Weighted Poincaré Inequalities for Non-local Dirichlet Forms

Xin Chen () and Jian Wang ()
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Xin Chen: Shanghai Jiao Tong University
Jian Wang: Fujian Normal University

Journal of Theoretical Probability, 2017, vol. 30, issue 2, 452-489

Abstract: 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

Keywords: Non-local Dirichlet forms (with large jumps); Weighted Poincaré inequality; Lyapunov functions; Fractional Dirichlet forms; 60G51; 60G52; 60J25; 60J75 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10959-015-0650-8

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