Related papers: Branching brownian motion conditioned on large lev…
We analyze the convergence to equilibrium of one-dimensional reflected Brownian motion (RBM) and compute a number of related initial transient formulae. These formulae are of interest as approximations to the initial transient for queueing…
We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM $X$ stays below a certain level until time $T$. Recently, Oshanin et al. study a physical model…
In this note we consider a branching Brownian motion (BBM) on $\mathbb{R}$ in which a particle at spatial position $y$ splits into two at rate $\beta y^2$, where $\beta>0$ is a constant. This is a critical breeding rate for BBM in the sense…
For a set $A\subset C[0,\infty)$, we give new results on the growth of the number of particles in a dyadic branching Brownian motion whose paths fall within A. We show that it is possible to work without rescaling the paths. We give large…
We prove large deviations principles in large time, for the Brownian occupation time in random scenery. The random scenery is constant on unit cubes, and consist of i.i.d. bounded variables, independent of the Brownian motion. This model is…
We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the…
This paper provides yet another look at the mixed fractional Brownian motion (fBm), this time, from the spectral perspective. We derive an approximation for the eigenvalues of its covariance operator, asymptotically accurate up to the…
We study the extremes of variable speed branching Brownian motion (BBM) where the time-dependent "speed functions", which describe the time-inhomogeneous variance, converge to the identity function. We consider general speed functions lying…
We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed…
The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes -- the conditionally…
We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson…
We study branching Brownian motion in hyperbolic space. As hyperbolic Brownian motion is transient, the normalised empirical measure of branching Brownian motion converges to a random measure $\mu_\infty$ on the boundary. We show that the…
Euclidean branching Brownian motion (BBM) has been intensively studied during many decades by renowned researchers. BBM on hyperbolic space has received less attention. A profound study of Lalley and Sellke (1997) provided insight on the…
Consider a branching Brownian motion (BBM). It is well known \cite{Bramson1983ConvergenceOS, Lalley1987ACL} that the rightmost particle is located near \( m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t \). Let $\mathcal{N}(t,x)$ be the set…
Let $B_{H}(t), t\geq [0,T], T\in(0,\infty)$ be the standard Multifractional Brownian Motion(mBm), in this contribution we are concerned with the exact asymptotics of \begin{eqnarray*} \mathbb{P}\left\{\sup_{t\in[0,T]}B_{H}(t)>u\right\}…
The long-term behavior of a supercritical branching random walk can be described and analyzed with the help of Biggins' martingales, parametrized by real or complex numbers. The study of these martingales with complex parameters is a rather…
The fractional Brownian motion (fBm) extends the standard Brownian motion by introducing some dependence between non-overlapping increments. Consequently, if one considers for example that log-prices follow an fBm, one can exploit the…
We present precise moderate deviation probabilities, in both quenched and annealed settings, for a recurrent diffusion process with a Brownian potential. Our method relies on fine tools in stochastic calculus, including Kotani's lemma and…
In this paper we study the moderate deviations principle (MDP) for slow-fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter $H\in(1/2,1)$. We derive conditions…
We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion, random walks in…