Brownian motion on stable looptrees

Random stable looptrees*

We introduce a class of random compact metric spaces Lα indexed by α ∈ (1, 2) and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can informally be viewed as dual graphs of α-stable Lévy trees. We study their properties and prove in particular that the Hausdorff dimension of Lα is almost surely equal to α. We also show tha...

Percolation on Random Triangulations and Stable Looptrees

We study site percolation on Angel & Schramm’s Uniform Infinite Planar Triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stab...

Limiting Behaviors for Brownian Motion Reflected on Brownian Motion

Suppose that g(t) and Wt are independent Brownian motions starting from g(0) = W0 = 0. Consider the Brownian motion Yt reflected on g(t), obtained from Wt by the means of the Skorohod lemma. The upper and lower limiting behaviors of Yt are presented. The upper tail estimate on exit time is computed via principal eigenvalue.

on Brownian Motion

We present an introduction to Brownian motion, an important continuous-time stochastic process that serves as a continuous-time analog to the simple symmetric random walk on the one hand, and shares fundamental properties with the Poisson counting process on the other hand. Throughout, we use the following notation for the real numbers, the non-negative real numbers, the integers, and the non-n...

Notes on Brownian Motion