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Compactness and Density Estimates for Weighted Fractional Heat Semigroups

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  • Jian Wang

    (Fujian Normal University)

Abstract

We prove that the operator $$L_0=-(1+|x|)^\beta (-\Delta )^{\alpha /2}$$ L 0 = - ( 1 + | x | ) β ( - Δ ) α / 2 with $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , $$d>\alpha $$ d > α and $$\beta \ge 0$$ β ≥ 0 generates a compact semigroup or resolvent on $$L^2(\mathbb {R}^d;(1+|x|)^{-\beta }\,\mathrm{d}x)$$ L 2 ( R d ; ( 1 + | x | ) - β d x ) , if and only if $$\beta >\alpha $$ β > α . When $$\beta >\alpha $$ β > α , we obtain two-sided asymptotic estimates for high-order eigenvalues, and sharp bounds for the corresponding heat kernel.

Suggested Citation

  • Jian Wang, 2019. "Compactness and Density Estimates for Weighted Fractional Heat Semigroups," Journal of Theoretical Probability, Springer, vol. 32(4), pages 2066-2087, December.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:4:d:10.1007_s10959-018-0838-9
    DOI: 10.1007/s10959-018-0838-9
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    References listed on IDEAS

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    1. Chen, Zhen-Qing & Wang, Jian, 2014. "Ergodicity for time-changed symmetric stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2799-2823.
    2. 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.
    3. Kumar, Rohini & Popovic, Lea, 2017. "Large deviations for multi-scale jump-diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1297-1320.
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