The scaling limit of the KPZ equation in space dimension 3 and higher

Abstract

We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $${\partial }_{t}h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x){|}^2+\sqrt{D}\eta (t,x),(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}^d$$ in $d\ge 3$ dimensions in the perturbative regime, i.e. for $\lambda >0$ small enough and a smooth, bounded, integrable initial condition $h_0=h(t=0,\cdot )$. The forcing term $\eta$ in the right-hand side is a regularized space-time white noise. The exponential of $h$ -- its so-called Cole-Hopf transform -- is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson's renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer-Magnen. Standard large deviation estimates for $\eta$ make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution $h$ may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards-Wilkinson model ($\lambda =0$) with renormalized coefficients ${\nu }_{eff}=\nu +O({\lambda }^2),D_{eff}=D+O({\lambda }^2)$.

65 pages, 4 figures -- see v1 for connection to perturbation theory

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