Related papers: Variable speed branching Brownian motion 1. Extrem…
The Brownian motion $(U^N_t)_{t\ge 0}$ on the unitary group converges, as a process, to the free unitary Brownian motion $(u_t)_{t\ge 0}$ as $N\to\infty$. In this paper, we prove that it converges strongly as a process: not only in…
The question how the extremal values of a stochastic process achieved on different time intervals are correlated to each other has been discussed within the last few years on examples of the running maximum of a Brownian motion, of a…
In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions with drift and are killed upon reaching the origin. We prove that the extremal process of this branching Brownian motion with…
We continue the study of the maximum of the scale-inhomogeneous discrete Gaussian free field in dimension two. In this paper, we consider the regime of weak correlations and prove the convergence in law of the centred maximum to a randomly…
Fractional Brownian motion is a self-affine, non-Markovian and translationally invariant generalization of Brownian motion, depending on the Hurst exponent $H$. Here we investigate fractional Brownian motion where both the starting and the…
We consider slow / fast systems where the slow system is driven by fractional Brownian motion with Hurst parameter $H>{1\over 2}$. We show that unlike in the case $H={1\over 2}$, convergence to the averaged solution takes place in…
Passive scalar motion in a family of random Gaussian velocity fields with long-range correlations is shown to converge to persistent fractional Brownian motions in long times.
It is well known that, under suitable regularity conditions, the normalized fractional process with fractional parameter $d$ converges weakly to fractional Brownian motion for $d>1/2$. We show that, for any non-negative integer $M$,…
In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is…
We study the precise large deviation probabilities for the sizes of intermediate level sets in branching Brownian motion (BBM). Our conclusions improve a result of A\"{i}dekon, Hu and Shi in [J. Math. Sci. \textbf{238}(2019)]. Additionally,…
We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and…
This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show…
We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of…
In this work, we characterize all the point processes $\theta=\sum_{i\in \mathbb{N}} \delta_{x_i}$ on $\mathbb{R}$ which are left invariant under branching Brownian motions with critical drift $-\sqrt{2}$. Our characterization holds under…
We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the…
We identify the fluctuations of the partition function of the continuous random energy model on a Galton-Watson tree in the so-called weak correlation regime. Namely, when the ``speed functions'', that describe the time-inhomogeneous…
This article presents various weak laws of large numbers for the so-called realised covariation of a bivariate stationary stochastic process which is not a semimartingale. More precisely, we consider two cases: Bivariate moving average…
The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed…
We prove a convergence theorem for a sequence of super-Brownian motions moving among hard Poissonian obstacles, when the intensity of the obstacles grows to infinity but their diameters shrink to zero in an appropriate manner. The…
We study the shape of the outer envelope of a branching Brownian motion (BBM) in $\mathbb{R}^d$, $d\geq 2$. We focus on the extremal particles: those whose norm is within $O(1)$ of the maximal norm amongst the particles alive at time $t$.…