Dispersion under finite mode Kolmogorov flow.

Speaker: 

Professor Michael Cranston

Institution: 

University of California, Irvine

Time: 

Tuesday, December 7, 2004 - 11:00am

Location: 

MSTB 254

We consider the rate of spread of a body of passive tracers moving under the influence of a random evolving vector field.
The vector field is of a type used as a model for ocean currents and was introduced by Kolmogorov. The rate of growth of the diameter of the body is of interest for practical reasons (such as in problems of pollution control) and we specify its rate of growth.

Random walks on percolation clusters

Speaker: 

Professor Martin Barlow

Institution: 

University of British Columbia

Time: 

Tuesday, October 19, 2004 - 11:00am

Location: 

MSTB 254

This talk will discuss random walks on percolation clusters.

The first case is supercritical ($p>p_c$) bond percolation in
$Z^d$. Here one can obtain Aronsen type bounds on the transition
probabilities, using analytic methods based on ideas of Nash.

For the critical case ($p=p_c$) one needs to study the incipient
infinite cluster (IIC). The easiest situation is the IIC on trees -
where the methods described above lead to an alternative approach to
results of Kesten (1986). (This case is joint work with T. Kumagai).

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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