Related papers: The parabolic Anderson model in a dynamic random e…

A metric measure space equipped with a Dirichlet form is called recurrent if its Hausdorff dimension is less than its walk dimension. In bounded domains of such spaces we study the parabolic Anderson models \[ \partial_{t} u(t,x) = \Delta…

We consider the parabolic Anderson problem $\partial_tu=\Delta u+\xi(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=\delta_0(x)$ and random i.i.d. potential $\xi$. Under the assumption that the…

Let $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $\delta_0$ and driven by space-time white noise on $\mathbb{R}_+\times\mathbb{R}$, and let $p_t(x):= (2\pi…

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb…

In this paper, we study the {\it parabolic Anderson model} starting from the Dirac delta initial data: \[ \left(\frac{\partial}{\partial t} -\frac{\nu}{2}\frac{\partial^2}{\partial x^2} \right) u(t,x) = \lambda u(t,x) \dot{W}(t,x), \qquad…

We consider the parabolic Anderson model (PAM) $\partial_t u = \frac12 \Delta u + \xi u$ in $\mathbb R^2$ with a Gaussian (space) white-noise potential $\xi$. We prove that the almost-sure large-time asymptotic behaviour of the total mass…

We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability…

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case when $\{\xi(z):z\in\mathbb{Z}^d\}$ is a collection of independent identically distributed…

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…

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…

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a L\'evy…

We consider the parabolic Anderson model $\frac{\partial}{\partial t} v_n=\kappa\Delta_n v_n + \xi_n v_n$ on the $n$-dimensional hypercube $\{-1,+1\}^n$ with random i.i.d. potential $\xi_n$. We parametrize time by volume and study $v_n$ at…

We consider random Schr\"odinger equations on $\bZ^d$ for $d\ge 3$ with identically distributed random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time variables…

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $\partial_t u=-(-\Delta)^{1/2}u+\xi u$, where $\xi$ is a periodic spatial white noise. To be precise, we construct limits as…

The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper we consider potentials which are constant in time and independent exponentially distributed in…

We consider the parabolic Anderson model driven by fractional noise: $$ \frac{\partial}{\partial t}u(t,x)= \kappa \boldsymbol{\Delta} u(t,x)+ u(t,x)\frac{\partial}{\partial t}W(t,x) \qquad x\in\mathbb{Z}^d\;,\; t\geq 0\,, $$ where…

The parabolic problem $u_t-\Delta u=\frac{\lambda f(x)}{(1-u)^2}+P$ on a bounded domain $\Omega$ of $R^n$ with Dirichlet boundary condition models the microelectromechanical systems(MEMS) device with an external pressure term. In this…

Consider a stochastic heat equation $\partial_t u = \kappa \partial^2_{xx}u+\sigma(u)\dot{w}$ for a space-time white noise $\dot{w}$ and a constant $\kappa>0$. Under some suitable conditions on the the initial function $u_0$ and $\sigma$,…

We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)\Delta u = g(u)\xi $$ The key idea of our approach is a simple…

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that…