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. |
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| 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] |
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| Database: | Engineering Source |
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