This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) $\xi$ is strictly less than $1$ and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and...

Consider a random environment in ${\mathbb{Z}}^d$ given by i.i.d. conductances. In this work, we obtain tail estimates for the fluctuations about the mean for the following characteristics of the environment: the effective conductance between opposite faces of a cube, the diffusion matrices of periodized environments and the spectral gap of the random walk in a finite cube.

We identify the limit of the internal DLA cluster generated by Sinai’s walk as the law of a functional of a brownian motion which turns out to be a new interpretation of the Arcsine law.

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed $(1+d)$-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the $d$ orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically....

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys.92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the...

In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa 𝛥u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, where the $u$-field and the $\xi$-field are $ℝ$-valued, $\kappa \in [0,\infty )$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $\xi$-field plays the role of adynamic random environmentthat drives the equation. The initial condition $u(x,0)=u_0\left(x\right)$, $x\in {\mathbb{Z}}^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump...

We derive a quenched invariance principle for random walks in random environments whose transition probabilities are defined in terms of weighted cycles of bounded length. To this end, we adapt the proof for random walks among random conductances by Sidoravicius and Sznitman (Probab. Theory Related Fields129 (2004) 219–244) to the non-reversible setting.

We consider a catalytic branching random walk on $\mathbb{Z}$ that branches at the origin only. In the supercritical regime we establish a law of large number for the maximal position $M_n$: For some constant $\alpha$, $\frac{M_n}{n}\to \alpha$ almost surely on the set of infinite number of visits of the origin. Then we determine all possible limiting laws for $M_n-\alpha n$ as $n$ goes to infinity.

Consider a variant of the simple random walk on the integers, with the following transition mechanism. At each site x, the probability of jumping to the right is ω(x)∈[½, 1), until the first time the process jumps to the left from site x, from which time onward the probability of jumping to the right is ½. We investigate the transience/recurrence properties of this process in both deterministic and stationary, ergodic environments {ω(x)}x∈Z. In deterministic environments, we also study the speed...

The asymptotics of the probability that the self-intersection local time of a random walk on ${\mathbb{Z}}^d$ exceeds its expectation by a large amount is a fascinating subject because of its relation to some models from Statistical Mechanics, to large-deviation theory and variational analysis and because of the variety of the effects that can be observed. However, the proof of the upper bound is notoriously difficult and requires various sophisticated techniques. We survey some heuristics and some recently elaborated...

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions,...

We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa gt;0$ that determines the fluctuations of the process. When $0lt;\kappa lt;2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for...