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

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We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation ∂ₜ h(t,x)=νΔh(t,x)+λV(|∇ h(t,x)|) +√D η(t,x), x∈Rᵈ in d≥ 1 dimensions. The forcing term η in the right-hand side is a regularized white noise. The deposition rate V is assumed to be isotropic and convex. Assuming V(0)≥ 0, one finds V(|∇ h|) |∇ 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 W-spaces of functions with "locally bounded averages", as in [24], implying in particular global existence and uniqueness of solutions.