| Date | Title | Speaker |
|---|---|---|
| Feb 11, 2012 | On the stochastical completeness of graphs | Jun Masamune |
| Mar 3, 2012 | Random homogenization of the fluid flow through leaky obstacles | Florin Maris |
| Apr 1, 2012 | The zeta-function of elliptic operators | Thomas Krainer |
| May 6, 2012 | Dirichlet and Neumann Laplacians on Infinite Graphs | Radoslaw Wojciechowski |
On the stochastical completeness of graphs
Presented by Jun Masamune, Penn State Altoona
Saturday, February 11, 2012 | 3:00 PM | 257 Hawthorn Building
Abstract: A space is called stochastically complete if the canonical stochastic process starting from an arbitrary point in the space stays in the space for any time. Determining the stochastic completeness is one of the most fundamental problems in probability and potential theory. For example, one finds that Euclidean spaces and Z^n are stochastically complete by Gaussian integral. On the other hand, it is known that the rapid volume growth of the space at infinity may violate the stochastic completeness. So, a challenge in this problem is to find the sharp volume growth condition.
For a Riemannian manifold, A. Grigori’yan found a sharp volume growth condition in 1986. This result was extended to a Dirichlet form of local type by K. Th. Sturm in 1994. However, the counter result for a discrete space; namely, a random walk on a graph, has not been known.
In this talk, I will talk about a new result on a sharp volume growth of a graph to be stochastically complete. First, I will talk about a volume growth condition for a general non-local operator to be stochastically complete, and next, I will apply it for a random walk on a graph.
This result was obtained in a collaboration with A. Grigori’yan and X.P. Haung (Bielefeld).
Random homogenization of the fluid flow through leaky obstacles
Presented by Florin Maris, WPI
Saturday, March 3, 2012 | 4:00 PM | 256 Hawthorn Building
Abstract: In this talk, I will present a homogenization result concerning the flow of a viscous fluid passing through permeable obstacles with random sizes and shapes distributed along a hyperplane. Leak boundary conditions of threshold type are considered on the obstacles, the normal velocity of the fluid is zero until the jump of the normal component of the stress reaches a certain limit.
The description of the obstacles is given in terms of a random set-valued variable defined on a probability space and a dynamical system acting on it. Effective boundary conditions for the fluid are derived. Depending on the relative size of the holes, there are two cases , in the first case we obtain a homogenized equation of the same type and in the second case a slip boundary condition of Navier type is obtained. If the dynamical system is assumed to be ergodic, the limiting behaviour of the fluid is deterministic.
The method of proof is based on the Mosco convergence, which allows us to pass from the stationary case to the time dependent case through the convergence of associated semigroups.
The zeta-function of elliptic operators
Presented by Thomas Krainer, Penn State Altoona
Sunday, April 1, 2012 | 3:00 PM | 257 Hawthorn Building
Abstract: The talk will have the character of a colloquium rather than a specialized research seminar. I plan to speak about some aspects of zeta-functions of elliptic operators and how they relate to heat kernels and trace expansions. I will explain the “standard” results, hint at how they are proved, and why the standard results are desirable. I will also discuss situations where the standard results fail. While these “nonstandard” phenomena are highly undesirable, they are nevertheless present and are still wanting proper explanation.
Dirichlet and Neumann Laplacians on Infinite Graphs
Presented by Radoslaw Wojciechowski, CUNY
Sunday, May 6, 2012 | 3:00 PM | 256 Hawthorn Building
Abstract: We will introduce a general framework for the study of operators on discrete measure spaces and then focus on the Dirichlet and Neumann Laplacians. In particular, we will characterize when these two operators agree and explore connections to other properties of infinite graphs such as stochastic and metric completeness and essential self-adjointness. We will then illustrate the various possibilities with a series of examples. This is joint work with Sebastain Haeseler, Matthias Keller, and Daniel Lenz from the Friedrich Schiller University in Jena, Germany.