Zhan Shi

Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees

We establish a second-order almost sure limit theorem for the minimal position in a one-dimensional super-critical branching random walk, and also prove a martingale convergence theorem which answers a question of Big-gins and Kyprianou [Electron. J. Probab. 10 (2005) 609-631]. Our method applies, furthermore, to the study of directed polymers on a disordered tree. In particular, we give a rigorous proof of a phase transition phenomenon for the partition function (from the point of view of convergence in probability), already described by Derrida and Spohn [J. Statist. Phys. 51 (1988) 817-840]. Surprisingly, this phase transition phenomenon disappears in the sense of upper almost sure limits.

The Seneta–Heyde scaling for the branching random walk

We consider the boundary case (in the sense of Biggins and Kyprianou [Electron. J. Probab. 10 (2005) 609--631] in a one-dimensional super-critical branching random walk, and study the additive martingale

Small Deviations for Some Multi-Parameter Gaussian Processes

(W_n)

Chung’s law for integrated Brownian motion

. We prove that, upon the systems survival,

Asymptotics for the survival probability in a killed branching random walk

n^{1/2}W_n

Many visits to a single site by a transient random walk in random environment

converges in probability, but not almost surely, to a positive limit. The limit is identified as a constant multiple of the almost sure limit, discovered by Biggins and Kyprianou [Adv. in Appl. Probab. 36 (2004) 544--581], of the derivative martingale.

Weak convergence for the minimal position in a branching random walk: A simple proof

We prove some general lower bounds for the probability that a multi-parameter Gaussian process has very small values. These results, when applied to a certain class of fractional Brownian sheets, yield the exact rate for their so-called small ball probability. We show by example how to use such results to compute the Hausdorff dimension of some exceptional sets determined by maximal increments.

Laws of the iterated logarithm for iterated Wiener processes

The small ball problem for the integrated process of a real–valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.

Rates of convergence of diffusions with drifted Brownian potentials

Considerons une marche aleatoire branchante surcritique a temps discret. Nous nous interessons a la probabilite quil existe un rayon infini du support de la marche aleatoire branchante, le long duquel elle croit plus vite quune fonction lineaire de pente y - e, ou γ designe la vitesse asymptotique de la position de la particule la plus a droite dans la marche aleatoire branchante. Sous des hypotheses generales peu restrictives, nous prouvons que, lorsque e → 0, cette probabilite decroit comme exp{—β+o(1) e 1/2 }, ou β est une constante strictement positive dont la valeur depend de la loi de la marche aleatoire branchante. Dans le cas special ou des variables aleatoires i.i.d. de Bernoulli(p) (avec 0 < p < 1 2 ) sont placees sur les aretes dun arbre binaire enracine, ceci repond a une question ouverte de Robin Pemantle (Ann. Appl. Probab. 19 (2009) 1273-1291).

Branching random walks

We consider a transient random walk on in random environment, and study the almost sure asymptotics of the supremum of its local time. Our main result states that if the random walk has zero speed, there is a (random) sequence of sites and a (random) sequence of times such that the walk spends a positive fraction of the times at these sites. This was known for a recurrent random walk in random environment (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990; Stochastic Process. Appl. 76 (1998) 231). Our method of proof is different and relies on the connection of random walk in random environment with branching processes in random environment used in Kesten et al. (Compositio Math. 30 (1975) 145).