Related papers: Spatial ergodicity and central limit theorems for …
We consider the linear stochastic heat equation on $\mathbb{R}^\ell$, driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any…
We study limit theorems for time-dependent averages of the form $X_t:=\frac{1}{2L(t)}\int_{-L(t)}^{L(t)} u(t, x) \, dx$, as $t\to \infty$, where $L(t)=\exp(\lambda t)$ and $u(t, x)$ is the solution to a stochastic heat equation on…
Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}\Delta u + \sigma(u)\eta$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $\sigma:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous and non random, and $\eta$…
In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time pure-jump L\'evy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial…
The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=\Delta u(t,z)+\xi(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(\xi(z):z\in{\mathbb{Z}}^d)$. We consider independent and…
In this paper, we present a quantitative central limit theorem for the d-dimensional stochastic heat equation driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. We show…
We show a priori bounds for solutions to $(\partial_t - \Delta) u = \sigma (u) \xi$ in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269--504, 2014]. We assume $\sigma \in C_b^2 (\mathbb{R})$ and that…
In this article, we study the hyperbolic Anderson model in dimension 1, driven by a time-independent rough noise, i.e. the noise associated with the fractional Brownian motion of Hurst index $H \in (1/4,1/2)$. We prove that, with…
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…
The parabolic Anderson model (PAM) is one of the most interesting and challenging SPDEs related to various physical phenomena, and can be described mathematically as a stochastic heat equation driven by linear multiplicative noise. In this…
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…
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 stochastic quantization equation associated to the $\Phi_2^4$ model on the torus in a spatial white noise environment. We study the long time behavior of this heat equation with independent multiplicative white…
In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension $d\leq 2$, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a…
We consider nonlinear parabolic SPDEs of the form $\partial_t u=\Delta u + \lambda \sigma(u)\dot w$ on the interval $(0, L)$, where $\dot w$ denotes space-time white noise, $\sigma$ is Lipschitz continuous. Under Dirichlet boundary…
We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=\Delta u(t,x) +\xi_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$. Here, the $\xi$-field is $\mathbb{R}$-valued, acting as a dynamic random environment, and…
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…
We consider a parabolic sine-Gordon model with periodic boundary conditions. We prove a fundamental maximum principle which gives a priori uniform control of the solution. In the one-dimensional case we classify all bounded steady states…
This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the…
We study one-dimensional nonlinear stochastic cable equations driven by a multiplicative space-time white noise. Using the Malliavin-Stein method, we prove a central limit theorem for the spatial average of the solution. The convergence is…