This work is related to statistics of diffusion and its application into math finance

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