ANISOTROPIC NONLOCAL DIFFUSION OPERATORS FOR NORMAL AND ANOMALOUS DYNAMICS

被引:13
作者
Deng, Weihua [1 ]
Wang, Xudong [1 ]
Zhang, Pingwen [2 ]
机构
[1] Lanzhou Univ, Sch Math & Stat, Gansu Key Lab Appl Math & Complex Syst, Lanzhou 730000, Peoples R China
[2] Peking Univ, Sch Math Sci, Lab Math & Appl Math, Beijing 100871, Peoples R China
关键词
jump processes; nonlocal normal diffusion; anisotropic anomalous diffusion; tempered Levy flight; multiple internal states; well-posedness; TIME RANDOM-WALKS; EQUATIONS; ADVECTION; GUIDE;
D O I
10.1137/18M1184990
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
The Laplacian Delta is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian Delta(beta/2) serves as the infinitesimal generator of the limit process of isotropic Levy process. Taking limit, in some sense, means that the operators can approximate the physical process well after sufficient long time. We introduce the nonlocal operators (being effective from the starting time), which describe the general processes undergoing anisotropic normal diffusion. For anomalous diffusion, we extend to the anisotropic fractional /2 0,A Laplacian Delta(beta/2)(m) and the tempered one Delta(beta/2,lambda )(m)in R-n. Their definitions are proved to be equivalent to an alternative one in Fourier space. Based on these new anisotropic diffusion operators, we further derive the deterministic governing equations of some interesting statistical observables of the very general jump processes with multiple internal states. Finally, we consider the associated initial and boundary value problems and prove their well-posedness of the Galerkin weak formulation in R-n.To obtain the coercivity, we claim that the probability density function Y should be nondegenerate.
引用
收藏
页码:415 / 443
页数:29
相关论文
共 46 条
[1]   Ergodicity, rejuvenation, enhancement, and slow relaxation of diffusion in biased continuous-time random walks [J].
Akimoto, Takuma ;
Cherstvy, Andrey G. ;
Metzler, Ralf .
PHYSICAL REVIEW E, 2018, 98 (02)
[2]  
[Anonymous], 2011, First Steps in Random Walks: From Tools to Applications
[3]  
Applebaum D., 2009, Lvy Processes and Stochastic Calculus, DOI [10.1017/CBO9780511809781, DOI 10.1017/CBO9780511809781]
[4]   From continuous time random walks to the fractional Fokker-Planck equation [J].
Barkai, E ;
Metzler, R ;
Klafter, J .
PHYSICAL REVIEW E, 2000, 61 (01) :132-138
[5]   RELATIVISTIC SCHRODINGER-OPERATORS - ASYMPTOTIC-BEHAVIOR OF THE EIGENFUNCTIONS [J].
CARMONA, R ;
MASTERS, WC ;
SIMON, B .
JOURNAL OF FUNCTIONAL ANALYSIS, 1990, 91 (01) :117-142
[6]   First passage and arrival time densities for Levy flights and the failure of the method of images [J].
Chechkin, AV ;
Metzler, R ;
Gonchar, VY ;
Klafter, J ;
Tanatarov, LV .
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 2003, 36 (41) :L537-L544
[7]   Subdiffusion in an external force field [J].
Chen, Yao ;
Wang, Xudong ;
Deng, Weihua .
PHYSICAL REVIEW E, 2019, 99 (04)
[8]   Localization and Ballistic Diffusion for the Tempered Fractional Brownian-Langevin Motion [J].
Chen, Yao ;
Wang, Xudong ;
Deng, Weihua .
JOURNAL OF STATISTICAL PHYSICS, 2017, 169 (01) :18-37
[9]   Two-sided eigenvalue estimates for subordinate processes in domains [J].
Chen, ZQ ;
Song, RM .
JOURNAL OF FUNCTIONAL ANALYSIS, 2005, 226 (01) :90-113
[10]   Continuous time random walks on moving fluids [J].
Compte, A .
PHYSICAL REVIEW E, 1997, 55 (06) :6821-6831