PERIODIC HOMOGENIZATION OF NONSYMMETRIC LEVY-TYPE PROCESSES

• 成果类型: Article
• 署名作者: Chen, Xin; Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian
• 署名单位: Shanghai Jiao Tong University; University of Washington; University of Washington Seattle; Kyoto University; Fujian Normal University; Fujian Normal University; Fujian Normal University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/21-AOP1518
• 发表日期: 2021
• 页码: 2874-2921
• 关键词: CENTRAL-LIMIT-THEOREM; integrodifferential equations; heat kernels; DIFFUSIONS; Operators

摘要

In this paper we study homogenization problem for strong Markov processes on R-d having infinitesimal generators Lf(x) = integral(Rd) (f(x + z) - f(x) - (del f(x), z)1({vertical bar z vertical bar <= 1}))k(x, z)Pi(dz) +< b(x), del f(x)>, f is an element of C-b(2)(R-d) in periodic media, where Pi is a nonnegative measure on R-d that does not charge the origin 0, satisfies f(Rd) (1 <^> vertical bar z vertical bar(2))Pi(dz) < infinity and can be singular with respect to the Lebesgue measure on R-d. Under a proper scaling we show the scaled processes converge weakly to Levy processes on R-d. The results are a counterpart of the celebrated work (Asymptotic Analysis for Periodic Structures (1978) North-Holland; Ann. Probab. 13 (1985) 385-396) in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when b(x) is a bounded continuous multivariate 1-periodic R-d-valued function, k(x, z) is a nonnegative bounded continuous function that is multivariate 1-periodic in both x and z variables and, in spherical coordinate z = (r, theta) is an element of R+ x Sd-1, 1({vertical bar z vertical bar>1})Pi(dz) = 1({r>1})partial derivative 0(d theta)dr/r(1+alpha) with alpha is an element of (0, infinity) and partial derivative(0) being any finite measure on the unit sphere Sd-1 in R-d. Different phenomena occur depending on the values of alpha; there are five cases alpha is an element of (0, 1), alpha = 1, alpha is an element of (1, 2), alpha = 2 and alpha is an element of (2, infinity).