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Time changes of symmetric diffusions and Feller measures. (English) Zbl 1076.60063
Let \((E,m,{\mathcal F},{\mathcal E})\) be a regular irreductible Dirichlet space such that the associated Markov process is a conservative diffusion. Let \(F\) be a closed subset of \(E\) such that \(m(E\setminus F)<\infty\), Cap\((F)>0\) and the energy measure \(\mu_{\langle u \rangle}(F)=0\) for each \(u\) in the extended Dirichlet space \({\mathcal F}_e\). The authors prove that, for any \(u\in L^2(F,\mu) \cap {\mathcal F}_e\), \[ {\mathcal E}(Hu,Hu) = \frac{1}{2} \int_{F\times F} (u(\xi)-u(\eta))^2 U(d\xi,d\eta) \] where \(H\) (resp. \(U\)) is the hitting (resp. Feller) kernel for \(F\) with respect to \(m\). This formula extends, with great generality, an original result of J. Douglas [Trans. Am. Math. Soc. 33, 263–321 (1931; Zbl 0001.14102)] and J. L. Doob [Ann. Inst. Fourier 12, 573–621 (1962; Zbl 0121.08604)]. Moreover, this result is applied for the reflecting Brownian motion on the closure of a bounded Lipschitz domain of \({\mathbb{R}}^d\).

MSC:
60J45 Probabilistic potential theory
60J50 Boundary theory for Markov processes
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