Tohoku Mathematical Journal
2022
June
SECOND SERIES VOL. 74, NO. 2

Tohoku Math. J.
74 (2022), 165-194

Title HEAT KERNEL ESTIMATES ON SPACES WITH VARYING DIMENSION

Author Takumu Ooi

(Received July 21, 2020, revised November 10, 2020)
Abstract. We obtain sharp two-sided heat kernel estimates for some process whose regular Dirichlet form is strongly local on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by Z.-Q. Chen and S. Lou (Ann. Probab. 2019), in which a 1-dimensional space and a 2-dimensional space are connected at one point.

Mathematics Subject Classification. Primary 60J60; Secondary 60J35, 31C25, 60H30, 60J45.
Key words and phrases. Space of varying dimension, Brownian motion, heat kernel estimates.

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