Heat kernels and analyticity of non-symmetric jump diffusion semigroups

• 成果类型: Article
• 署名作者: Chen, Zhen-Qing; Zhang, Xicheng
• 署名单位: University of Washington; University of Washington Seattle; Wuhan University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0631-y
• 发表日期: 2016
• 页码: 267-312
• 关键词: fractional laplacian; EQUATIONS

摘要

Let d >= 1 and alpha epsilon (0, 2). Consider the following non-local and nonsymmetric Levy-type operator on R-d : L-alpha(kappa) f(x) := p.v. integral(Rd) (f (x + z) - f (x))kappa(x, z)/vertical bar z vertical bar(d+alpha) dz, where 0 < kappa(0) <= kappa(x, z) <= kappa(1), kappa(x, z) = kappa(x,-z), and vertical bar kappa(x, z) - kappa(y, z) <= kappa(2)vertical bar x -y vertical bar(beta) for some beta is an element of(0, 1). Using Levi's method, we construct the fundamental solution (also called heat kernel) p(alpha)(kappa)(t, x, y) of L-alpha(kappa), and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that p(alpha)(kappa) (t, x, y) is jointly Hlder continuous in (t, x). The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of L-alpha(kappa) gives rise a Feller process {X, Px, x. Rd} on Rd. We determine the LEvy system of X and show that Px solves themartingale problem for (L-alpha(kappa), C-b(2)(R-d)). Furthermore, we show that the C0-semigroup associated with L-alpha(kappa) is analytic in L-p(R-d) for every p is an element of [1, infinity). A maximum principle for solutions of the parabolic equation partial derivative(t)u = L-alpha(kappa) 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 dX(t) = A(Xt-)dY(t) is derived, where Y is a ( rotationally) symmetric stable process on R-d and A(x) is a Holder continuous d x d matrix-valued function on R-d that is uniformly elliptic and bounded.