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Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Author

Listed:
  • Xin Chen

    (Shanghai Jiao Tong University)

  • Jian Wang

    (Fujian Normal University)

Abstract

Let V be a locally bounded measurable function on $${\mathbb {R}}^d$$ R d such that $$\mu _V(\mathrm{d}x)=C_V \mathrm{e}^{-V(x)}\,\mathrm{d}x$$ μ V ( d x ) = C V e - V ( x ) d x is a probability measure. Explicit criteria are presented for weighted Poincaré inequalities of the following non-local Dirichlet form $$\begin{aligned} \hat{D}_{\rho ,V}(f,f)=\iint _{\{|x-y|>1\}}(f(y)-f(x))^2\rho (|y-x|)\,\mathrm{d}y\, \mu _V(\mathrm{d}x). \end{aligned}$$ D ^ ρ , V ( f , f ) = ∫ ∫ { | x - y | > 1 } ( f ( y ) - f ( x ) ) 2 ρ ( | y - x | ) d y μ V ( d x ) . Taking $$\rho (r)={\mathrm{e}^{-\delta r}}{r^{-(d+\alpha )}}$$ ρ ( r ) = e - δ r r - ( d + α ) with $$0

Suggested Citation

  • Xin Chen & Jian Wang, 2017. "Weighted Poincaré Inequalities for Non-local Dirichlet Forms," Journal of Theoretical Probability, Springer, vol. 30(2), pages 452-489, June.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:2:d:10.1007_s10959-015-0650-8
    DOI: 10.1007/s10959-015-0650-8
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    References listed on IDEAS

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    1. Renming Song, 2006. "Estimates on the Transition Densities of Girsanov Transforms of Symmetric Stable Processes," Journal of Theoretical Probability, Springer, vol. 19(2), pages 487-507, June.
    2. Feng-Yu Wang & Jian Wang, 2015. "Functional Inequalities for Stable-Like Dirichlet Forms," Journal of Theoretical Probability, Springer, vol. 28(2), pages 423-448, June.
    3. Chen, Xin & Wang, Jian, 2014. "Functional inequalities for nonlocal Dirichlet forms with finite range jumps or large jumps," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 123-153.
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