Remarks on non-linear noise excitability of some stochastic heat equations

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 conditions and a linear growth condition on $\sigma$, we show that the expected $L^2$-energy is of order $\exp[\text{const}\times\lambda^4]$ as $\lambda\rightarrow \infty$. This significantly improves a recent result of Khoshnevisan and Kim. Our method is very different from theirs and it allows us to arrive at the same conclusion for the same equation but with Neumann boundary condition. This improves over another result of Khoshnevisan and Kim

Almost Sure Invariance Principle for Continuous-Space Random Walk in Dynamic Random Environment

We consider a random walk on $\R^d$ in a polynomially mixing random environment that is refreshed at each time step. We use a martingale approach to give a necessary and sufficient condition for the almost-sure functional central limit theorem to hold.Comment: minor typos fixe

Strong invariance and noise-comparison principles for some parabolic stochastic PDEs

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.Comment: 26 page

A non-Gaussian continuous state space model for asset degradation

The degradation model plays an essential role in asset life prediction and condition based maintenance. Various degradation models have been proposed. Within these models, the state space model has the ability to combine degradation data and failure event data. The state space model is also an effective approach to deal with the multiple observations and missing data issues. Using the state space degradation model, the deterioration process of assets is presented by a system state process which can be revealed by a sequence of observations. Current research largely assumes that the underlying system development process is discrete in time or states. Although some models have been developed to consider continuous time and space, these state space models are based on the Wiener process with the Gaussian assumption. This paper proposes a Gamma-based state space degradation model in order to remove the Gaussian assumption. Both condition monitoring observations and failure events are considered in the model so as to improve the accuracy of asset life prediction. A simulation study is carried out to illustrate the application procedure of the proposed model

On the chaotic character of the stochastic heat equation, before the onset of intermitttency

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 establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $\sigma$, $\sup_{x\in \mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_t(x)/({\log}|x|)^{1/6}>0$ when $u_0$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.Comment: Published in at http://dx.doi.org/10.1214/11-AOP717 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org