Let $Z$ be the transient reflecting Brownian motion on the closure of anunbounded domain $D\subset {\mathbb R}^d$ with $N$ number of Liouvillebranches. We consider a diffusion $X$ on $\overline D$ having finite lifetimeobtained from $Z$ by a time change. We show that $X$ admits only a finitenumber of possible symmetric conservative diffusion extensions $Y$ beyond itslifetime characterized by possible partitions of the collection of $N$ ends andwe identify the family of the extended Dirichlet spaces of all $Y$ (which areindependent of time change used) as subspaces of the space ${\rm BL}(D)$spanned by the extended Sobolev space $H_e^1(D)$ and the approachingprobabilities of $Z$ to the ends of Liouville branches.
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机译：假设$ Z $是瞬态反应布朗运动的结果，该运动是在无边界域$ D \ subset {\ mathbb R} ^ d $的Liouvillebranches数为$ N $的情况下完成的。我们认为通过时间变化从$ Z $获得的有限寿命是$ \ overline D $上的扩散$ X $。我们表明$ X $仅允许$ N $超出其生存期的有限数量的对称对称保守扩散扩展$ Y $以其$ N $端集合的可能分区为特征，并且我们确定了所有$ Y $的扩展Dirichlet空间的族（它们是独立的的时间变化）作为空间$ {\ rm BL}（D）$的子空间，再由扩展的Sobolev空间$ H_e ^ 1（D）$和$ Z $到Liouville分支末端的接近概率所跨越。
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