We study angular variation in the frontier of branching Brownian motion (BBM) in multiple dimensions. The time average of this frontier is tied to the long-time limit of the critical derivative martingale, which is a random integrable function on the sphere. We show that this function is nowhere locally bounded. As a consequence, BBM exhibits strong persistent anisotropy: there are dense arbitrarily large gaps between the BBM frontier in different directions.
We give a full characterization of the fixed points of Branching Brownian motion with critical and supercritical drifts under no additional assumptions besides it being locally finite almost surely. In particular, we do not assume finite intensity (Kabluchko, 2012) or the finite top particle (Chen-Garban-Shekhar, 2023)...
Xin-Xin Chen, Arnab Chowdhury, A. Shekhar et al.· 0 citations
In Brownian yet non-Gaussian diffusion (BnGD) the mean squared displacement grows linearly in time. However, the displacement statistics do not follow a normal distribution throughout. Typically, they are non-Gaussian at intermediate times, before they cross over to Gaussian in the long-time regime. We demonstrate that...
We consider fractional Brownian motion with D-dimensional time in the domain G(T)=(-T<ti<T,tj>0 if j<d+1). Let P(T) be the probability that the process will not exceed a fixed level in G(T). We show that for large T, logP(T)=(D-dH)logT(1+o(1)) where H is the Hurst parameter of the process
We study a random walk in $\mathbb{Z}^d$ in a dynamical random environment constituted of independent null recurrent birth-and-death chains, and establish a scaling limit result for the position of the walker in terms of a sublinear time change of Brownian motion, thus establishing subdiffusivity, in contrast to a diff...
We study a Brownian particle in $\mathbb{R}^d$, $d\geq2$, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The pr...
Scott Armstrong, Ahmed Bou-Rabee, Tuomo Kuusi· 0 citations
We consider scaling limits of integer-valued random walks whose transition probabilities are translation-invariant everywhere except on a set that we will call a membrane or an interface. If the jumps outside the membrane are i.i.d. mean-zero random variables with finite variance, then possible examples of the correspo...
A. Pilipenko· 0 citations
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