On the density of the supremum of nonlinear SPDEs

Abstract

We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -\kappa \frac{\partial^4 u}{\partial x^4}(t,x) + \rho \frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) + \sigma(u(t,x))\, \dot W(t,x), \end{equation*} posed on a bounded spatial domain, where $u$ is understood in the random field sense and $\dot W(t,x)$ is space-time white noise. Depending on the value of $\kappa$, this equation includes the nonlinear stochastic heat equation with Dirichlet or Neumann boundary conditions, as well as the linearized stochastic Cahn-Hilliard equation with Neumann boundary conditions. We prove that the supremum of the solution admits a density with respect to Lebesgue measure. Our approach is based on Malliavin calculus, and in particular on the version of the Bouleau-Hirsch criterion for suprema developed by Nualart and Vives. One of the main difficulties lies in the analysis of the argmax set of the solution and in showing that the Malliavin derivative is almost surely nondegenerate on this set. As a byproduct of our arguments, we also establish H\"older continuity properties for the Malliavin derivative of the solution as an $L^2-$valued process in the regimes considered in this work.