Related papers: An asymptotic theory for randomly forced discrete …

Consider the semilinear heat equation $\partial_t u = \partial^2_x u + \lambda\sigma(u)\xi$ on the interval $[0\,,1]$ with Dirichlet zero boundary condition and a nice non-random initial function, where the forcing $\xi$ is space-time white…

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the…

Consider the stochastic heat equation $\partial_tu=\mathscr{L}u+\lambda\sigma(u)\xi$, where $\mathscr{L}$ denotes the generator of a L\'{e}vy process on a locally compact Hausdorff Abelian group $G$, $\sigma:\mathbf{R}\to\mathbf{R}$ is…

We approximate the white-noise driven stochastic heat equation by replacing the fractional Laplacian by the generator of a discrete time random walk on the one dimensional lattice, and approximating white noise by a collection of i.i.d.…

This is a preliminary announcement of results in the PhD. thesis of the first author concerning the nonlinear stochastic heat equation in the spatial domain $\R$, driven by space-time white noise. A central special case is the parabolic…

We study the nonlinear stochastic heat equation in the spatial domain $\mathbb {R}$, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on $\mathbb {R}$,…

We study the stochastic growth process in discrete time $x_{i+1} = (1 + \mu_i) x_i$ with growth rate $\mu_i = \rho e^{Z_i - \frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - \gamma Z_t dt +…

In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general…

We analyze the nonlinear stochastic heat equation driven by heavy-tailed noise in free space and arbitrary dimension. The existence of a solution is proved even if the noise only has moments up to an order strictly smaller than its…

For a stochastic difference equation $D_n=A_nD_{n-1}+B_n$ which stabilises upon time we study tail distribution asymptotics of $D_n$ under the assumption that the distribution of $\log(1+|A_1|+|B_1|)$ is heavy-tailed, that is, all its…

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 consider the semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1} u \ln ^{\alpha}( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ \alpha \in \mathbb{R}$. Unlike the standard case…

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,…

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textless{}H\textless{}1/2 in…

We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}\Delta u+b(u)u+\sigma(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and…

In this paper, we study a nonlinear one spatial dimensional stochastic heat equations driven by Gaussian noise: $\frac{\partial u }{\partial t}=\frac{\partial^2 u }{\partial x^2}+\sigma(u )\dot{W} $, where $\dot{W} $ is white in time and…

In this paper, we discuss the asymptotic stability of singular steady states of the nonlinear heat equation in the weighted Lebesgue norms.

We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this…

We consider nonlinear parabolic stochastic equations of the form $\partial_t u=\sL u + \lambda \sigma(u)\dot \xi$ on the ball $B(0,\,R)$, where $\dot \xi$ denotes some Gaussian noise and $\sigma$ is Lipschitz continuous. Here $\sL$…

In this paper, we observe how the heat equation in a non-cylindrical domain can arise as the asymptotic limit of a parabolic problem in a cylindrical domain, by adding a potential that vanishes outside the limit domain. This can be seen as…