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Zbl 0769.60069
Kazumi, Tetsuya; Shigekawa, Ichiro
Measures of finite $(r,p)$-energy and potentials on a separable metric space. (English)
[A] Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 415-444 (1992). ISBN 3-540-56021-1/pbk

[For the entire collection see Zbl 0754.00008.]\par Let $\{T\sb t\}$ be an $\Vert\ \Vert\sb 2$-contraction, strongly continuous and Markovian semigroup on $L\sp 2(X;m)$, where $X$ is a separable metric space and $m$ be a finite Borel measure on $(X,{\cal B}(X))$, and suppose furthermore that the dual semigroup $\{T\sp*\sb t\}$ is also Markovian. In Section 2 the authors recall fundamental notions and properties of the Sobolev space, the $(r,p)$-capacity, the $(r,p)$- energy and the generalized function associated with $\{T\sb t\}$. Section 3 is devoted to the identification of a positive generalized function as finite tight measure on $X$. In Section 4 equivalent conditions are stated such that a function $u$ belonging to the Sobolev space ${\cal F}\sb{r,p}$ could be identified as an $(r,p)$-potential, i.e., $u=U\mu$, where $U$ is the Maz'ya-Khavin nonlinear potential operator and $\mu$ is a positive generalized function in $({\hat{\cal F}}\sb{r,q})\sb +$, $(1/p+1/q=1)$. In the last section some results of Feyel-De la Pradelle on capacity of functions for Gaussian measures are generalized to the case studied by the authors.
[X.L.Nguyen (Hanoi)]

MSC 2000:: *60J45 Probabilistic potential theory
47D07 Markov semigroups of linear operators

Keywords: Markovian semigroup; dual semigroup; tight measure; capacity of functions for Gaussian measures

Citations: Zbl 0754.00008

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