SFB 701, project A6
Abstract: We introduce the class of spherically symmetric graphs of unbounded geometry. We will then discuss their stochastic completeness and basic spectral properties. We also show how to use this class to obtain some comparison results. |
Abstract: I will show the Mosco convergence of the Dirichlet forms which is defined on a cylinder with highly conductive boundary. The tightness of the associated Wiener measures will be shown, and as a conclusion, we will obtain the weak-convergence of those Wiener measures. |
Abstract: In this talk, I would like to show the conservation property of a non-local regular Dirichlet form on an Euclidean space with Radon measure under a volume growth condition. Our approach is to develop the integral deviation property and to apply the method by M.P. Gaffney. If the time permits, I would like to talk also on the L^p Liouville property for the same forms. Both results were obtained in joint works with Toshihiro Uemura. |
Abstract: We discuss the problem of description of real hypersurfaces that are (locally) homogeneous with respect to different structures (holomorphic, affine, projective) of the ambient complex spaces of small dimensions. The history of this problem goes back to 1932 when E. Cartan gave a complete solution of the holomorphic homogeneity problem in the case of 2-dimensional complex spaces. It was not until 1995 when a comprehensive description of affinely homogeneous surfaces of 3-dimensional real space was obtained by
Doubrov, Komrakov and Rabinovich. We describe a coefficient approach to the problem, that is connected to the use of the canonical equations of the manifolds under consideration and to the description of Lie algebras of vector fields on these surfaces. Recent results of the author about the homogeneity in 2- and 3-dimensional complex spaces will be presented. |
Abstract: Given an electrical network (a finite set of nodes and resistances between pairs of nodes), there is, for every pair of nodes, a number called effective resistance, so that the network with respect to this pair of nodes is equivalent to a single wire equipped with the effective resistance. Given the $\binom{n}{2}$ resistances we can compute the set of $\binom{n}{2}$ effective resistances (e.g. using Schur complements). We are interested in the inverse problem. Given effective resistances, can we recover the original resistances? Kigami showed that there is at most one network solving this inverse problem without giving an explicit solution. We will give an explicit solution using an old determinant identity due to Sylvester and point out a connection to the problem of counting spanning trees. |
Abstract: The recent development of Analysis on fractal spaces has motivated the study of abstract heat kernels on metric measure spaces. An abstract heat kernel is a function on such a space that by definition satisfies certain standard properties of the classical heat kernels, such as positivity, symmetry, the semi-group property, approximation of identity, and stochastic completeness. Such a heat kernel is associated with a Markov process on the space in question and is, in fact, its transition density. In many cases of interest, heat kernels satisfy two-sided estimates of two types: (1) Gaussian or, more generally, sub-Gaussian estimate, which is typical for diffusion processes; (2) estimate with a polynomial tail, which is typical for symmetric stable processes. One of the main results is that, under certain mild assumptions, these two cases exhaust all possible heat kernel estimates of homogeneous type. |
Abstract: The talk will consider the simple random walk on the supercritical clusters of bond percolation in the d-dimensional integer lattice. It is now well known that there are Gaussian heat kernel estimates and a quenched invariance principle for this process. We establish a parabolic Harnack inequality and show that this leads to a local limit theorem for the convergence of the transition kernel for the random walk to the Gaussian heat kernel. These ideas can be extended to other random walks on sequences of graphs. |
Abstract: In 1975, Kesten, Kozlov and Spitzer proved a limit theorem for one
dimensional random walks in random environments with null speed. They proved that the RWRE properly renormalised converges to a stable law, but they left open the explicit description of the parameters of this law. In this talk, based on a joint work with N. Enriquez and O. Zindy, I will present a different proof of this limit theorem, based on the analysis of the potential associated with the environment, which leads to a description of the parameters of the law. The case of beta environments appears to be particularly explicit. |
Abstract: I will talk about the relationships between the parabolic Harnack
inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks
with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates. This is a joint work with M.T. Barlow, A. Grigor'yan and with M.T. Barlow, R.F. Bass. |
Di 14:30-17:00 T2-213
17.04.07 Jiaxin Hu (Beijing) Domains of Dirichlet forms and the effective resistance estimates on p.c.f. fractals 08.05.07 Gilles Carron (Nantes) Riesz transform and connected sumAbstract: We will present recent results (both positive and negative) on the following question. Assume that M is a complete Riemannian manifold such that the Riesz transform D^{-1/2} is bounded on L^{p }for a certain p in ]1,∞[. Is it true that the Riesz transform is bounded on the L^{p} on the connected sum M#M? This is partially a joint work with T. Coulhon (Cergy) and A. Hassell (Canberra). |
Mi 15:30-18:00 U5-133
08.11.06 Andras Telcs (Budapest) Heat kernels on graphs, isoperimetric inequalities, and Einstein relation 15.11.06 Wolfhard Hansen (Bielefeld) Convexity of limits of harmonic measures 22.11.06 Laurent Saloff-Coste (Cornell) Heat kernels, Neumann and Dirichlet, in intrinsically uniform domainsAbstract: There are many great work dealing with the Dirichlet problem and the Martin boundary of Euclidean domains, starting with the paper by Hunt and Wheeden (1970) on the Martin Boundary of Lipschitz domains, the papers of Dahlberg (1977), Ancona (1978) and Wu (1978), the work of Jerison and Kenig (1982) and more recent work of Aikawa. These works are concerned with the elliptic theory. Aspects of the parabolic case appears in work by E. Fabes and M. Safonov and their collaborators including Garofalo, Salsa, and Yuan. Strangely enough, the heat kernel is absent of most (if not all) of these works and only appeared quite recently in work of N. Varopoulos (and also Q.S. Zhang and R. Song). In this talk, I will discuss a recent joint work with my graduate student Pavel Gyrya where we study the heat kernel (with either Neumann or Dirichlet boundary conditions) in a large class of domains that includes the domain above the graph of any Lipschitz function and the exterior of any convex set in Euclidean n-space. |
Abstract: There is a simple classification of the metric triples (=metric spaces with measure) up to measure preserving isometries (Gromov-V), which is a particular case of the classification of the measurable functions of several variables. The invariant is a measure on the space of so called distance matrices. There are many open problems related to this subject. |
Abstract: Geometric evolution equations for hypersurfaces
can be used to prove several sharp isoperimetric inequalities. The lecture concentrates on the analytical aspects of this method |
Abstract: We introduce and analyze generalized Ricci curvature bounds for metric measure spaces (M;d;m), based on convexity properties of the relative entropy Ent(:jm). For Riemannian manifolds, Curv(M;d;m) ¸ K if and only if Ric(M) ¸ K; for the Wiener space, Curv(M;d;m) = 1. One of the main results is that these lower curvature bounds are stable under (e.g. measured Gromov-Hausdorff) convergence. This solves one of the basic problems in this field, open for many years. Furthermore, we introduce a (more restrictive) curvature-dimension condition CD(K;N) which implies sharp versions of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows to construct a canonical Dirichlet form with Gaussian bounds for the corresponding heat kernel. Finally, we indicate how to construct a canonical reversible process on the L^{2}-Wasserstein space of probability measures P(R), regarded as an infinite dimensional Riemannian manifold. This process has an invariant measure P^{¯} which may be characterized as the 'uniform distribution' on P(R) with weight function 1/Z exp(¡¯ ¢Ent(:jm)) where m denotes a given finite measure on R. One of the key results is the quasi-invariance of this measure P^{¯} under push forwards ¹ 7! h_{¤}¹ by means of smooth di®eomorphisms h of R. |