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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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