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

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SRQW02 - Quantum field theory, renormalisation and stochastic partial differential equations

We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $$ \partial_t h(t,x)=\nu\Del h(t,x)\lambda |\nabla h(t,x)|2 \sqrt{D}\, \eta(t,x), \qquad (t,x)\in{\mathbb{R}}+\times{\mathbb{R}}d $$ 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 $\nu{eff}=\nu+O(\lambda2),D_{eff}=D+O(\lambda2)$. This is joint work with J. Magnen.

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