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On unique extension of time changed reflecting Brownian motions. (English) Zbl 1189.60141
Let \(D\) be an unbounded unbounded uniform domain in \(\mathbb{R}^d\), \(d\geq 3\) (or, equivalently, \(D\) is an \((\varepsilon, \delta)\) with \(\delta=\infty\), see, for example, [P. W. Jones, Acta Math. 147, 71–88 (1981; Zbl 0489.30017), and J. Väisälä, Tôhoku Math. J., II. Ser. 40, No. 1, 101–118 (1988; Zbl 0627.30017)]), which has either continuous boundary or is extendable. Then the reflected symmetric Brownian motion \(X\) on \(\overline{D}\) is transient. Further, assuming that \(X\) is transient, there is in some sense unique extension of the the time changed process \(Y\) on \(\overline{D}\) (time changed with respect to the Revuz measure \(1_D(x) m(x)dx\), where \(m\) is strictly positive continuous integrable function) or, equivalently, the associated Dirichlet form \((\mathcal{E}^*, \mathcal{F}^*)\)), with following properties: Under the additional assumption that the space of harmonic functions with finite Dirichlet integral consists of constants only, the Dirichlet \((\mathcal{E}^*, \mathcal{F}^*)\) on \(L_2(\overline{D}^*, m^*)\), where \(\overline{D}^*:=D\cup \{\partial\}\) is the one-point compactification, ad \(m^*\) is the extension of \(m\) to \(\overline{D}^*\) with \(m^*(\partial D\cap \{\partial\})=0\), is recurrent, strongly local, regular Dirichlet form, and the associate process \(Y^*\) has finite lifetime with probability 1; such an extension is unique.

MSC:
60J50 Boundary theory for Markov processes
60J60 Diffusion processes
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