\(C_{r,p}\)-capacity associated with Hunt processes (Q1581875)

Let \(E\) be a Hausdorff topological space and \(m\) a \(\sigma\)-finite Borel measure on \(E\). For a strongly continuous Markovian semigroup \((T_t)_{t>0}\) on \(L^p(E;m)\), put \({\mathcal F}_{r,p}=V_r(L^p)\) and \(\|V_r f\|_{r,p}=\|f\|_{L^p}\) by using the \(\Gamma\)-transformation \(V_r f\) of \(f \in L^p\). \textit{H.~Kaneko} [Osaka J. Math. 23, 325-336 (1986; Zbl 0633.60090)] and \textit{M.~Fukushima} [in: Probability theory and mathematical statistics, 96-103 (1992; Zbl 0817.60082)] introduced the notion of \((r,p)\)-capacity and, under the analyticity condition of the semigroup, constructed the associated Hunt process for locally compact and general \(E\), respectively. The main purpose of this paper is to release the analyticity condition. More precisely, assuming that the tightness of the capacity, the existence of a dense quasi-continuous subfamily \({\mathcal F}^0_{r,p}\) of \({\mathcal F}_{r,p}\) and the existence of a countable \(Q\)-algebra \(D\subset {\mathcal F}^0_{r,p}\) separating points of \(E\) except negligible set and \(1 \in D\), the author shows that there exists a Hunt process whose transition function \(Q_t f\) is a \(C_{r,p}\)-quasi-continuous \(m\)-version of \(T_t f\) for any bounded \(f \in {\mathcal F}_{r,p}\). For the proof, a metric \(d\) which makes any function of \(D\) uniformly continuous is introduced. Then, after constructing a Hunt process on the \(d\)-compactified space \(\overline{E}\), the tightness property is used to restrict the process to \(E\).