Brownian motion on ℝ-trees

Athreya, Siva; Eckhoff, Michael; Winter, Anita

doi:10.1090/s0002-9947-2012-05752-7

Cited by 20 publications

(43 citation statements)

References 27 publications

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“…We will refer to this Markov process as speed-ν motion on (T, r) or variable speed motion associated to ν on (T, r). If (T, r) is path-connected, then X has continuous paths and equals the so-called ν-Brownian motion on (T, r), which was recently constructed in [AEW13]. On the other hand, if (T, r) is discrete, X is a continuous time nearest neighbor Markov chain which jumps from v to v ′ ∼ v at rate (1.5) γ vv ′ := 1 2 · ν({v}) · r(v, v ′ ) −1 (see Lemma 2.11).…”

Section: Introduction and Main Results (Theorem 1)mentioning

confidence: 99%

Invariance principle for variable speed random walks on trees

Athreya¹,

Löhr²,

Winter³

2017

Ann. Probab.

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We consider stochastic processes on complete, locally compact tree-like metric spaces $(T,r)$ on their "natural scale" with boundedly finite speed measure $\nu$. Given a triple $(T,r,\nu)$ such a speed-$\nu$ motion on $(T,r)$ can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all $x,y\in T$ and all positive, bounded measurable $f$, \[ \mathbb{E}^x [ \int^{\tau_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_T\nu(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty, \] where $c(x,y,z)$ denotes the branch point generated by $x,y,z$. If $(T,r)$ is a discrete tree, $X$ is a continuous time nearest neighbor random walk which jumps from $v$ to $v'\sim v$ at rate $\tfrac{1}{2}\cdot (\nu(\{v\})\cdot r(v,v'))^{-1}$. If $(T,r)$ is path-connected, $X$ has continuous paths and equals the $\nu$-Brownian motion which was recently constructed in [AthreyaEckhoffWinter2013]. In this paper we show that speed-$\nu_n$ motions on $(T_n,r_n)$ converge weakly in path space to the speed-$\nu$ motion on $(T,r)$ provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology introduced recently in [AthreyaLohrWinter2016].Comment: 45 page

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Section: Introduction and Main Results (Theorem 1)mentioning

confidence: 99%

Invariance principle for variable speed random walks on trees

Athreya¹,

Löhr²,

Winter³

2017

Ann. Probab.

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“…The existence of such a process follows from techniques of resistance forms (see [44] for an introduction on resistance forms and [6,26] for more details on the existence of Brownian motions on real trees). More specifically, the following was proved in section 6 of [22]: PROPOSITION 2.9.…”

Section: The Brownian Motion On the Ise: B Isementioning

confidence: 99%

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Arous

Cabezas

Fribergh

2019

Comm Pure Appl Math

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We study here a detailed conjecture regarding one of the most important cases of anomalous diffusion, i.e., the behavior of the “ant in the labyrinth.” It is natural to conjecture that the scaling limit for random walks on large critical random graphs exists in high dimensions and is universal. This scaling limit is simply the natural Brownian motion on the integrated super‐Brownian excursion. We give here a set of four natural, sufficient conditions on the critical graphs and prove that this set of assumptions ensures the validity of this conjecture. The remaining future task is to prove that these sufficient conditions hold for the various classical cases of critical random structures, like the usual Bernoulli bond percolation, oriented percolation, and spread‐out percolation in high enough dimension. In a companion paper, we do precisely that in a first case, the random walk on the trace of a large critical branching random walk. We verify the validity of these sufficient conditions and thus obtain the scaling limit mentioned above in dimensions larger than 14. © 2019 Wiley Periodicals, Inc.

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“…Here, C ∞ (T ) is the space of continuous functions on T that vanish at infinity. By [6], Propositions 2.4 and 4.1, and the proof of Theorem 1, (E T , F T ) is a local, regular Dirichlet form on L 2 (T , µ T ). The Brownian motion on (T , d T , µ T ) is the continuous, µ T -symmetric, strong Markov process ((X T t ) t≥0 , (P T x ) x∈T ) associated with this Dirichlet form (see [27]).…”

mentioning

confidence: 78%

“…Heat kernel bounds for the scaling limit. As in Section 5 and the proof of Theorem 1.4(a), (b), let (P δn ) n≥1 be a convergent sequence with limitP, and suppose that (T , d T , µ T , φ T , ρ T ) is a random variable with law P. It follows from [6], Remark 3.1 and [27], Theorem 1.5.2, that the Dirichlet form (E T , F T ) given in Section 6 is the same as that of [32], Section 5. In particular, this is the form associated with the natural "resistance form" on (T , d T ), and so we can apply [33], Theorem 10.4, to deduce the existence of a jointly continuous transition density (p T t (x, y)) x,y∈T ,t>0 for the process X T .…”

Section: Random Walk On the Uniform Spanning Treementioning

confidence: 98%

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Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

Barlow¹,

Croydon²,

Kumagai³

2017

Ann. Probab.

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The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8/5. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are diffusions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.

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Brownian motion on ℝ-trees

Cited by 20 publications

References 27 publications

Invariance principle for variable speed random walks on trees

Invariance principle for variable speed random walks on trees

Scaling Limit for the Ant in High‐Dimensional Labyrinths

Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree

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