| Date | Title | Speaker |
|---|---|---|
| Feb 11, 2011 | On the stochastical completeness of graphs | Jun Masamune |
On the stochastical completeness of graphs
Presented by Jun Masamune, Penn State Altoona
Friday, February 11, 2011 | 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).