Related papers: Lyapunov exponents in a slow environment
We continue our study of the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ for the space-time field $u\colon\,\Z^d\times [0,\infty)\to\R$, where $\kappa \in [0,\infty)$ is the diffusion constant,…
We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete…
In this paper we study the parabolic Anderson equation \partial u(x,t)/\partial t=\kappa\Delta u(x,t)+\xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where the u-field and the \xi-field are \R-valued, \kappa \in [0,\infty) is the diffusion constant, and…
In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to…
We consider the parabolic Anderson model (PAM) which is given by the equation $\partial u/\partial t = \kappa\Delta u + \xi u$ with $u\colon\, \Z^d\times [0,\infty)\to \R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$…
The Lyapunov exponents for Anderson localization are studied in a one dimensional disordered system. A random Gaussian potential with the power law decay $\sim 1/|x|^q$ of the correlation function is considered. The exponential growth of…
The parabolic Anderson model is defined as the partial differential equation \partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete…
We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the…
We consider the solution $u$ to the one-dimensional parabolic Anderson model with homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence…
We consider the parabolic Anderson model which is driven by a Gaussian noise fractional in time and having certain scaling property in the spatial variables. Recently, Xia Chen has obtained exact Lyapunov exponent for all moments of…
We consider the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ with $u\colon\, \Z^d\times R^+\to \R^+$, where $\kappa\in\R^+$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in\R^+$ is…
We provide a complete and self-contained proof of spectral and dynamical localization for the one-dimensional Anderson model, starting from the positivity of the Lyapunov exponent provided by F\"urstenberg's theorem. That is, a…
For a pendulum suspended below a vibrating block with white noise forcing, the solution in which the pendulum remains vertical is called the single mode solution. When this solution becomes unstable there is energy transfer from the block…
We consider the statistical properties of a non-falling trajectory in the Whitney problem of an inverted pendulum excited by an external force. In the case when the external force is white noise, we recently found the instantaneous…
In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension $d$, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures:…
We consider a mechanism for area preserving Hamiltonian systems which leads to the enhanced probability, $P(\lambda, t)$, to find small values of the finite time Lyapunov exponent, $\lambda$. In our investigation of chaotic dynamical…
In this paper we derive a quantitative dichotomy for the top Lyapunov exponent of a class of non-dissipative SDEs on a compact manifold in the small noise limit. Specifically, we prove that in this class, either the Lyapunov exponent is…
Positive and negative Lyapunov exponents for a dilute, random, two-dimensional Lorentz gas in an applied field, $\vec{E}$, in a steady state at constant energy are computed to order $E^{2}$. The results are:…
We investigate Lyapunov exponents of Brownian motion in a nonnegative Poissonian potential $V$. The Lyapunov exponent depends on the potential $V$ and our interest lies in the decay rate of the Lyapunov exponent if the potential $V$ tends…
We study the largest Lyapunov exponent $\lambda$ and the finite size effects of a system of N fully-coupled classical particles, which shows a second order phase transition. Slightly below the critical energy density $U_c$, $\lambda$ shows…