# Stochastic calculus with respect to the fractional Brownian motion

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

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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