Related papers: An asymptotic theory for randomly forced discrete …

We consider the stochastic heat equation of the following form \frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) + \sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where $\sL$ is the generator of a L\'evy process and…

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…

In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic…

We consider a family of nonlinear stochastic heat equations of the form $\partial_t u=\mathcal{L}u + \sigma(u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric L\'evy process on $\R$, and…

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 study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u, \quad \hbox{in} \ \real^N\times(0,\infty). $$ The asymptotic behavior proves to…

In this paper we investigate the asymptotic behavior and decay of the solution of the discrete in time $N$-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the…

Consider the nonlinear stochastic heat equation $$ \frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ \sigma(u (t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white…

We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations \nu u_{tt}+u_t=\D u+f(u)+\nu^\alpha\dot{W} on a bounded spatial domain.…

n this paper we study the asymptotic behavior for a nonlocal heat equation in an inhomogenous medium: $$\rho(x)u_t=J\ast u-u \text{in}\mathbb{R}^N\times (0,\infty)\,,$$ where $\rho$ is a continous positive function, $u$ is nonnegative and…

In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space. After an introductory review of the main facts on the existence and properties of solutions, we…

Consider the heat equation driven by a smooth, Gaussian random potential: \begin{align*} \partial_t u_{\varepsilon}=\tfrac12\Delta u_{\varepsilon}+u_{\varepsilon}(\xi_{\varepsilon}-c_{\varepsilon}), \ \ t>0, x\in\mathbb{R}, \end{align*}…

In this article, we study the asymptotic behavior of the stochastic heat equation for large times.

We consider time fractional stochastic heat type equation $$\partial^\beta_tu(t,x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$,…

We consider a nonlinear stochastic heat equation $\partial_tu=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space-time white noise and $\sigma:\mathbf {R}\to \mathbf {R}$ is Lipschitz continuous. We…

This is a survey on the intermittent behavior of the parabolic {Anderson} model, which is the Cauchy problem for the heat equation with random potential on the lattice $\Z^d$. We first introduce the model and give heuristic explanations of…

We consider non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$…

Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order $2 + \delta$ for some $\delta>0$. For any starting point $x\in \mathbb R$,…

This paper considers the asymptotic behaviour of deterministically and stochastically forced linear pantograph equations. The asymptotic behaviour is studied in the case when all solutions of the pantograph equation without forcing tend to…

We study the simple hypothesis testing problem for the drift coefficient for stochastic fractional heat equation driven by additive noise. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such…