Generalized PDE estimates for KPZ equations through Hamilton-Jacobi-Bellman formalism

Abstract

We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=\nu\Delta h(t,x)+\lambda V(|\nabla h(t,x)|) +\sqrt{D}\, \eta(t,x), \qquad x\in{\mathbb{R}}^d $$ in $d\ge 1$ dimensions. The forcing term $\eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)\ge 0$, one finds $V(|\nabla h|)\ltimes |\nabla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article, a continuation of [24], is dedicated to a generalization of the PDE estimates obtained in the previous article to the case of a deposition rate $V$ with polynomial growth of arbitrary order at infinity, for which in general the Cole-Hopf transformation does not allow any more a comparison to the heat equation. The main tool here instead is the representation of $h$ as the solution of some minimization problem through the Hamilton-Jacobi-Bellman formalism. This sole representation turns out to be powerful enough to produce local or pointwise estimates in ${\cal W}$-spaces of functions with "locally bounded averages", as in [24], implying in particular global existence and uniqueness of solutions.