Heat kernels and analyticity of non-symmetric jump diffusion semigroups

• Published in: Probability theory and related fields Vol. 165; no. 1-2; pp. 267 - 312
• Main Authors: Chen, Zhen-Qing, Zhang, Xicheng
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2016; Springer Nature B.V
• Subjects: Continuity (mathematics); Derivatives; Differential equations; Diffusion; Economics; Estimates; Finance; Insurance; Kernels; Lower bounds; Management; Markov analysis; Martingales; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Maximum principle; Operations Research/Decision Theory; Operators (mathematics); Partial differential equations; Probability; Probability theory; Probability Theory and Stochastic Processes; Quantitative Finance; Semigroups; Statistics for Business; Stochastic models; Studies; Texts; Theoretical; 60J75; Fractional derivative estimate; Primary 60J35; Heat kernel estimate; Non-symmetric stable-like operator; Discontinuous Markov process; Levi’s method; 47G20; Lévy system; Stable process; Martingale problem; Stochastic differential equation; Secondary 47D07
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-015-0631-y

Let d ⩾ 1 and α ∈ ( 0 , 2 ) . Consider the following non-local and non-symmetric Lévy-type operator on R d : L α κ f ( x ) : = p.v. ∫ R d ( f ( x + z ) - f ( x ) ) κ ( x , z ) | z | d + α d z , where 0 < κ 0 ⩽ κ ( x , z ) ⩽ κ 1 , κ ( x , z ) = κ ( x , - z ) , and | κ ( x , z ) - κ ( y , z ) | ⩽ κ 2 | x - y | β for some β ∈ ( 0 , 1 ) . Using Levi’s method, we construct the fundamental solution (also called heat kernel) p α κ ( t , x , y ) of L α κ , and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that p α κ ( t , x , y ) is jointly Hölder continuous in ( t , x ) . The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of L α κ gives rise a Feller process { X , P x , x ∈ R d } on R d . We determine the Lévy system of X and show that P x solves the martingale problem for ( L α κ , C b 2 ( R d ) ) . Furthermore, we show that the C 0 -semigroup associated with L α κ is analytic in L p ( R d ) for every p ∈ [ 1 , ∞ ) . A maximum principle for solutions of the parabolic equation ∂ t u = L α κ u is also established. As an application of the main result of this paper, sharp two-sided estimates for the transition density of the solution of d X t = A ( X t - ) d Y t is derived, where Y is a (rotationally) symmetric stable process on R d and A ( x ) is a Hölder continuous d × d matrix-valued function on R d that is uniformly elliptic and bounded.