## 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.

### MSC:

 60J45 Probabilistic potential theory 60J75 Jump processes (MSC2010) 60G18 Self-similar stochastic processes 60J99 Markov processes 60J65 Brownian motion

### Citations:

Zbl 0631.60073; Zbl 0651.60081
Full Text:

### References:

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