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Diffusive limit for 3-dimensional KPZ equation: the Cole-Hopf case | | Jacques Magnen
; Jérémie Unterberger
; | | Date: |
10 Feb 2017 | | Abstract: | We study in the present article the Kardar-Parisi-Zhang (KPZ) equation $$
partial_t h(t,x)=
uDelta h(t,x)+lambda |
abla h(t,x)|^2 +sqrt{D},
eta(t,x), qquad (t,x)inmathbb{R}_+ imesmathbb{R}^d $$ in $dge 3$
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 cite{IagMag}. 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 $
u_{eff}=
u+O(lambda^2),D_{eff}=D+O(lambda^2)$. | | Source: | arXiv, 1702.3122 | | Services: | Forum | Review | PDF | Favorites | |
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