yields an ordering on the set of positive roots. The set of positive roots with respect to this ordering is called a root poset. The root posets have attracted a lot of interest in the last years. The set of antichains (with a suitable ordering) in a root poset turns out to be a lattice, it is called lattice of (generalized) non-crossing partitions. As Ingalls and Thomas have shown, this lattice is isomorphic to the lattice of thick subcategories of a hereditary abelian category of Dynkin type. The isomorphism can be used in order to provide conceptual proofs of several intriguing counting results for non-crossing partitions.

Mon, 21 Jan 2013

15:45 - 16:45
Oxford-Man Institute

The stochastic quasi-geostrophic equation

RONGCHAN ZHU
(Bielefeld University)
Abstract
In this talk we talk about the 2D stochastic quasi-geostrophic equation on T2 for general parameter _ 2 (0; 1) and multiplicative noise. We
prove the existence of martingale solutions and Markov selections for multiplicative noise for all _ 2 (0; 1) . In the subcritical case _ > 1=2, we prove existence and uniqueness of (probabilistically) strong solutions. We obtain the ergodicity for _ > 1=2 for degenerate noise. We also study the long time behaviour of the solutions tothe 2D stochastic quasi-geostrophic equation on T2 driven by real linear multiplicative noise and additive noise in the subcritical case by proving the existence of a random attractor.
Mon, 06 Jun 2011
17:00
Oxford-Man Institute

tba

Sasha Grigoryan
(Bielefeld University)
Tue, 23 Feb 2010
14:15
DH 1st floor SR

Stopping with Multiple Priors and Variational Expectations in Contiuous Time

Frank Riedel
(Bielefeld University)
Abstract

We develop a theory of optimal stopping problems under ambiguity in continuous time. Using results from (backward) stochastic calculus, we characterize the value function as the smallest (nonlinear) supermartingale dominating the payoff process. For Markovian models, we derive a Hamilton–Jacobi–Bellman equation involving a nonlinear drift term that describes the agent’s ambiguity aversion. We show how to use these general results for search problems and American Options.

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