Related papers: Precise asymptotics for the parabolic Anderson mod…

We study an active Brownian run-and-tumble particle (ABRTP) model, that consists of an active Brownian run state during which the active velocity of the particle diffuses on the unit circle, and a tumble state during which the active…

Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $\gamma$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^\gamma]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an…

We derive the exact asymptotics of \[ P\left( \sup_{t\ge 0} \Bigl( X_1(t) - \mu_1 t\Bigr)> u, \ \sup_{s\ge 0} \Bigl( X_2(s) - \mu_2 s\Bigr)> u \right), \ \ u\to\infty, \] where $(X_1(t),X_2(s))_{t,s\ge0}$ is a correlated two-dimensional…

In this paper, we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\xi u$, where $u:\mathbb{Z}^d\times [0,\infty)\to\mathbb{R}$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete…

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 give a new example of a measure-valued process without a density, which arises from a stochastic partial differential equation with a multiplicative noise term. This process has some unusual properties. We work with the heat equation…

We derive rigorously the leading asymptotics of the so-called Anderson integral in the thermodynamic limit for one-dimensional, non-relativistic, spin-less Fermi systems. The coefficient, $\gamma$, of the leading term is computed in terms…

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in $Z^d$. We use i.i.d. potentials $\xi: Z^d \to \R$ in the third universality class, namely the class of almost bounded potentials, in…

We consider a random walk among i.i.d. obstacles on the one dimensional integer lattice under the condition that the walk starts from the origin and reaches a remote location y. The obstacles are represented by a killing potential, which…

In this paper we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is $$ \begin{cases} u_{t}-\Delta u = \mu & \text{in}\ (0,T)\times\Omega,\\[0.7 ex] u(0,x)=u_0 & \text{in}\…

Let $\xi$ be a Gaussian white noise on $\mathbb R^d$ ($d=1,2,3$). Let $(\xi_\varepsilon)_{\varepsilon>0}$ be continuous Gaussian processes such that $\xi_\varepsilon\to\xi$ as $\varepsilon\to0$, defined by convolving $\xi$ against a…

We investigate the long time behavior of solutions to semilinear hyperbolic equation (E$_{\alpha}$): $ u^{\prime\prime}(t)+\gamma(t)u^{\prime}(t)+Au(t)+f(u(t))=g(t),~t\geq0, $ where $A$ is a self-adjoint nonnegative operator, $f$ a function…

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated…

In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem \[ u_{t}=\Delta u+\displaystyle\frac{\lambda f(u)}{\big(\int_{\Omega}f(u)dx\big)^{p}}, x\in \Omega, t>0, \] with homogeneous Dirichlet boundary condition,…

We establish explicit quenched asymptotics for pure-jump symmetric L\'evy processes in general Poissonian potentials, which is closely related to large time asymptotic behavior of solutions to the nonlocal parabolic Anderson problem with…

A hybrid asymptotic-numerical method is presented for obtaining the full probability distribution of capture times of a random walker by multiple small traps located inside a bounded two-dimensional domain with reflective boundaries. As…

Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous…

In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is…

We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the fundamental solution to the heat equation with a compactly supported potential. It is assumed that the corresponding stationary operator has at least one positive…

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$ \partial_t u-\Delta u=a u-b(x) u^p \text{in} \Omega\times \R^+, u(0)=u_0, u(t)|_{\partial \Omega}=0 $$ as $p\to +\infty$, where…