SFB 1283, project
IRTG 2235, area D
Di 10:15-11:45 V4-11229.10.19 10:15 V4-112 Meng Yang (Bielefeld) Metrics and uniform Harnack Inequality on the Strichartz hexacarpet 19.11.19 10:15 V4-112 Shilei Kong (Bielefeld) Compact quasi-metric spaces as hyperbolic boundaries 26.11.19 10:15 V4-112 Satoshi Ishiwata (Yamagata University, Japan) Poincare constant on manifolds with ends
Abstract. It is a well-known fact that the Poincare inequality does not hold on a connected sum Rn#Rn for n>2. In this talk, we discuss the estimates of the Poincare constant on manifolds with ends, where the Poincare constant is given by the supremum of variance over the Dirichlet energy form of functions on balls. This talk is based on a joint work with A. Grigor'yan and L. Saloff-Coste.
Abstract. We study removable sets (in the sense of Mazja) and Lp-uniqueness problems on manifolds and metric measure spaces. The Laplacian, considered on suitable spaces of test functions vanishing in a neighborhood of a 'removed' null set, has a unique extension if and only if this set has vanishing second order capacity. If the semigroup is strong Feller and satisfies a specific gradient inequality then two types of capacities are comparable and the critical size of the removed set can be expressed in terms of Hausdorff measures. As an application we show that from the point of view of essential self-adjointness the Laplacian on certain Gromov-Hausdorff limits of non-collapsing spaces 'ignores' the singular part of the limit space. This is joint work with Jun Masamune (Sapporo) and Kohei Suzuki (Pisa).
Abstract. By classical results of G.C.Evans and G.Choquet on ''good kernels in potential theory'', for every polar -set , there exists finite measure on such that on , and a set admits a finite measure on such that
if and only if is a polar -set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions.
In this talk it is shown that, by elementary ''metric'' considerations and without using any potential theory, such results can be obtained for general kernels satisfying a local triangle property. The particular case, on where solves an open problem from the book of N.S.Landkoff on ''Foundations of Modern Potential Theory''.
Di 10:15-11:45 V4-11916.04.19 10:15 V4-119 Andrey Piatnitski (Narvik) Homogenization of non-symmetric convolution type operators with integrable kernels in periodic media 23.04.19 10:15 V4-119 Simon Nowak (Bielefeld) Hs,p regularity theory for a class of nonlocal elliptic equations
30.04.19 10:15 V4-119 Rostislav Grigorchuk (Texas A&M University) On spectra of graphs and groups 07.05.19 10:15 V4-119 Jun Cao (Bielefeld) Heat kernels and Besov spaces on metric measure spaces 14.05.19 10:15 V4-119 Jun Cao (Bielefeld) Heat kernels and Besov spaces on metric measure spaces II 21.05.19 10:15 V4-119 Jun Cao (Bielefeld) Construction of wavelets on metric measure spaces 04.06.19 10:15 V4-119 Mathav Murugan (University of British Columbia) A bridge between elliptic and parabolic Harnack inequalities
Abstract. The notion of conformal walk dimension serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that the finiteness of the conformal walk dimension characterizes the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension is the infimum of all possible values of the walk dimension that can be attained by a time-change of the process and by a quasisymmetric change of the metric. Two natural questions arise
(a) What are the possible values of the conformal walk dimension?
(b) When is the infimum attained?
In this talk, I will explain the answer to (a), and mention partial progress towards (b).
This talk is based on joint work with Naotaka Kajino.
Abstract. In this talk we introduce (in some sense natural) diffusion processes on a parametric family of fractals called generalized diamond fractals. These spaces arise as scaling limits of diamond hierarchical lattices, which are studied in the physics literature in relation to random polymers, Ising and Potts models among others. In the case of constant parameters, diamond fractals are self-similar sets. This property was exploited in earlier investigations by Hambly and Kumagai to study the properties of the corresponding diffusion process and its associated heat kernel. These questions are of interest in particular because in this setting some usual assumptions like volume doubling are not satisfied. Alternatively, a diamond fractal can also be regarded as an inverse limit of metric measure graphs. Through a procedure proposed by Barlow and Evans, one can construct a canonical diffusion process for more general parameters, also in the absence of self-similarity. It turns out that it is possible to give a rather explicit expression of the associated heat kernel, which is in particular uniformly continuous and admits an analytic continuation.
Abstract.The Zakharov system is a system of coupled Schrödinger-wave equations which was originally derived as a model in plasma physics. We show that in dimensions d>3 for large wave data, and small Schrödinger data, solutions to the Zakharov system exist globally in time and scatter. Moreover, we extend the regularity region for well-posedness to the sharp region. The key step is to prove a Strichartz estimate for the Schrödinger equation with a potentially large free wave potential. In contrast to previous work on the Zakharov system, we avoid the use of normal forms, and instead work with spaces which are carefully adapted to control bilinear interactions between solutions to the Schrödinger and wave equations. Avoiding the use of normal forms allows us to consider data with regularity in the extended full region of local well-posedness. This is joint work with Sebastian Herr and Kenji Nakanishi.
Abstract. We consider symmetric Dirichlet forms that consist of strongly local (diffusion) part and non-local (jump) part on a metric measure space. Under general volume doubling condition and some mild assumptions on scaling functions, we establish stability of two-sided heat kernel estimates in terms of Poincare inequalities, jumping kernels and generalized capacity inequalities. We also discuss characterizations of the associated parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2. This is a joint work with Z.Q. Chen (Seattle) and J. Wang (Fuzhou).
Abstract. If a random variable X has an absolutely continuous density f(x), its score is defined to be the random variable
\rho(X) = f'(X)/f(X), where f'(x) is the derivative of f. We will discuss upper bounds on the moments of the scores, especially in the case when X represents the sum of independent random variables.
Abstract. Within the framework of balayage spaces (the analytical equivalent of nice Hunt processes), we prove equicontinuity of bounded families of harmonic functions and apply it to obtain criteria for compactness of potential kernels.
Abstract. An important problem in the theory of random processes and random fields (Dynkin, Dobrushin etc.) is to describe the probability measures with given co-transition distributions, e.g. to find all Markov processes with the same co-transition distributions as a given Markov process. The set of all such measures is referred to as the Absolute. This problem is solved on certain groups, including commutative groups, nilpotent groups, trees. Connections with the theory of harmonic functions, Poisson-Furstenberg boundaries and others will be explained.
Abstract. The Duckman's function and distribution are well-known in physical and applied probabilistic literature. They are practically unknown among "pure probabilists". My talk will contain the introduction to the subject and several extensions of the theory. The applications include the member theory ( the works of Dickman and de Bruijn), cell-growth models, limit theorems on solvable Lie groups, differential-functional equations etc.
Abstract. We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the work, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest because it may yield isoperimetric type inequalities and Bounded Variation (BV) function spaces. This is a joint work with Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam.
Di 10:15-11:45 V4-11623.10.18 10:15 V4-116 Wolfhard Hansen (Bielefeld) Nearly hyperharmonic functions are infima of excessive functions 30.10.18 10:15 V4-116 Wolfhard Hansen (Bielefeld) Nearly hyperharmonic functions are infima of excessive functions II 20.11.18 10:15 V4-116 Jun Cao (Zhejiang University of Technology und Bielefeld) Heat kernels and Besov spaces associated with second order divergence form elliptic operators 27.11.18 10:15 V4-116 Jun Cao (Zhejiang University of Technology und Bielefeld) Heat kernels and Besov spaces associated with second order divergence form elliptic operators II 04.12.18 10:15 V4-116 Jian Wang (Fujian Normal University, China) Heat kernel and Harnack inequalities for random walks among random conductances with stable-like jumps 18.12.18 10:15 V4-116 Michael Hinz (Bielefeld) Hydrodynamic limits of exclusion processes on the Sierpinski gasket 15.01.19 10:15 V4-116 Shilei Kong (Bielefeld) On a class of hyperbolic graphs arising from iterations 22.01.19 10:15 V4-116 Melissa Meinert (Bielefeld) On the viscous Burgers equation on metric graphs and fractals 29.01.19 10:15 V4-116 Meng Yang (Bielefeld) Resistance Estimates and Heat Kernel Estimates
Di 10:15-11:45 V4-11624.04.18 10:15 V4-116 Shilei King (Bielefeld) Random walks and induced energy forms on compact doubling spaces 08.05.18 10:15 V4-116 Philipp Sürig (Bielefeld) Regularity results for fully nonlinear equations 15.05.18 10:15 V4-116 Shilei King (Bielefeld) Random walks and induced energy forms on compact doubling spaces II 22.05.18 10:15 V4-116 Shilei King (Bielefeld) Random walks and induced energy forms on compact doubling spaces III 29.05.18 10:15 V4-116 Meng Yang (Bielefeld) Construction of a local Dirichlet form on Sierpinski gasket 05.06.18 10:15 V4-116 Jun Cao (Bielefeld) The lifting-transference method for Grushin operators in Hardy spaces 19.06.18 10:15 V4-116 Michael Hinz (Bielefeld) Canonical diffusions on the pattern spaces of aperiodic Delone sets 26.06.18 10:15 V4-116 Wolfhard Hansen (Bielefeld) Nearly hyperharmonic functions and Jensen measures 03.07.18 10:15 V4-116 Andrey Piatnitski (Narvik) and Elena Zhizhina (Moscow) Pointwise estimates for heat kernels of convolution type operators 10.07.18 10:15 V4-116 Stanislav Molchanov (University of North Carolina, Charlotte, USA) Global limit theorems for the moderate tails 17.07.18 10:15 V4-116 Stanislav Molchanov (University of North Carolina, Charlotte, USA) Survey on the spectral theory of fractals
SFB 701, projects A6, A10
Di 10:15-11:45 V4-1192.05.17 10:15 V4-119 Jun Cao (Zhejiang University of Technology, China) Differential operators, semigroup and Hardy spaces 16.05.17 10:15 V4-119 Martin Barlow (University of British Columbia) Stability of the elliptic Harnack Inequality
|Abstract: Following the work of Moser, as well as de Giorgi and Nash, Harnack inequalities have proved to be a powerful tool in PDE as well as in the study of the geometry of spaces. In the early 1990s Grigor'yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI). This characterisation implies that the PHI is stable under bounded perturbation of weights, as well as rough isometries. In this talk we prove the stability of the EHI. This is joint work with Mathav Murugan (UBC).|
|Abstract: Motivated by the literature we consider several peculiar conditions on the heat kernel and the characteristic exponent of a Levy process in Rd and we show that they are equivalent. Next, we discuss (local) lower bounds for the heat kernel under those conditions. Assuming comparability with an isotropic unimodal Levy process on the level of Levy measures we complement the lower bound and also prove upper bound. The talk is based on a joint work with Tomasz Grzywny.|
Di 10:15-11:45 V3-204/V3-20125.10.16 10:15 V3-204 Eryan Hu (Bielefeld) Two-sided estimates of heat kernels of jump type Dirichlet forms 08.11.16 10:15 V3-204 Eryan Hu (Bielefeld) Two-sided estimates of heat kernels of jump type Dirichlet forms II 15.11.16 10:15 V3-204 Yuhua Sun (Nankai University) On nonnegative solutions of semilinear elliptic inequalities on Riemannian manifolds 06.12.16 10:15 V3-204 Wolfhard Hansen (Bielefeld) Reduced functions and Jensen measures 13.12.16 10:15 V3-204 Michael Hinz (Bielefeld) First order calculus for Dirichlet forms and some applications to PDE 20.12.16 10:15 V3-204 Pavlo Tkachev (Bielefeld) Acceleration and constant speed of propagation in non-local mono-stable reaction-diffusion equations 10.01.16 10:15 V3-204 Delio Mugnolo (Hagen) The Airy equation on a quantum graph 17.01.16 10:15 V3-204 Moritz Kassmann (Bielefeld) On Li-Yau estimates on graphs 24.01.16 10:15 V3-204 Elena Zhizhina (Bielefeld) Nonlocal operators with bounded kernels and homogenization 31.01.16 10:15 V3-204 Olaf Post (Trier) Norm resolvent convergence for operators in varying spaces and applications
Di 10:15-11:45 V3-204/V3-20119.04.16 10:15 V3-204 Boguslaw Zegarlinski (Imperial College London) Application of log-Sobolev inequality to reaction-diffusion system 26.04.16 10:15 V3-204 Jun Masamune (Tohoku University, Japan) Existence of non-constant integrable harmonic functions on Riemannian manifolds 10.05.16 10:15 V3-204 Jun Masamune (Tohoku University, Japan) Existence of non-constant integrable harmonic functions on Riemannian manifolds II 13.05.16 10:15 V3-201 Jun Kigami (Kyoto University, Japan) Time change of Brownian motion - Poincaré inequality, protodistance and heat kernel estimates 24.05.16 10:15 V3-204 Dimitri Volchenkov (Bielefeld) Diffusion metrics and geometrization of finite graphs and relational databases 07.06.16 10:15 V3-204 Meng Yang (Bielefeld) Jump processes on Sierpinski gasket 14.06.16 10:15 V3-204 Meng Yang (Bielefeld) Jump processes on Sierpinski gasket II 21.06.16 10:15 V3-201 Meng Yang (Bielefeld) Jump processes on Sierpinski gasket and carpet 28.06.16 10:15 V3-201 Jiaxin Hu (Tsinghua University, China) The Davies method for heat kernel upper bounds of regular Dirichlet forms on metric spaces 19.07.16 10:15 V3-201 Michael Hinz (Bielefeld) Some questions related to metric cohomology
Di 10:15-11:45 V4-11603.11.15 10:15 V4-116 Bartosz Trojan (Wroclaw) Random Walks on Grids
|Abstract: The aim of the talk is to present the asymptotic of the heat kernel $p(n, x)$ for a finitely supported random walk on $\ZZ^d$, uniform in $n$ and $x$ on a large region.|
|Abstract: We review the main ideas of the article "Li-Yau inequality on graphs" by Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoubi, and Shing-Tung Yau. We present the idea in the simplest context of finite and infinite but flat graphs.|
present the asymptotic formulas and estimates for the transition densities
of isotropic unimodal convolution semigroups of probability measures on
R^d under the assumption that its Levy
exponent varies slowly. The talk is based on the joint project with M. Ryznar and B. Trojan.
Di 10:15-11:45 V4-11614.04.15 10:15 V4-116 Christian Rose (Chemnitz) Schrödinger operators on manifolds: the role of curvature 16.04.15 15:00-16:00 U2-147 Naotaka Kajino (Kobe University, Japan) Heat kernel analysis for Brownian motion of 2-dimensional Liouville quantum gravity 21.04.15 10:15 V4-116 Alexander Bendikov (Wroclaw/Bielefeld) Random perturbations of the hierarchical Laplacian 05.05.15 10:15 V4-116 Wolfhard Hansen (Bielefeld) A general approach to Harnack inequalities and Hölder continuity for harmonic functions 19.05.15 10:15 V4-116 Olaf Müller (Regensburg) Existence of a metric of bounded geometry in every conformal class and implications for the Yamabe flow 26.05.15 10:15 V4-116 Tomasz Grzywny (Wroclaw University of Technology) Asymptotics of heat kernels of unimodal convolution semigroups
In this talk we investigate behaviour of densities for isotropic
unimodal Lévy processes.
The main result is a description of the asymptotics under an assumption that the Lévy-Khinchine
exponent varies regularly of index between 0 and 2. Moreover, we show that for unimodal Lévy processes, the regular variation of the characteristic exponent is equivalent to the asymptotic behaviour for the transition density.
|Abstract: The talk deals with the exterior derivative operator defined on 1-forms on topologically one dimensional spaces with a strongly local regular Dirichlet form. It is proved that exterior derivative operator taking 1-forms into 2-forms is not closable if the martingale dimension is larger than one. Although the main results are applicable to general diffusions, some of the most interesting examples include the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. For these carpets we prove that not only the curl operator is not closable, but that its adjoint operator has a trivial domain.|
|Abstract: After the pioneer works on cloaking (electromagnetic invisibility) simultaneously publisched by Leonahrdt (Science 312, 2006, 1777-1780) and Pendry, Shurig and Smith (Science 312, 2006, 1780-1782), several papers gave a general definition of cloaking in closely related setting of electric impedance tomography and for the Helmoltz Equation. Among others R.V.Kohn et al (Inverse Problems 2007 and Comm. Pure and Appl. Math. Vol LX III, 2010) gave the following definition: a region of space is cloaked for a particular class of measurements if its contents and even the existence of the cloak are invisible using such measurements. In a joint project with Michael Roeckner we will present a preliminary version for the cloaking related to a quasilinear elliptic differential equation.|
Di 10:15-11:45, V4-11228.10.14 10:15 V4-112 Satoshi Ishiwata (Yamagata University, Japan) A central limit theorem for non-symmetric random walk on crystal lattices 04.11.14 10:15 V4-112 Matthias Keller (Jena) Spectral theory and intrinsic metrics on graphs 11.11.14 10:15 V4-112 Shun-Xiang Ouyang (Bielefeld) Volume growth and escape rate of diffusion processes 18.11.14 10:15 V4-112 Shiping Liu (Durham, UK) Eigenvalue ratios on closed Riemannian manifolds with nonnegative Ricci curvature 27.11.14 10:15 V3-201 Asma Hassannezhad (MPI Bonn) Eigenvalue bounds in Riemannian and sub-Riemannian geometry 13.01.15 10:15 V4-112 Michael Hinz (Bielefeld) Magnetic fields on resistance spaces 20.01.15 10:15 V4-112 Wolfgang Hansen (Bielefeld) Hunt's hypothesis (H) and triangle property for the Green function 03.02.15 10:15 V4-112 Moritz Kaßmann (Bielefeld) Intrinsic scaling for jump processes
Di 10:15-11:45, V4-11608.04.14 10:15 V4-116 Thierry Coulhon (Australian National University, Canberra) New approaches to Gaussian heat kernel upper and lower bounds 15.04.14 10:15 V4-116 Wolfhard Hansen (Bielefeld) Potential theory for processes with isotropic unimodal Green function 29.04.14 10:15 V4-116 Wolfhard Hansen (Bielefeld) Unavoidable collections of balls for processes with isotropic unimodal Green function 06.05.14 10:15 V4-116 Wolfhard Hansen (Bielefeld) Unavoidable collections of balls for processes with isotropic unimodal Green function II 13.05.14 10:15 V4-116 Michael Hinz (Bielefeld) Feynman-Kac-Ito formulas for local regular Dirichlet forms 26.05.14 10:15 D2-136 Tomasz Grzywny (Wroclaw University of Technology) Exit time and survival probability for unimodal Levy processes
|Abstract: The basic object of interest in this talk is the expected exit time from a bounded smooth domain for arbitrary starting point of an isotropic unimodal Levy process. We derive sharp estimates up to the boundary of the set by giving barriers for the ball of arbitrary radius and subharmonic functions in the complement of the ball. Next we discus applications of those for instance estimates of the survival probability in bounded smooth domains or exteriors sets.|
|Abstract: A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors. Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators. In this talk, we will construct a family of geometrically natural sub-elliptic Laplacian operators and discuss the trouble with defining one which is canonical. We will also construct a random walk on M which converges weakly to a process whose infinitesimal generator is one of our sub-elliptic Laplacian operators. This is joint work with Tom Laetsch.|
Di 10:15-11:45, V3-204
|Abstract: I will present recently obtained results about isotropic unimodal Levy processes. These include: the Harnack inequality, the boundary Harnack inequality, estimates of the expected exit time (up to the boundary of set), the survival probabilities and Dirichlet heat kernel of a ball, a half-space and the complement of a ball.|
|Abstract: The related heat maximal operator and the first-order Riesz transform will be seen to be of weak type 1,1 for the appropriate measure.|
|Abstract: Fourier restriction conjecture is one of the most well-known open problems in harmonic analysis. It is closely related to many other problems. In this talk, I will focus on its role in PDEs. The content of the talk is: (I) Introduction on the Fourier restriction conjecture (II) Weak Fourier restriction estimate (III) Connection to the Strichartz estimates and generalizations (IV) Application to Zakharov system.|
(2n-2)/3 and n/3<D<n/2 21.11.13 10:15 V3-201 Peter Stollmann (Chemnitz) The complex Laplacian and its heat semigroup 26.11.13 10:15 V3-204 Elton P. Hsu (Northwestern University, USA) Geometric Deviation from Levy's arcsine law 03.12.13 10:15 V3-204 (joint with B8) Junfeng Li (Beijing Normal University / Bonn) Some results in the well posedness of KP II problems
In this talk, I will present some recent results on the well posedness of
KP II problems. In these results, we find that the Galilean
invariant are very important. By decomposing the nonlinear
part of the problems into some Galilean invariant terms, we could obtain some more interesting bilinear estimates which we thought be very nature in the context of KP.
|Abstract: We will review the classical Strichartz estimates for wave and Schroedinger equations and related tools from harmonic analysis. For the Dirac equation and the Klein-Gordon equation we will present new endpoint estimates in dimension three, which have been obtained recently in collaboration with Ioan Bejenaru (UC San Diego).|
|Abstract: We discuss a subclass of (integro-)differential operators of fractional order. These
operators are related to semi-groups and stochastic processes in a natural way. In the talk we present definitions, basic results as well as recent developments for linear and nonlinear equations. The main
emphasis is on new intrinsic scaling properties.