Functional inequalities for time-changed symmetric \alpha-stable processes

Abstract

We establish sharp functional inequalities for time-changed symmetric \alpha-stable processes on {\mathrm{ℝ}}^d with d\ge \mathrm{1} and \alpha \in (\mathrm{0},\mathrm{2}), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x)={(\mathrm{1}+\left|x\right|)}^{\beta } with \beta >\alpha we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.

Keywords

Symmetric \alpha-stable process / time change, functional inequality

References

[1] Bakry D, Gentil I, Ledoux M. Analysis and Geometry of Markov Diffusion Operators. Grundlehren Math Wiss, Vol 348. Berlin: Springer, 2014

[2] Carlen E A,Kusuoka S, Stroock D W.Upper bounds for symmetric Markov transition functions. Ann Inst Henri Poincaré Probab Stat, 1987, 23: 245–287

[3] ChenM F.Eigenvalues, Inequalities, and Ergodic Theory.London: Springer-Verlag, 2005

[4] Chen X, Wang J.Intrinsic ultracontractivity for general Lévy processes on bounded open sets. Illinois J Math, 2014, 58: 1117–1144

[5] Chen Z Q, Fukushima M. Symmetric Markov Processes, Time Change, and Boundary Theory. London Math Soc Monogr Ser, Vol 35. Princeton: Princeton Univ Press, 2011

[6] Chen Z Q, Kim P, Kumagai T. Weighted Poincaré inequality and heat kernel estimates for finite range jump processes. Math Ann, 2008, 342: 833–883

[7] Chen Z Q, Kumagai T. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl, 2003, 108: 27–62

[8] Chen Z Q, Wang J.Ergodicity for time changed symmetric stable processes. Stochastic Process Appl, 2014, 124: 2799–2823

[9] Demengel F,Demengel G.Functional Spaces for the Theory of Elliptic Partial Differential Equations. Universitext. London: Springer-Verlag, 2012

[10] Di Nezza E, PalatucciG,Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136: 521–573

[11] Hurri-Syrjänen R, Vähäkangas A V. On fractional Poincaré inequalities. J Anal Math, 2013, 120: 85–104

[12] Kumar R, Popovic L. Large deviations for multi-scale jump-diffusion processes. Stochastic Process Appl, 2017, 127: 1297–1320

[13] Metafune G, Spina C. Elliptic operators with unbounded diffusion coefficients in L_p-spaces. Ann Sc Norm Super Pisa Cl Sci (5), 2012, 11(2): 303–340

[14] Nguyen H M, Squassina M. Fractional Caffarelli-Kohn-Nirenberg inequalities. J Funct Anal, 2018, 274: 2661–2672

[15] Spina C. Heat kernel estimates for an operator with unbounded diffusion coefficients in ℝ and ℝ2: Semigroup Forum, 2013, 86: 67–82

[16] Wang F Y. Functional Inequalities, Markov Processes and Spectral Theory. Beijing: Science Press, 2005

[17] Wang J. Compactness and density estimates for weighted fractional heat semigroups. J Theoret Probab, 2019, 172: 301–376

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