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On the equivalence of heat kernels of second-order parabolic operators

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Abstract

Let P be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold M, and let V be a real valued function which belongs to the class of small perturbation potentials with respect to the heat kernel of P in M. We prove that under some further assumptions (satisfied by large classes of P and M) the positive minimal heat kernels of P - V and of P on M are equivalent. Moreover, the parabolic Martin boundary is stable under such perturbations, and the cones of all nonnegative solutions of the corresponding parabolic equations are affine homeomorphic.

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Acknowledgements

The authors wish to thank Professor Baptiste Devyver and Professor Alexander Grigor’yan for valuable discussions. They acknowledge the support of the Israel Science Foundation (grants No. 970/15) founded by the Israel Academy of Sciences and Humanities. D. G. was supported in part at the Technion by a fellowship of the Israel Council for Higher Education.

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  1. Debdip Ganguly

    Present address: Department of Mathematics, Indian Institute of Technology Delhi, IIT Campus, Hauz Khas, New Delhi, Delhi, 110016, India

Authors and Affiliations

  1. Department of Mathematics, Technion Israel Institute of Technology, Haifa, 32000, Israel

    Debdip Ganguly & Yehuda Pinchover

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  1. Debdip Ganguly
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  2. Yehuda Pinchover
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Correspondence to Yehuda Pinchover.

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Ganguly, D., Pinchover, Y. On the equivalence of heat kernels of second-order parabolic operators. JAMA 140, 549–589 (2020). https://doi.org/10.1007/s11854-020-0097-4

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  • DOI: https://doi.org/10.1007/s11854-020-0097-4

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