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      Heat kernel estimates on spaces with varying dimension

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          Abstract

          Z.-Q. Chen and S. Lou (Ann. Probab. 2019) constructed Brownian motion on a space with varying dimension, in which a 1-dimensional space and a 2-dimensional space are connected at one point, and derived sharp two-sided estimates for its transition density (heat kernel). In this paper, we obtain sharp two-sided heat kernel estimates 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.

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          Author and article information

          Journal
          15 March 2020
          Article
          2003.06760
          00d45b8b-acc4-4e44-85c1-68f6b0649953

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          History
          Custom metadata
          60J60, 60J35, 31C25, 60H30, 60J45
          41 pages
          math.PR

          Probability
          Probability

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