  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.

Analysis of PDEs