2021/03/21 by D. Kinzebulatov, Kinzebulatov, D., Yu. A. Semënov +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2103.11482
openalex publication_date 2021/03/21 · openalex created_date 2022/07/13 · openalex updated_date 2026/07/28
We consider Kolmogorov operator -∇ ⋅ a ⋅ ∇ + b ⋅ ∇ with measurable uniformly elliptic matrix a and prove Gaussian lower and upper bounds on its heat kernel under minimal assumptions on the vector field b and its divergence \rm div b. More precisely, we prove: (1) Gaussian lower bound, provided that \rm div b ≥ 0, and b is in the class of form-bounded vector fields (containing e.g. the class Ld, the weak Ld class, as well as some vector fields that are not even in L\rm loc2+ε, ε>0); in these assumptions, the Gaussian upper bound is in general invalid; (2) Gaussian upper bound, provided that b is form-bounded, and the positive part of \rm div b is in the Kato class; in these assumptions, the Gaussian lower bound is in general invalid; (3) Gaussian upper and lower bounds, provided that b is form-bounded, \rm div b is in the Kato class; (4) A priori Gaussian upper and lower bounds, provided that b is in a large class containing the class of form-bounded vector fields, \rm div b is in the Kato class.