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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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References:
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