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FUNDAMENTAL SOLUTIONS TO KOLMOGOROV-FOKKER-PLANCK EQUATIONS WITH ROUGH COEFFICIENTS: EXISTENCE, UNIQUENESS, UPPER ESTIMATES.

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Title: FUNDAMENTAL SOLUTIONS TO KOLMOGOROV-FOKKER-PLANCK EQUATIONS WITH ROUGH COEFFICIENTS: EXISTENCE, UNIQUENESS, UPPER ESTIMATES.
Authors: AUSCHER, PASCAL1 pascal.auscher@universite-paris-saclay.fr, IMBERT, CYRIL2 cyril.imbert@ens.psl.eu, NIEBEL, LUKAS3 lukas.niebel@uni-muenster.de
Source: SIAM Journal on Mathematical Analysis. 2025, Vol. 57 Issue 2, p2114-2137. 24p.
Subjects: Differential operators, Operator equations, Cauchy problem, Integrals, Equations
Abstract: We show the existence and uniqueness of fundamental solution operators to Kolmogorov-Fokker-Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is, when the diffusion operator is of differential type, we prove L² decay using Davies's method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's L² Lâž estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay. [ABSTRACT FROM AUTHOR]
Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: We show the existence and uniqueness of fundamental solution operators to Kolmogorov-Fokker-Planck equations with rough (measurable) coefficients and local or integral diffusion on finite and infinite time strips. In the local case, that is, when the diffusion operator is of differential type, we prove L² decay using Davies's method and the conservation property. We also prove that the existence of a generalized fundamental solution with the expected pointwise Gaussian upper bound is equivalent to Moser's L² Lâž estimates for local weak solutions to the equation and its adjoint. When coefficients are real, this gives the existence and uniqueness of such a generalized fundamental solution and a new and natural way to obtain pointwise decay. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of SIAM Journal on Mathematical Analysis is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – TitleFull: FUNDAMENTAL SOLUTIONS TO KOLMOGOROV-FOKKER-PLANCK EQUATIONS WITH ROUGH COEFFICIENTS: EXISTENCE, UNIQUENESS, UPPER ESTIMATES.
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              Text: 2025
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