Functional inequalities for nonlocal Dirichlet forms with finite range jumps or large jumps

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Chen, Xin¹  Wang, Jian² 

[1] Univ Lisbon, Grp Fis Matemat, P-1649003 Lisbon, Portugal

[2] Fujian Normal Univ, Sch Math & Comp Sci, Fuzhou 350007, Peoples R China

关键词: Non-local Dirichlet form with finite range jump;    Non-local Dirichlet form with large jump;    (super) Poincare inequality;    Local (super) Poincare inequality;    Lyapunov condition;    Concentration of measure;   

DOI  :  10.1016/j.spa.2013.07.001

来源: Elsevier

【 摘 要 】

The paper is a continuation of our paper, Wang and Wang (2013) [13], Chen and Wang [4], and it studies functional inequalities for non-local Dirichlet forms with finite range jumps or large jumps. Let alpha is an element of (0, 2) and mu v (dx) = C-ve(-V(x)) dx be a probability measure. We present explicit and sharp criteria for the Poincare inequality and the super Poincare inequality of the following non-local Dirichlet form with finite range jump E-alpha,E-v (f, f) := 1/2 integral integral{vertical bar x-y vertical bar <= 1} (f(x) -f(y))(2) /vertical bar x-y vertical bar(d +alpha) on the other hand, we give sharp criteria for the Poincare inequality of the non-local Dirichlet form with large jump as follows D-alpha,D-v(f, f ) : = 1/2 integral integral ((x-y)) f (Y))2 d and also derive that the super Poincare inequality does not hold for v. To obtain these results above, some new approaches and ideas completely different from Wang and Wang (2013), Chen and Wang (0000) are required, e.g. the local Poincare inequality for E-alpha,(v) and D-alpha,(v), and the Lyapunov condition for E-alpha,E-v. In particular, the results about E-alpha,E-V show that the probability measure fulfilling the Poincare inequality and the super Poincare inequality for non-local Dirichlet form with finite range jump and that for local Dirichlet form enjoy some similar properties; on the other hand, the assertions for D-alpha,D-v indicate that even if functional inequalities for non-local Dirichlet form heavily depend on the density of large jump in the associated Levy measure, the corresponding small jump plays an important role for the local super Poincare inequality, which is inevitable to derive the super Poincare inequality. (C) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

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