Intrinsic ultracontractivity and conditional gauge for symmetric stable processes. (English) Zbl 0886.60072

The theories of Green functions and Poisson kernels are systematically extended to a symmetric stable process with index \(\alpha\), killed at the boundary of a \(C^{1,1}\) domain \(D\). The intrinsic ultracontractivity, i.e. the ultracontractivity of Doob’s \(h\)-process with \(h=\varphi\), where \(\varphi\) is the principal eigenfunction of the generator of this killed process, is obtained by using a logarithmic Sobolev inequality. This \(\varphi\)-process and its behavior are the analogue of the killed conditioning process in the diffusion case, see G. Gong, M. Qian and Z. Zhao [Probab. Theory Relat. Fields 80, No. 1, 151-167 (1988; Zbl 0631.60073)]. Then for the Feynman-Kac semigroup of the killed process with the potential term \(q\in K_{n,\alpha}\) (the Kato class): \(\lim_{r\to 0}\sup_{x\in R^n}\int_{|x-y|\leq r}\frac{|q(y)|}{|x-y|^{n-\alpha}}dy=0\), the following conditional gauge theorem \[ 1/c\leq\inf_{x,y\in D} E_y^x[ e_q (\tau_{D \setminus \{ y\}})]\leq\sup_{x,y \in D} E_y^x[ e_q (\tau_{D \setminus \{ y \}})]\leq c \] with \(e_q(t)=e^{\int _0^t |q(X_s)|ds}\), where \(\tau_{D\setminus{\{ y \}}}\) is the life time of the killed process, is proven.


60J45 Probabilistic potential theory
60J75 Jump processes (MSC2010)
60G18 Self-similar stochastic processes
60J99 Markov processes
60J65 Brownian motion
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[1] Aizenman, M.; Simon, B., Brownian motion and Harnack inequality for Schrödinger operators, Commun. pure appl. math., 35, 209-273, (1982) · Zbl 0459.60069
[2] Bakry, D., L’hypercontractivité et son utilisation en théorie des semigroups, Ecole d’été de saint flour 1992, Lect. notes math., 1581, (1994), Springer-Verlag Berlin/New York · Zbl 0856.47026
[3] Bañuelos, R., Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. funct. anal., 100, 181-206, (1991) · Zbl 0766.47025
[4] Bertoin, J., Lévy processes, (1996), Cambridge Univ. Press Cambridge
[5] Blanchard, Ph.; Ma, Z.M., New results on the Schrödinger semigroups with potentials given by signed smooth measures, Lecture notes math., 1444, 213-243, (1991)
[6] Blumenthal, R.M.; Getoor, R.K., Some theorems on stable processes, Trans. amer. math. soc., 95, 263-273, (1960) · Zbl 0107.12401
[7] Blumenthal, R.M.; Getoor, R.K.; Ray, D.B., On the distribution of first hits for the symmetric stable processes, Trans. amer. math. soc., 99, 540-554, (1961) · Zbl 0118.13005
[8] Carmona, R.; Masters, W.C.; Simon, B., Relativistic Schrödinger operators: asymptotic behavior of the eigenvalues, J. functional anal., 91, 1-10, (1990)
[9] Z.-Q. Chen, R. Song, Estimates on Green functions and Poisson kernels of symmetric stable processes
[10] Chung, K.L.; Rao, K.M., Feynman – kac functional and the Schrödinger equation, Seminar on stochastic processes, (1981), Birkhäuser Boston, p. 1-29 · Zbl 0492.60073
[11] Chung, K.L.; Rao, K.M., General gauge theorem for multiplicative functionals, Trans. amer. math. soc., 306, 819-836, (1988) · Zbl 0647.60083
[12] Chung, K.L.; Zhao, Z., From Brownian motion to Schrödinger’s equation, (1995), Springer-Verlag Berlin
[13] Cranston, M.; Fabes, E.; Zhao, Z., Conditional gauge and potential theory for the Schrödinger operator, Trans. amer. math. soc., 307, 174-194, (1988) · Zbl 0652.60076
[14] Davies, E.B., Heat kernels and spectral theory, (1989), Cambridge Univ. Press Cambridge · Zbl 0699.35006
[15] Davies, E.B.; Simon, B., Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. funct. anal., 59, 335-395, (1984) · Zbl 0568.47034
[16] Elliot, J., Dirichlet spaces associated with integral-differential operators, part 1, Illinois J. math., 9, 87-98, (1965)
[17] Falkner, N., Feynman – kac functional and positive solutions of \(12\)δuqu, Z. wahrsch. verw. gebiete, 65, 19-34, (1983)
[18] Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet forms and symmetric Markov processes, (1994), Walter de Gruyter Berlin · Zbl 0838.31001
[19] Gross, L., Logarithmic Sobolev inequalities, Amer. J. math., 97, 1061-1083, (1975) · Zbl 0318.46049
[20] Gross, L., Logarithmic Sobolev inequalities and contractivity properties of semigroups, Lecture notes math., 1563, (1993) · Zbl 0812.47037
[21] Janicki, A.; Weron, A., Simulation and chaotic behavior ofα, (1994), Dekker New York
[22] Klafter, J.; Shlesinger, M.F.; Zumofen, G., Beyond Brownian motion, Physics today, 49, 33-39, (1996)
[23] Landkof, N.S., Foundations of modern potential theory, (1972), Springer-Verlag Berlin · Zbl 0253.31001
[24] Port, S.; Stone, C., Brownian motion and classical potential theory, (1978), Academic Press New York · Zbl 0413.60067
[25] Simon, B., Schrödinger semigroups, Bull. amer. math. soc., 7, 447-536, (1982) · Zbl 0524.35002
[26] Song, R., Probabilistic approach to the Dirichlet problem of perturbed stable processes, Probab. theory relat. fields, 95, 371-389, (1993) · Zbl 0792.60067
[27] Song, R., Feynman – kac semigroup with discontinuous additive functionals, J. theoret. probab., 8, 727-762, (1995) · Zbl 0831.60085
[28] Stein, E., Singular integrals and differential properties of functions, (1970), Princeton Univ. Press Princeton
[29] Zhao, Z., Uniform boundedness of conditional gauge and Schrödinger equations, Commun. math. phys., 93, 19-31, (1984) · Zbl 0545.35087
[30] Zhao, Z., Green function for Schrödinger operator and conditioned feynman – kac gauge, J. math. anal. appl., 116, 309-334, (1986) · Zbl 0608.35012
[31] Zhao, Z., A probabilistic principle and generalized Schrödinger perturbation, J. funct. anal., 101, 162-176, (1991) · Zbl 0748.60069
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