### A condition for weak disorder for directed polymers in random environment.

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In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.

We consider the continuous time, one-dimensional random walk in random environment in Sinai’s regime. We show that the probability for the particle to be, at time $t$ and in a typical environment, at a distance larger than ${t}^{a}$ ($0\<a\<1$) from its initial position, is $exp\{-\mathrm{Const}\xb7{t}^{a}/[(1-a)lnt](1+o\left(1\right))\}$.

We consider the continuous time, one-dimensional random walk in random environment in Sinai's regime. We show that the probability for the particle to be, at time t and in a typical environment, at a distance larger than ta (0<a<1) from its initial position, is exp{-Const ⋅ ta/[(1 - a)lnt](1 + o(1))}.

The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time,...

We consider transient one-dimensional random walks in a random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of “valleys“ of height $logt$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.

We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.

We study a one-dimensional brownian motion conditioned on a self-repelling behaviour. Given a nondecreasing positive function f(t), t≥0, consider the measures μt obtained by conditioning a brownian path so that Ls≤f(s), for all s≤t, where Ls is the local time spent at the origin by time s. It is shown that the measures μt are tight, and that any weak limit of μt as t→∞ is transient provided that t−3/2f(t) is integrable. We conjecture that this condition is sharp and present a number of open problems....