@article{29914e234b564c46bebbcf8e4a31f6bf,
title = "Heat kernel estimates for stable-like processes on d-sets",
abstract = "The notion of d-set arises in the theory of function spaces and in fractal geometry. Geometrically self-similar sets are typical examples of d-sets. In this paper stable-like processes on d-sets are investigated, which include reflected stable processes in Euclidean domains as a special case. More precisely, we establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such stable-like processes. Results on the exact Hausdorff dimensions for the range of stable-like processes are also obtained.",
keywords = "Besov spaces, Heat kernels, Jump processes, L{\'e}vy systems, Parabolic Harnack inequality, Stable-like processes",
author = "Chen, {Zhen Qing} and Takashi Kumagai",
note = "Funding Information: This research was initiated while the second author was visiting University of Washington with the support by Grant-in-Aid for Scientific Research (B)(2) 13440030 Japan Society for the Promotion of Science. He thanks the University of Washington for its hospitality. Note added in revision. After the manuscript was submitted, we were informed of the papers Bogdan et al. (2002a, b) . Although these two papers are related to the general topic of this paper, where Bogdan et al. (2002a) is the announcement of Bogdan et al. (2002b) , their contents and starting point are different from ours. The processes considered in Bogdan et al. (2002a, b) are the subordinations of a fractional diffusion on a d-set that is assumed to have two-sided heat kernel estimate (1.7) . The main purpose of Bogdan et al. (2002a, b) is to obtain Harnack inequality for these subordinated processes. In fact, a stronger result can be established. Under the assumption of Bogdan et al. (2002a, b) , a direct calculation using subordination and (1.7) shows that the heat kernels for the subordinated processes have estimate (1.6) and so, by the second paragraph following Theorem 1.2 in Section 1 of our paper the parabolic Harnack inequality follows. Furthermore, it can be shown, as it is done in Theorem 4.14 of this paper that any parabolic functions are H{\"o}lder continuous.",
year = "2003",
month = nov,
day = "1",
doi = "10.1016/S0304-4149(03)00105-4",
language = "English",
volume = "108",
pages = "27--62",
journal = "Stochastic Processes and their Applications",
issn = "0304-4149",
publisher = "Elsevier",
number = "1",
}