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SUMMARY:The scaling limit of the KPZ equation in space dimension 3 and  hi
 gher - Jeremie Unterberger (Université de Lorraine)
DTSTART:20181025T090000Z
DTEND:20181025T100000Z
UID:TALK113014@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We study in the present article the Kardar-Parisi-Zhang  (KPZ)
  equation $$ \\partial_t h(t\,x)=\\nu\\Del h(t\,x)+\\lambda  |\\nabla h(t\
 ,x)|^2 +\\sqrt{D}\\\, \\eta(t\,x)\, \\qquad (t\,x)\\in{\\mathbb{R}}_+\\tim
 es{\\mathbb{R}}^d $$ in $d\\ge 3$ dimensions in the perturbative regime\, 
 i.e. for $\\lambda>0$ small enough and a  smooth\, bounded\, integrable  i
 nitial condition $h_0=h(t=0\,\\cdot)$. The forcing term $\\eta$ in the rig
 ht-hand side is  a regularized space-time white noise. The exponential of 
 $h$ -- its so-called Cole-Hopf  transform --  is known to satisfy a  linea
 r PDE with multiplicative noise.  We  prove a large-scale diffusive limit 
 for the solution\, in particular a  time-integrated heat-kernel behavior f
 or the covariance in a parabolic scaling.  The proof is  based on  a rigor
 ous implementation of K. Wilson&#39\;s renormalization group scheme. A dou
 ble cluster/momentum-decoupling expansion  allows for  perturbative estima
 tes of the bare resolvent of the Cole-Hopf linear PDE in the small-field r
 egion where the noise is not too large\, following the broad lines of Iago
 lnitzer-Magnen. Standard large deviation estimates for $\\eta$ make it pos
 sible to extend the above estimates to the large-field region. Finally\,  
 we show\, by resumming all the by-products of the expansion\, that the sol
 ution $h$  may be written in the large-scale limit (after a suitable Galil
 ei transformation) as a small perturbation of the  solution  of the underl
 ying linear Edwards-Wilkinson model ($\\lambda=0$) with renormalized coeff
 icients $\\nu_{eff}=\\nu+O(\\lambda^2)\,D_{eff}=D+O(\\lambda^2)$.   This i
 s joint work with J. Magnen.
LOCATION:Seminar Room 1\, Newton Institute
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