2019/02/28 by Eleanor Archer
Economics, Econometrics and Finance · Mathematics · #Brownian motion #Compact space #Dimension (graph theory) #Fractional Brownian motion #Limit (mathematics) #Random Matrices and Applications #Random walk #Scaling #Scaling limit #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:60F17
paper · pdf · doi:10.1214/20-ejp413
published as Electron. J. Probab. 25 (2020),1-48 · 45 pages (some further proof details added to earlier version). arXiv admin note: text overlap with arXiv:1902.01713
openalex created_date 2019/03/02 · openalex publication_date 2020/01/01 · arxiv created 2020/01/15 · arxiv updated 2020/05/19 · openalex updated_date 2026/08/05
We give a construction of an infinite stable looptree, which we denote by L ∞ α , and prove that it arises both as a local limit of the compact stable looptrees of Curien and Kortchemski (2015), and as a scaling limit of the infinite discrete looptrees of Richier (2017), and Björnberg and Stefánsson (2015). As a consequence, we are able to prove various convergence results for volumes of small balls in compact stable looptrees, explored more deeply in a companion paper. We also establish the spectral dimension of L ∞ α , and show that it agrees with that of its discrete counterpart. Moreover, we show that Brownian motion on L ∞ α arises as a scaling limit of random walks on discrete looptrees, and as a local limit of Brownian motion on compact stable looptrees, which has similar consequences for the limit of the heat kernel.