On Calabi's strong maximum principle via local Dirichlet forms

Speaker: 

Prof. Kazuhiro Kuwae

Institution: 

Yokohama, visiting UCSD

Time: 

Tuesday, February 3, 2004 - 11:00am

Location: 

MSTB 254

I will talk on a generalization of classical Calabi's strong maximum (1957) in the framework of Dirichlet forms associated with strong Feller diffusion processes.
The proof is stochastic and the result can be applicable to a singular geometric space appeared in the measured Gromov-Hausdorff convergence (precisely in the convergence by spectral distance by Kasue Kumura) of compact Riemannian manifolds with uniform lower Ricci curvature and uniform upper diameter.

Heat kernel estimates for jump processes of mixed types on metric measure spaces

Speaker: 

Professor Zhenqing Chen

Institution: 

University of Washington

Time: 

Tuesday, May 2, 2006 - 1:00pm

Location: 

MSTB 254

Alfors $d$-regular set is a class of fractal sets which
contains geometrically self-similar sets.
In this paper, we investigate symmetric jump-type processes
on $d$-sets with jumping intensities comparable
to radially symmetric functions on $d$-regular sets.
A typical example is the symmetric jump process with jumping intensity
$$
\int_{\alpha_1}^{\alpha_2} \frac{c(\alpha, x, y)}{|x-y|^{d+\alpha}} \,
\nu (d\alpha),
$$
where $\nu$ is a probability measure on $[\alpha_1, \alpha_2]\subset (0, 2)$, and $c(\alpha, x, y)$ is a jointly measurable function that is symmetric in $(x, y)$ and is bounded between two positive constants.
We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

This is a joint work with Takashi Kumagai

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