Abstract
The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form \({\mathscr{L}}_{1} + {\mathscr{L}}_{2} - \partial _{t}\), but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators \({\mathscr{L}}_{1} - \partial _{t}\) and \({\mathscr{L}}_{2} - \partial _{t}\). Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander’s 1967 groundbreaking paper on hypoellipticity.
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23 August 2022
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Both authors are supported in part by a Progetto SID: “Non-local Sobolev and isoperimetric inequalities”, University of Padova, 2019.
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Garofalo, N., Tralli, G. Heat Kernels for a Class of Hybrid Evolution Equations. Potential Anal 59, 823–856 (2023). https://doi.org/10.1007/s11118-022-10003-2
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DOI: https://doi.org/10.1007/s11118-022-10003-2