Invariance principle for variable speed random walks on trees
Invariance principle for variable speed random walks on trees
复制标题
树上变速随机游走的不变性原理
DOI：
10.1214/15-aop1071
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发表时间：
2014
影响因子：
2.3
通讯作者：
A. Winter
中科院分区：
文献类型：
--
作者：
S. Athreya;Wolfgang Lohr;A. Winter
考虑具有有限速度测度νν的完全的、局部紧化的树状度量空间（T,r）（T,r）上的随机过程。给定一个三重体（T,r,ν）（T,r,ν），在（T,r）（T,r）上的速度ν运动可以表征为唯一的强马尔可夫过程，如果将其限制为紧子树，它满足所有x，y∈Tx，y∈T和所有正的有界可测ff，
We consider stochastic processes on complete, locally compact tree-like metric spaces (T,r)(T,r) on their “natural scale” with boundedly finite speed measure νν. Given a triple (T,r,ν)(T,r,ν) such a speed-νν motion on (T,r)(T,r) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all x,y∈Tx,y∈T and all positive, bounded measurable ff,
Ex[∫τy0dsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞,
Ex[∫0τydsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞,
where c(x,y,z)c(x,y,z) denotes the branch point generated by x,y,zx,y,z. If (T,r)(T,r) is a discrete tree, XX is a continuous time nearest neighbor random walk which jumps from vv to v′∼vv′∼v at rate 12⋅(ν({v})⋅r(v,v′))−112⋅(ν({v})⋅r(v,v′))−1. If (T,r)(T,r) is path-connected, XX has continuous paths and equals the νν-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. In this paper, we show that speed-νnνn motions on (Tn,rn)(Tn,rn) converge weakly in path space to the speed-νν motion on (T,r)(T,r) provided that the underlying triples of metric measure spaces converge in the Gromov–Hausdorff-vague topology introduced in [Stochastic Process. Appl. 126 (2016) 2527–2553].