## Diffusive limit for 3-dimensional KPZ equation. (1) PDE estimates    [PDF]

Jeremie Unterberger
We study in the present series of articles the Kardar-Parisi-Zhang (KPZ) equation $$\partial_t h(t,x)=\nu\Del h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\eta(t,x), \qquad x\in\Z^d$$ in $d\ge 3$ dimensions in the perturbative regime, i.e. for $\lambda>0$ small enough and an initial condition $h_0=h(t=0,\cdot)$ such that $\lambda|\nabla h_0|=o(1)$. The forcing term $\eta$ in the right-hand side is a regularized white noise. We prove a large-scale diffusive limit for the solution, in particular a heat-kernel behaviour for the covariance in a parabolic scaling. The proof is generally based on perturbative estimates obtained by a multi-scale cluster expansion, and a rigorous implementation of K. Wilson's renormalization group scheme; it extends to equations in the KPZ universality class, $$\partial_t h(t,x)=\nu\Del h(t,x)+\lambda V(\nabla h(t,x)) +\sqrt{D}\eta(t,x), \qquad x\in\Z^d$$ for a large class of isotropic, smooth deposition rates $V$ with at most quadratic growth at infinity. Our expansion is meant to be a rigorous substitute for the response field formalism whenever the latter makes predictions. An important part of the proof relies however more specifically on a priori bounds for the solutions of the KPZ equation, coupled with large deviation estimates. The present article is dedicated to PDE estimates. Our results extend those previously obtained for the noiseless equation. We prove in particular a comparison principle for sub- and supersolutions of the KPZ equation in an adapted functional space containing unbounded functions, which may be of independent interest for the study of viscous Hamilton-Jacobi equations in general.
View original: http://arxiv.org/abs/1307.1980