2008/06/23 by Krzysztof Bogdan, Bogdan, Krzysztof, Wolfhard Hansen +3
Computer Science · Engineering · Mathematics · #47D08 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Probability (math.PR) #Stability and Controllability of Differential Equations #math.FA #math.PR #msc:47D08
paper · pdf · doi:10.48550/arxiv.0806.3549
22 pages, to appear in Studia Mathematica
openalex publication_date 2008/06/23 · arxiv created 2008/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct the fundamental solution of ∂t-Δy- q(t,y), for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure spaces. We also discuss specific applications to Schrödinger perturbations of the fractional Laplacian in view of the fact that the 3P Theorem holds for the fundamental solution corresponding to the operator.