Yueyun Hu

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.

Laws of the iterated logarithm for iterated Wiener processes

The recent interest in iterated Wiener processes was motivated by apparently quite unrelated studies in probability theory and mathematical statistics. Laws of the iterated logarithm (LIL) were independently obtained by Burdzy(2) and Révész(17). In this work, we present a functional version of LIL for a standard iterated Wiener process, in the spirit of functional asymptotic results of an ℝ2-valued Gaussian process given by Deheuvels and Mason(9) in view of Bahadur-Kiefer-type theorems. Chungs “liminf sup” LIL is established as well, thus providing further insight into the asymptotic behavior of iterated Wiener processes.

Rates of convergence of diffusions with drifted Brownian potentials

We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.

Almost sure convergence for stochastically biased random walks on trees

We are interested in the biased random walk on a supercritical Galton–Watson tree in the sense of Lyons (Ann. Probab. 18:931–958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249–264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system’s non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)3, converges almost surely to a known positive constant.

The Local time of Simple Random Walk in Random Environment

Two integral tests are established, which characterize respectively Lévys upper and lower classes for the local time of Sinais simple random walk in random environment. The weak convergence of the local time is also studied, and the limiting distribution determined. Our results can be applied to a class of diffusion processes with random potentials which asymptotically behave like Brownian motion.

The problem of the most visited site in random environment

Abstract. We prove that the process of the most visited site of Sinais simple random walk in random environment is transient. The rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or for Brownian motion remains open.

Moderate deviations for diffusions with Brownian potentials

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 Kotanis lemma and Lampertis representation for exponential functionals. In particular, our result for quenched moderate deviations is in agreement with a recent theorem of Comets and Popov [Probab. Theory Related Fields 126 (2003) 571-609] who studied the corresponding problem for Sinais random walk in random environment.

Einstein relation for biased random walk on Galton-Watson trees

We prove the Einstein relation, relating the velocity under a small perturbation to the diffusivity in equilibrium, for certain biased random walks on Galton--Watson trees. This provides the first example where the Einstein relation is proved for motion in random media with arbitrary deep traps.

Tightness of localization and return time in random environment

Consider a class of diffusions with random potentials which behave asymptotically as Brownian motion. We study the tightness of localization around the bottom of some Brownian valley, and determine the limit distribution of the return time to the origin after a typical time. Via the Skorokhod embedding in random environment, we also solve the return time problem for Sinais walk.

The Csörgö-Révész modulus of non-differentiability of iterated Brownian motion

Khoshnevisan and Lewis (1994a) have recently established Levys modulus of continuity of iterated Brownian motion. In this note, we obtain the Csorgo-Revesz modulus of non-differentiability.