Intermittency and chaos for a stochastic non-linear wave equation in dimension 1

We consider a non-linear stochastic wave equation driven by space-time white noise in dimension 1. First of all, we state some results about the intermittency of the solution, which have only been carefully studied in some particular cases so far. Then, we establish a comparison principle for the solution, following the ideas of Mueller. We think it is of particular interest to obtain such a result for a hyperbolic equation. Finally, using the results mentioned above, we aim to show that the solution exhibits a chaotic behavior, in a similar way as was established by Conus, Joseph, and Khoshnevisan for the heat equation. We study the two cases where 1. the initial conditions have compact support, where the global maximum of the solution remains bounded and 2. the initial conditions are bounded away from 0, where the global maximum is almost surely infinite. Interesting estimates are also provided on the behavior of the global maximum of the solution.

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 $sigma$, $sup_{xin mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $limsup_{|x|toinfty}u_t(x)/({log}|x|)^{1/6}>0$ 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.

On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

We study a family of non-linear stochastic heat equations in (1+1) dimensions, driven by the generator of a Levy process and space-time white noise. We assume that the underlying Levy process has finite exponential moments in a neighborhood of the origin and that the initial condition has exponential decay at infinity. Then we prove that under natural conditions on the non-linearity: (i) The absolute moments of the solution to our stochastic heat equation grow exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the non-linear stochastic heat equation under the present setting. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.