Aging for interacting diffusion processes

Abstract

We study the aging phenomenon for a class of interacting diffusion processes {X t (i), i ∈ Z d }. In this framework we see the effect of the lattice dimension d on aging, as well as that of the class of test functions f (X t) considered. We further note the sensitivity of aging to specific details, when degenerate diffusions (such as super random walk, or parabolic Anderson model), are considered. We complement our study of systems on the infinite lattice, with that of their restriction to finite boxes. In the latter setting we consider different regimes in terms of box size scaling with time, as well as the effect that the choice of boundary conditions has on aging. The key tool for our analysis is the random walk representation for such diffusions.

Localisation and ageing in the parabolic Anderson model with Weibull potential

“…The ageing phenomenon has been extensively studied for disordered systems such as trap models and spin glasses; see [3] and references therein. In the context of the parabolic Anderson model, a certain form of ageing based on correlations was studied for some time-dependent potentials in [2,6], and it was shown that such systems exhibit no ageing. The recent paper [11] dealt with potentials from class (1) and studied the correlation ageing (which gives only indirect information about the evolution of localisation) and more explicit annealed ageing (which, in contrast to the quenched setting, is based on the evolution of the islands contributing to the solution averaged over the environment).…”

Persistence Probability for a Class of Gaussian Processes Related to Random Interface Models

“…The term φ t (x) corresponds to the height of the interface at position x ∈ Z d and time t ≥ 0. In particular, we have a random walk representation of space-time correlations of this model (see [8], [12,Proposition 1.3]). Since the energy of the interface φ is determine by its height differences, this model is called the ∇φ interface model and its studies have been active in both of static and dynamic aspects (see [14] and references therein).…”

“…Namely, asymptotics of the correlation ( (s, s + t)/ √ (s, s) √ (s + t, s + t)) as s, t → ∞ depends on the choice of s, t (see [8]). Namely, asymptotics of the correlation ( (s, s + t)/ √ (s, s) √ (s + t, s + t)) as s, t → ∞ depends on the choice of s, t (see [8]).…”

Persistence of Gaussian processes: non-summable correlations

“…Further, there exists a random walk representation for the space-time correlations of (1.15) (c.f. [DD,Deu2]; see also the references therein for other interacting diffusion processes admitting a random walk representation for their correlations). From this random walk representation we have that the covariance of the centered Gaussian process g t := φ t (0) is…”