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

$$\begin{aligned} \partial _t h(t,x)=\nu \Delta h(t,x)+\lambda |\nabla h(t,x)|^2 +\sqrt{D}\, \eta (t,x), \qquad (t,x)\in \mathbb {R}_+\times \mathbb {R}^d \end{aligned}$$

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 and Magnen (Commun Math Phys 162(1):85–121, 1994). 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 $\nu _{eff}=\nu +O(\lambda ^2),D_{eff}=D+O(\lambda ^2)$.

Appendices

Appendix 1: Cluster Expansions

1.1 Horizontal Cluster Expansion

The cluster expansion between boxes of scale 0 is performed according to the classical Bridges-Kennedy-Abdesselam-Rivasseau (BKAR) procedure (see [1, 58], or [45], §2.1 and 2.2), which we now briefly describe, following [45]. We apply it to the $A^0$ and $B^0$ kernels, and also to the covariance kernel $C_{\eta }(\cdot ,\cdot ):= \langle \eta (\cdot )\eta (\cdot )\rangle$ of the noise. The effect of the cluster expansion on the A’s and B’s is to “cut” all propagators between scale 0 boxes belonging to different polymers. The effect of the cluster expansion on the $\eta$’s is to make independent the $\eta$-fields produced in scale 0 boxes belonging to different polymers. As a result of those two operations, different polymers have been made totally independent, which makes it possible to extract averaged quantities such as counterterms. Since the covariance kernel of $\eta$ has finite range (with our cut-off conventions $\langle \eta (t,x)\eta (t',x')\rangle =0$ except if $(t,x),(t',x')$ belong to the same unit box in $\Delta ^0$ or to neighboring boxes), the cluster expansion on the $\eta$’s is hardly noticeable—in particular when it comes to bounds—, yet necessary.

Let $\mathcal{O}\subset \mathbb {D}^0$, and $|\mathcal{O}|:= \cup _{\Delta \in \mathcal{O}} \Delta \subset \mathbb {R}_+\times \mathbb {R}^d$ its support. We say that two boxes $\Delta ,\Delta '\in \mathcal{O}$, $\Delta \not =\Delta '$, are linked if (i) either $\Delta =[k,k+1)\times \bar{\Delta }$, $\Delta '=[k,k+1)\times \bar{\Delta }'$, $\bar{\Delta },\bar{\Delta }'\in \bar{\mathbb {D}}^0$, or (ii) $\Delta =[k,k+1)\times \bar{\Delta }$, $\Delta '=[k-1,k)\times \bar{\Delta }'$ or conversely $\Delta =[k-1,k)\times \bar{\Delta }$, $\Delta '=[k,k+1)\times \bar{\Delta }'$. By construction, there exists $(t,x)\in \Delta$, $(t',x')\in \Delta '$, such that $A^0((t,x),(t',x'))\rangle \not =0$ or $A^0((t',x'),(t,x))\not =0$ if and only if $\Delta =\Delta '$ or $\Delta ,\Delta '$ are linked. Similarly, if $\langle \eta (t,x)\eta (t',x')\rangle \not =0$, then $\Delta =\Delta '$ or $\Delta ,\Delta '$ are linked (and, furthermore, $d(\Delta ,\Delta ')=O(1)$). Denote by $L(\mathcal{O})$ the set of linked pairs $\{\Delta ,\Delta '\}$. Then, for every link weakening of $\mathcal{O}$, i.e. for every function $\varvec{s} \ : \ L(\mathcal{O})\rightarrow [0,1]$, extended trivially on the diagonal by letting $\varvec{s}_{\Delta ,\Delta }\equiv 1$ ($\Delta \in \mathcal{O}$), we define

$$\begin{aligned} B^0(\varvec{s})((t,x),(t',x'))= & {} s_{\Delta ^0_{t,x}, \Delta ^0_{t',x'}} B^0((t,x),(t',x')), \end{aligned}$$

(7.1)

$$\begin{aligned} A^{0}(\varvec{s}) ((t,x),(t',x'))= & {} s_{\Delta ^0_{t,x}, \Delta ^0_{t',x'}} A^0((t,x),(t',x')) \end{aligned}$$

(7.2)

$$\begin{aligned} \langle \eta (t,x)\eta (t',x')\rangle _{\varvec{s}}:= & {} s_{\Delta ^0_{t,x}, \Delta ^0_{t',x'}} \langle \eta (t,x)\eta (t',x')\rangle \end{aligned}$$

(7.3)

if $(\Delta _{t,x},\Delta _{t',x'})\in L(\mathcal{O})$, 0 else. Thus the effect of the function $\varvec{s}$ is to weaken off-diagonal elements of the propagator/covariance kernel.

We need some terminology before we get to the point. In the following discussion, $\mathcal{O}$ is fixed. A scale 0 forest ${\mathbb {F}}^0$ is a finite number of boxes $\Delta \in \mathcal{O}$, seen as vertices, connected by links, without loops. A (non-oriented) link $\ell$ connects $\Delta _{\ell }$ to $\Delta '_{\ell }$. Space-time variables ranging in $\Delta _{\ell }$, resp. $\Delta '_{\ell }$ are generally denoted as $(t_{\ell },x_{\ell })$, resp. $(t'_{\ell },x'_{\ell })$, or for short $z_{\ell }$, resp. $z'_{\ell }$. Non-isolated components of ${\mathbb {F}}^0$, i.e. connected components of ${\mathbb {F}}^0$ containing $\ge 2$ boxes are called trees, or (specifically in this statistical physics context) polymers. The (finite) set of vertices of polymers is denoted by $V({\mathbb {F}}^0)$. The set of all 0-th scale cluster forests is denoted by $\mathcal{F}^0(\mathcal{O})$, or simply $\mathcal{F}^0$ if $\mathcal{O}=\mathbb {D}^0$. If there exists a link between $\Delta$ and $\Delta '$, then we write $\Delta \sim _{{\mathbb {F}}^0}\Delta '$, or simply (if no ambiguity may arise) $\Delta \sim \Delta '$.

Now the following formula—called BKAR formula—holds: let $F=F(A^0,B^0|\eta )$ be some random function of the $A^0$ ’s and $B^0$ ’s, then

Proposition 7.1

(BKAR formula) (see [45], Proposition 2.6)

$$\begin{aligned} \langle F(A^0,B^0|\eta ) \rangle =\sum _{{\mathbb {F}}^0\in \mathcal{F}^0} \left( \prod _{\ell \in L({\mathbb {F}}^0)}\int _0^1 dw_{\ell }\right) \left( \left( \prod _{\ell \in L({\mathbb {F}}^0)} \frac{d}{ds_{\ell }}\right) \langle F(A^0(\varvec{s}(\varvec{w})),B^0(\varvec{s}(w)))|\eta \rangle _{\varvec{s}(\varvec{w})} \right) \nonumber \\ \end{aligned}$$

(7.4)

$s_{\Delta ,\Delta '}(\varvec{w})$, $\Delta \not =\Delta '$ being the infimum of the $w_{\ell }$ for $\ell$ running over the unique path from $\Delta$ to $\Delta '$ in ${\mathbb {F}}^0$ if $\Delta \sim _{{\mathbb {F}}^0}\Delta '$, and $s_{\Delta ,\Delta '}(\varvec{w})=0$ else.

The above formula is obtained by iterating the following step-by-step procedure. Choose some box $\Delta _1\in \mathbb {D}^0$, and Taylor-expand simultaneously with respect to the parameters $(s_{\ell })_{\ell }$ where $\ell$ ranges in the set $L_1(\mathbb {D}^0)$ of all pairs $\{\Delta _{\ell },\Delta '_{\ell }\}$ such that $\Delta _1=\Delta _{\ell }$ or $\Delta '_{\ell }$. One obtains:

$$\begin{aligned} F(\varvec{s}\big |_{L(\mathbb {D}^0)\setminus L_1(\mathbb {D}^0)};\varvec{s}\big |_{L_1(\mathbb {D}^0)}=1)&=F(\varvec{s}\big |_{L(\mathbb {D}^0)\setminus L_1(\mathbb {D}^0)};\varvec{s}\big |_{L_1(\mathbb {D}^0)}=0)\nonumber \\&\quad + \sum _{\ell _1\in L_1(\mathcal{O})} \int _0^1 dw_1\, \partial _{s_{\ell _1}}F(\varvec{s}\big |_{L(\mathbb {D}^0)\setminus L_1(\mathbb {D}^0)};\varvec{s}\big |_{L_1(\mathbb {D}^0)}=w_1).\nonumber \\ \end{aligned}$$

(7.5)

The following elementary relation is shown in [58],

$$\begin{aligned} \frac{d}{ds_{\ell }}\langle F(\eta )\rangle _{\varvec{s}(\varvec{w})} = \int _{\Delta _{\ell }} dz_{\ell } \int _{\Delta '_{\ell }} dz'_{\ell } \, \langle \eta (z_{\ell })\eta (z'_{\ell })\rangle _{\varvec{s}=1} \ \cdot \ \left\langle \frac{\delta }{\delta \eta (z_{\ell })} \frac{\delta }{\delta \eta (z'_{\ell })} F(\eta ) \right\rangle _{\varvec{s}(\varvec{w})}. \nonumber \\ \end{aligned}$$

(7.6)

In other words, an s-derivative acting on an averaged quantity $\langle F(\eta )\rangle _{\varvec{s}(\varvec{w})}$ has the effect of producing an explicit pairing $\langle \eta (z_{\ell })\eta (z'_{\ell })\rangle _{\varvec{s}=1}$, with the original covariance kernel, between two arbitrary points belonging resp. to one box and to the other box.

As explained before, each choice of forest ${\mathbb {F}}^0$ yields an explicit connection through $A^0$, $B^0$- or $\eta$-pairings of all boxes within a given connected component (tree), and disconnects boxes $\Delta ,\Delta '$ lying in different connected components since $B^{0}(\varvec{s}(w)) ((t,x),(t',x'))=A^{0}(\varvec{s}(w)) ((t,x),(t',x'))= \langle \eta (t,x)\eta (t',x')\rangle _{\varvec{s}(\varvec{w})}=0$ for $(t,x)\in \Delta , (t',x')\in \Delta '$.

1.2 Mayer Expansion

For the Mayer expansion (see § 5.1), we choose another set of objects $\mathcal O$ and a different way of implementing the s-dependence, and apply a slightly different formula. Namely, we let $\mathcal{O}\equiv \mathcal{O}({\mathbb {F}}^0, \{\mu _{\Delta }\}_{\Delta })$ be the set of scale 0 polymers, i.e. of non-isolated connected components of ${\mathbb {F}}^0$ with their external structure, depending on the differentiation orders $\{\mu _{\Delta }\}_{\Delta }$, produced by the vertical cluster expansion. Among these polymers, there are polymers with exactly two external legs, making up a subset $\mathcal{O}_1\equiv \mathcal{O}_1({\mathbb {F}}^0,\{\mu _{\Delta }\}_{\Delta })\subset \mathcal{O}$. The complementary set $\mathcal{O}_2\equiv \mathcal{O}_2({\mathbb {F}}^0,\{\mu _{\Delta }\}_{\Delta }):=\mathcal{O}\setminus \mathcal{O}_1$ is made up of polymers with $>2$ external legs, which require no renormalization. The following variant of BKAR’s formula, found originally in [1], is stated in the present form in [45]. We now denote by $\{\mathbb {P}_{\ell },\mathbb {P}'_{\ell }\}$ a pair of polymers connected by a link $\ell \in L(\mathcal{O})$.

Proposition 7.2

(Restricted 2-type cluster or BKAR2 formula) Assume $\mathcal{O}=\mathcal{O}_1\amalg \mathcal{O}_2$. Choose as initial object an object $o_1\in \mathcal{O}_1$ of type 1, and stop the Brydges-Kennedy-Abdesselam-Rivasseau expansion as soon as a link to an object of type 2 has appeared. Then choose a new object of type 1, and so on. This leads to a restricted expansion, for which only the link variables $z_{\ell }$, with $\ell \not \in \mathcal{O}_2\times \mathcal{O}_2$, have been weakened. The following closed formula holds. Let $\varvec{S}:L(\mathcal{O})\rightarrow [0,1]$ be a link weakening of $\mathcal{O}$, and $F=F((\varvec{S}_{\ell })_{\ell \in L(\mathcal{O})})$ a smooth function. Let $\mathcal{F}_{res}(\mathcal{O})$ be the set of forests ${\mathbb {G}}^0$ on $\mathcal O$, each component of which is (i) either a tree of objects of type 1, called unrooted tree; (ii)or a rooted tree such that only the root is of type 2. Then

$$\begin{aligned} F(1,\ldots ,1)=\sum _{{\mathbb {G}}^0\in \mathcal{F}_{res}(\mathcal{O})} \left( \prod _{\ell \in L({\mathbb {G}}^0)} \int _0^1 dW_{\ell } \right) \left( \left( \prod _{\ell \in L({\mathbb {G}}^0)} \frac{\partial }{\partial S_{\ell } } \right) F(S_{\ell }(\varvec{W})) \right) , \end{aligned}$$

(7.7)

where $S_{\ell }(\varvec{W})$ is either 0 or the minimum of the w-variables running along the unique path in $\bar{{\mathbb {G}}}^0$ from $\mathbb {P}_{\ell }$ to $\mathbb {P}'_{\ell }$, and $\bar{{\mathbb {G}}}^0$ is the forest obtained from ${\mathbb {G}}^0$ by merging all roots of ${\mathbb {G}}^0$ into a single vertex.

The way functions of the type $\langle F(A^0(\varvec{s}(\varvec{w})),B^0(\varvec{s}(\varvec{w}))|\eta )\rangle _{\varvec{s}(\varvec{w})}$ are made S-dependent is explained in 5.1. Differentiating w.r. to an S-parameter $S_{\mathbb {P}_1,\mathbb {P}_2}$ produces a factor

$\Big [ \prod _{\Delta _1\in \mathbb {P}_1, \Delta _2\in \mathbb {P}_2,(\Delta _1,\Delta _2)\not \in \mathbf{\Delta }_{ext}(\mathbb {P}_1)\times \mathbf{\Delta }_{ext}(\mathbb {P}_2)} \mathbf{1}_{\Delta _1\not =\Delta _2} \Big ]-1$, which upon expansion yields a sum over all overlap possibilities between boxes of $\mathbb {P}_1$ and boxes of $\mathbb {P}_2$ except those containing the external legs. Each contribution comes with a sign $(-1)^n$, where n is the number of overlapping boxes (see Fig. 3 for a representation of this rule). See also Fig. 4 below illustrating a more elaborate case with $n=2$.

Fig. 3

Mayer subtraction rule for one overlapping box

Fig. 4

Mayer expansion

The above procedure leads, as discussed in a much more involved, multi-scale context e.g. in [45], Proposition 2.12, to some mild combinatorial factors, which we discuss briefly. Recall that (by Cayley’s theorem) the number of trees over $\mathbb {P}_1,\ldots ,\mathbb {P}_n$ with fixed coordination numbers $(n(\mathbb {P}_i))_{i=1,\ldots ,n}$ equals $\frac{n!}{\prod _i (n(\mathbb {P}_i)-1)!}$. Choose a tree ${\mathbb {T}}$ component of ${\mathbb {G}}^0$. Start from the leaves of ${\mathbb {T}}$ and go down the branches inductively. Let $\mathbb {P}_1,\ldots ,\mathbb {P}_{n(\mathbb {P}')-1}$ be the leaves attached onto one and the same vertex $\mathbb {P}'$. Choose $n(\mathbb {P}')-1$ (possibly non distinct) boxes $\Delta _1,\ldots , \Delta _{n(\mathbb {P}')-1}\in \mathbb {D}^0$ of $\mathbb {P}'$ (there are $|\mathbb {P}'|^{n(\mathbb {P}')-1}$ possibilities), and assume that $\Delta _i\in \mathbb {P}_i$. For each choice of polymer $\mathbb {P}'$, this gives a supplementary factor $O((C |\mathbb {P}'|)^{n(\mathbb {P}')-1})$, to be multiplied by $\frac{1}{(n(\mathbb {P}')-1)!}$ coming from Cayley’s theorem. Summing over $n(\mathbb {P}')=2,3,\ldots$ yields $e^{C |\mathbb {P}'|}-1\le e^{C|\mathbb {P}'|}$. Summing over all boxes takes care automatically of the sum over all permutations of the polymers, which takes down the n! factor. Since bounds produced in Sect. 6 are in $O((g^{(0)})^m)$, where $m=\sum _{\mathbb {P}\in \mathcal{O}} |\mathbb {P}|$ is the number of boxes obtained by the cluster expansion, the latter large factor is compensated by a simple redefinition of coupling constant $g^{(0)}\rightsquigarrow e^C g^{(0)}=O(g^{(0)})$ in the bounds.

Appendix 2: The Effective Propagator

The effective propagator $\tilde{G}_{eff}$ obtained in §6.4 by resumming $\nu$-counterterms along a string, see (8.9) below, is shown in this section to be very well approximated at large scale by the Green kernel $(\partial _t-\nu _{eff}\Delta )^{-1}$.

We first need a technical lemma.

Lemma 8.1

There exists some constant $C>0$ such that, for every $\varvec{\kappa }=(\kappa _1,\ldots , \kappa _d)$ and $t>t'$, $x,x'$:

$$\begin{aligned}&\Big |\nabla ^{\varvec{\kappa }} G_{\nu ^{(0)}}((t,x),(t',x'))\Big | \le C^{|\varvec{\kappa }|+1}\, \Big (\lambda \sqrt{\nu ^{(0)}(t-t')} \Big )^{-|\varvec{\kappa }|} \, \Gamma (|\varvec{\kappa }|/2))\,\nonumber \\&\quad G_{\nu ^{(0)}-O(\lambda ^2)}((t,x),(t',x')). \end{aligned}$$

(8.1)

Proof

The spatial Fourier transform of $\nabla ^{\varvec{\kappa }} G\equiv \nabla ^{\varvec{\kappa }}G_{\nu ^{(0)}}$ is

$$\begin{aligned} \widehat{\nabla ^{\varvec{\kappa }} G}(t-t',\varvec{\xi })=(\mathrm{i}\varvec{\xi })^{\varvec{\kappa }} \, {\hat{K}}_{t-t'}(\varvec{\xi }), \qquad {\hat{K}}_{t-t'}(\varvec{\xi }):=e^{-(t-t')\nu ^{(0)}(\varvec{\xi },\varvec{\xi })}. \end{aligned}$$

(8.2)

Let $\varvec{\xi }_0:=\frac{x-x'}{2\nu ^{(0)}(t-t')}$. Then

$$\begin{aligned}&\nabla ^{\varvec{\kappa }}G(t-t',x-x')=(2\pi )^{-d} \int _{\mathbb {R}^d} d\varvec{\xi }\, (\mathrm{i}\varvec{\xi })^{\varvec{\kappa }} \, {\hat{K}}_{t-t'}(\varvec{\xi }) e^{\mathrm{i}(x-x',\xi )} \nonumber \\&\quad =(2\pi )^{-d} \int _{\mathbb {R}^d+\mathrm{i}\varvec{\xi }_0} d\varvec{\xi }\, (\mathrm{i}\varvec{\xi })^{\varvec{\kappa }} \, {\hat{K}}_{t-t'}(\varvec{\xi }) e^{\mathrm{i}(x-x',\xi )} \nonumber \\&\quad = (2\pi )^{-d} e^{-|x-x'|^2/4\nu ^{(0)}(t-t')} \int _{\mathbb {R}^d} d\varvec{\xi }\, (\mathrm{i}\varvec{\xi }- \varvec{\xi }_0)^{\varvec{\kappa }} \, e^{-\nu ^{(0)}(t-t')|\varvec{\xi }|^2} \nonumber \\&\quad =(\nu ^{(0)}(t-t'))^{-|\varvec{\kappa }|/2} G_{\nu ^{(0)}}((t,x),(t',x')) \int _{\mathbb {R}^d} d\varvec{\zeta }\, (\mathrm{i}\varvec{\zeta }-\varvec{\zeta }_0)^{\kappa } \, e^{-|\varvec{\zeta }|^2} \end{aligned}$$

(8.3)

where now $\varvec{\zeta }:=\sqrt{\nu ^{(0)}(t-t')}\, \varvec{\xi }$, $\varvec{\zeta }_0:=\frac{x-x'}{2\sqrt{\nu ^{(0)}(t-t')}}$ are non-dimensional parameters. Rewrite $G_{\nu ^{(0)}}((t,x),(t',x'))$ as $G_{\nu ^{(0)}+O(\lambda ^2)}((t,x),(t',x')) \, \cdot \, O(1)\, e^{-O(\lambda ^2)|\varvec{\zeta }_0|^2}$. Then, for all $\varvec{\kappa }'\le \varvec{\kappa }$,

$$\begin{aligned} |\varvec{\zeta }_0|^{|\varvec{\kappa '}|} e^{-\lambda ^2|\varvec{\zeta }_0|^2}\lesssim \frac{|\varvec{\zeta }_0|^{\varvec{\kappa '}}}{\lambda ^{|\varvec{\kappa }|'}|\varvec{\zeta }_0|^{|\varvec{\kappa }'|}/\Gamma ( \frac{|\varvec{\kappa '}|}{2}+1)} =\lambda ^{-|\varvec{\kappa }'|} \Gamma (\frac{|\varvec{\kappa '}|}{2}+1) \end{aligned}$$

and $\int _{\mathbb {R}^d} d\varvec{\zeta }\, \varvec{\zeta }^{\varvec{\kappa }-\varvec{\kappa }'} \, e^{-|\varvec{\zeta }|^2} = O(C^{|\varvec{\kappa }|}) \Gamma (|\varvec{\kappa }-\varvec{\kappa }'|/2).$ One concludes by using the binomial formula. $\square$

Let us now come to the point. Recall $\Delta ^{\rightarrow 0}= \bar{\chi }^{(0)}*\Delta$ [see (4.6)] is an ultra-violet regularization of $\Delta$.

Lemma 8.2

Let $\delta \nu :=\nu _{eff}-\nu ^{(0)}$,

$$\begin{aligned} G_{eff}:= & {} (\partial _t-\nu _{eff}\Delta )^{-1}, \end{aligned}$$

(8.4)

$$\begin{aligned} \tilde{G}_{eff}:= & {} A^{\rightarrow 1} \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B^{\rightarrow 1}=(1-\delta \nu \, G^{\rightarrow 1}\Delta ^{\rightarrow 0})^{-1} G^{\rightarrow 1}.\nonumber \\ \end{aligned}$$

(8.5)

Then:

• 1. There exists $\tilde{\nu }^{(0)}=\nu ^{(0)}+O(\lambda ^2)$ and a constant $C>0$ such that

  $$\begin{aligned} \tilde{G}_{eff}((t,x),(t',x'))\le C G_{\tilde{\nu }^{(0)}}((t,x),(t',x')). \end{aligned}$$

  (8.6)

  Furthermore, if $1\le j\le j'\le j''$,

  $$\begin{aligned}&\left( \nabla ^{\kappa '} A^j \langle j| \, \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} \nabla ^{\kappa ''} B^{j'} \, |j'\rangle \right) ((t,x),(t',x')) \nonumber \\&\quad \lesssim 2^{-\frac{j'}{2}(|\kappa '|+|\kappa ''|)} 2^{-(j'-j)} G_{\tilde{\nu }^{(0)}/c}((t,x),(t',x')), \qquad t-t'\approx 2^{j'} \end{aligned}$$

  (8.7)
  $$\begin{aligned}&\left( \nabla ^{\kappa '} A^j \langle j| \, \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} \nabla ^{\kappa ''} B^{j'} \, |j'\rangle \right) ((t,x),(t',x')) \nonumber \\&\quad \lesssim 2^{-\frac{j'}{2}(|\kappa '|+|\kappa ''|)}2^{-(j''-j)} 2^{-(j''-j')} G_{\tilde{\nu }^{(0)}/c}((t,x),(t',x')), \qquad t-t'\approx 2^{j''} \nonumber \\ \end{aligned}$$

  (8.8)

  with $c=1$ if $\kappa '=\kappa ''=0$, and $c={1\over 2}$ else.

• 2. For every $\kappa '<{1\over 2}$, the following holds: if $t-t'\approx 1$,

  $$\begin{aligned}&(\tilde{G}_{eff}-G_{eff})((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x')) \sim _{\varepsilon \rightarrow 0} O(\varepsilon ^{2\kappa '})\nonumber \\&\quad G_{\tilde{\nu }^{(0)}}((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x')) \end{aligned}$$

  (8.9)

  uniformly for

  $$\begin{aligned} |x-x'|^2=o(\varepsilon ^{-1/2}(t-t')) \end{aligned}$$

  (8.10)

  [see (8.14) below] with $\tilde{\nu }^{(0)}=\nu ^{(0)}+O(\lambda ^2)$.

Thus $\tilde{G}_{eff}$ is equal to $G_{eff}$ with an excellent approximation at large scale which holds well beyond the normal regime $\frac{|x|^2}{t} \lesssim 1$ (one can compare with [23] where extended heat-kernel asymptotics are shown for a lattice regularization instead). Equation (8.7,8.8) show that the bounds on $\tilde{G}_{eff}=\sum _{j,j'\ge 1} A^{j} \, \langle j| \, \cdot \, \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} \, \cdot \, B^{j'}\, |j'\rangle$, expressed as a product of a resolvent by two propagators A, B as in (3.33), decrease exponentially with the difference of the “scales” of these three operators, thus yielding bounds that can be resummed adequately.

Proof

We concentrate on 2., with 1. proved on the way. Introduce $G_{1,eff}:=(\partial _t-\nu ^{(0)}\Delta - \delta \nu \Delta ^{\rightarrow 0})^{-1}=A (1-\delta \nu B \Delta ^{\rightarrow 0}A)^{-1} B$, and write for short $\chi ^0$ instead of $\bar{\chi }^{(0)}$.

• (i) First, $G_{1,eff}((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x'))=(1+O(\varepsilon )) G_{eff}((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x'))$ if (8.10) holds. Namely, the spatial Fourier transform of $G_{eff}-G_{1,eff}$ is

  $$\begin{aligned}&{\hat{G}}_{eff}(\varepsilon ^{-1}(t-t'),\varvec{\xi })-{\hat{G}}_{1,eff}(\varepsilon ^{-1}(t-t'),\varvec{\xi })= {\hat{K}}_{\varepsilon ^{-1}(t-t')}(\varvec{\xi }), \end{aligned}$$

  (8.11)
  $$\begin{aligned}&\text{ with }\, \qquad {\hat{K}}_{t-t'}(\varvec{\xi })= e^{-(t-t')\nu _{eff}(\varvec{\xi },\varvec{\xi })} \ \cdot \ \left( 1-e^{-(t-t')\delta \nu \, (\widehat{\chi ^0}(\xi )-1)\, (\varvec{\xi },\varvec{\xi })} \right) . \end{aligned}$$

  (8.12)

  Since $\chi ^0$ is compactly supported, its Fourier transform $\widehat{\chi ^0}$ extends to an entire function satisfying: $|\widehat{\chi ^0}(\xi )|\lesssim e^{C|\mathrm{Im\ }(\xi )|}$. If $\chi ^0(\cdot )$ is chosen to be isotropic (which we assume), then $\nabla (\widehat{\chi ^0})(0)=0$. Since $\int \chi ^0=1$ and $\chi ^0$ is smooth, $|\widehat{\chi ^0}(\varvec{\xi })-1|=O_{\varvec{\xi }\rightarrow 0}(|\varvec{\xi }|^2)= O_{\varvec{\xi }\rightarrow 0}((t-t')|\varvec{\xi }|^2)$ and $|\widehat{\chi ^0}(\varvec{\xi })|\, |\varvec{\xi }|^2= O_{|\mathrm{Re\ }(\varvec{\xi })|\rightarrow \infty }(e^{C|\mathrm{Im\ }(\varvec{\xi })|})=O((t-t')e^{C\sqrt{t-t'}\, |\mathrm{Im\ }(\varvec{\xi })|}).$ Let

  $$\begin{aligned} \varvec{\xi }_0:= \frac{x}{2\nu _{eff}(t-t')}, \qquad \rho _0:= \frac{1}{\sqrt{2\nu _{eff}(t-t')}} \left( 1+\frac{|x|}{\sqrt{2\nu _{eff}(t-t')}}\right) . \end{aligned}$$

  (8.13)

  Note that, provided

  $$\begin{aligned} |x|^2=O(\varepsilon ^{-{1\over 2}+2({1\over 2}-\kappa )}(t-t'))\qquad \text{ with }\ \frac{1}{4}<\kappa <{1\over 2} \end{aligned}$$

  (8.14)

  – which is compatible with our hypothesis (8.10) if one lets $\kappa \rightarrow ({1\over 2})^-$—, and $|\varvec{\xi }|\lesssim \varepsilon ^{\kappa } \rho _0$ – whence $|\xi |\ll 1$—the error term in the exponential, $\frac{t-t'}{\varepsilon }\delta \nu \, |1-\widehat{\chi ^0}(\varvec{\xi })|\, (\varvec{\xi },\varvec{\xi })=O(\lambda ^2) \varepsilon ^{4\kappa -1} \rho _0^4$ is a O(1). Hence

  $$\begin{aligned}&\int _{B(0,\varepsilon ^{\kappa }\rho _0)} d\varvec{\xi }\, {\hat{K}}_{\varepsilon ^{-1}(t-t')}(\varvec{\xi }) e^{\mathrm{i}(x-x',\varvec{\xi })}\nonumber \\&\quad = \int _{B(0,\varepsilon ^{\kappa }\rho _0)+\mathrm{i}\varepsilon ^{1/2}\varvec{\xi }_0} d\varvec{\xi }\, {\hat{K}}_{\varepsilon ^{-1}(t-t')}(\varvec{\xi }) e^{\mathrm{i}(x-x',\varvec{\xi })} + \partial I(x-x') \nonumber \\&\quad = O(\frac{t-t'}{\varepsilon }\delta \nu ) \ e^{-|x-x'|^2/4\nu _{eff}(t-t')} \int _{B(0,\varepsilon ^{\kappa }\rho _0)} d\varvec{\xi }\, |\varvec{\xi }|^4\, e^{-\varepsilon ^{-1}\nu _{eff}(t-t')(\varvec{\xi },\varvec{\xi })} + \partial I(x-x') \nonumber \\&\quad \sim _{\varepsilon \rightarrow 0} O(\lambda ^2 \varepsilon ^{2\kappa '})\, G_{eff}((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x')) + \partial I(x-x') \end{aligned}$$

  (8.15)

  where $2\kappa '=-1-\frac{d}{2}+ \kappa (d+4) \rightarrow _{\kappa \rightarrow {1\over 2}} 1$ and

  $$\begin{aligned} \partial I(x-x'):=\int _{\partial B(0,\varepsilon ^{\kappa }\rho _0)\times [0,\mathrm{i}\varepsilon ^{1/2}\varvec{\xi }_0]} d\varvec{\xi }\, {\hat{K}}_{\varepsilon ^{-1}(t-t')}(\varvec{\xi }) e^{\mathrm{i}(x-x',\varvec{\xi })} \end{aligned}$$

  (8.16)

  and

  $$\begin{aligned}&|\partial I(x-x')|,\ \Big |\int _{|\varvec{\xi }|\gtrsim \varepsilon ^{\kappa } \rho _0} d\varvec{\xi }\, K_{\varepsilon ^{-1}(t-t')}(\varvec{\xi }) e^{\mathrm{i}(x-x',\varvec{\xi })} \Big | \nonumber \\&\quad =O\Big (\varepsilon ^{d/2} \int _{|\varvec{\zeta }|\gtrsim \varepsilon ^{-({1\over 2}-\kappa )} (1+\frac{|x|}{2\nu _{eff}(t-t')})} d\varvec{\zeta }\, e^{-\nu _{eff}|\varvec{\zeta }|^2} \Big ) \end{aligned}$$

  (8.17)

  are negligible with respect to $G_{eff}((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x')).$ On the other hand,

  $$\begin{aligned}&A(1-\delta \nu B\Delta ^{\rightarrow 0}A)^{-1} B^0|0\rangle = A^0\, \langle 0| \, B^0 |0\rangle +\delta \nu AB\Delta ^{\rightarrow 0}A^0 \, \langle 0|\, B^0 |0\rangle + \cdots \nonumber \\&\quad = G^0 + \delta \nu \, G_{1,eff}\, \Delta ^{\rightarrow 0}G^0. \end{aligned}$$

  (8.18)

• (ii) Next, $\left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} \simeq 1$ at large scale. Namely, expanding $\Big (1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \Big )^{-1}$ into a series, we get a geometric series in $\delta \nu \, \Delta ^{\rightarrow 0} A^{\rightarrow 1}B^{\rightarrow 1}$. Write $A^{\rightarrow 1}B^{\rightarrow 1}\equiv G^{\rightarrow 1}$ as $\int _0^{+\infty } \tilde{\chi }^{\rightarrow 1}(t) e^{\nu ^{(0)} t\Delta } dt$, where (in the notations of Definition 3.3) $\tilde{\chi }^{\rightarrow 1}:=\sum _{j=1}^{+\infty } (\chi *\chi )^j$ is “one minus a bump function”, i.e. $\tilde{\chi }^{\rightarrow 1}\big |_{[0,c]}=0, \tilde{\chi }^{\rightarrow 1}\big |_{[c^{-1},+\infty )}=1$ for some $c>0$. Then, since $\Delta ^{\rightarrow 0}$ commutes with the $G^{\rightarrow 1}$’s,

  $$\begin{aligned}&( \Delta ^{\rightarrow 0}G^{\rightarrow 1})^2((t,x),(t',x')) \nonumber \\&\quad =\int _{t'}^{(t+t')/2} dt''\, \int dx''\, (\Delta ^{\rightarrow 0})^2 G^{\rightarrow 1}((t,x),(t'',x'')) G^{\rightarrow 1}((t'',x''),(t',x')) \nonumber \\&\qquad + \int _{(t+t')/2}^{t} dt''\, \int dx''\, G^{\rightarrow 1}((t,x),(t'',x'')) (\Delta ^{\rightarrow 0})^2 G^{\rightarrow 1}((t'',x''),(t',x')) \nonumber \\&\quad =O(1)\ (t-t') (\Delta ^{\rightarrow 0})^2 G^{\rightarrow 1}((t,x),(t',x')). \end{aligned}$$

  (8.19)

  We call this the commutation trick. Recall $|\delta \nu |=O(\lambda ^2)$. Iterating yields by using Lemma 8.1

  $$\begin{aligned}&\sum _{n\ge 1} (\delta \nu )^n\ \Big | (\Delta ^{\rightarrow 0}G^{\rightarrow 1})^n ((t,x),(t',x')) \Big | \lesssim \sum _{n\ge 1} (\delta \nu )^n \frac{(t-t')^{n-1}}{(n-1)!} \ \Big |(\Delta ^{\rightarrow 0})^n G^{\rightarrow 1}((t,x),(t',x')) \Big | \nonumber \\&\quad \lesssim \sum _{n\ge 1} \frac{(t-t')^{n-1}}{(n-1)!} 2^{-n} \Gamma (n) (t-t')^{-n} \tilde{G}^{\rightarrow 1}((t,x),(t',x')) \nonumber \\&\quad = O(1) \ (t-t')^{-1} \tilde{G}^{\rightarrow 1}((t,x),(t',x')) \end{aligned}$$

  (8.20)

  where $\tilde{G}^{\rightarrow 1}=G^{\rightarrow 1}_{\nu ^{(0)}+O(\lambda ^2)}$, and

  $$\begin{aligned}&\Big [\left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} -1\Big ] ((t,x),(t',x')) \nonumber \\&\quad =\delta \nu B^{\rightarrow 1} \Big ( \sum _{n\ge 0} (\delta \nu )^n (\Delta ^{\rightarrow 0}G^{\rightarrow 1})^n \Big ) \Delta ^{\rightarrow 0} A^{\rightarrow 1} ((t,x),(t',x')) \nonumber \\&\quad =O( \delta \nu )\, (t-t')^{-1}\, \tilde{G}^{\rightarrow 1}((t,x),(t',x') \end{aligned}$$

  (8.21)

  where $\tilde{G}^{\rightarrow 1}$ has again been possibly rescaled. Using for a third time the commutation trick, one finally gets

  $$\begin{aligned} \Big |\Big (A \Big [\left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} -1\Big ] B\Big ) ((t,x),(t',x')) \Big | = O(1)\ G^{\rightarrow 1}((t,x),(t',x')). \nonumber \\ \end{aligned}$$

  (8.22)

  On the other hand, the orthonormality of the basis $(|j\rangle )_{j\ge 0}$ implies immediately

  $$\begin{aligned} \Big (A^0 \langle 0|\, \Big [\left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} -1\Big ] B\Big ) ((t,x),(t',x'))= G^0((t,x),(t',x')).\nonumber \\ \end{aligned}$$

  (8.23)

  Point 1. is a particularization of (8.22). If $t-t'\approx 2^{j'}$ and $j\not =j'$, then

  $$\begin{aligned}&A^j \langle j| \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B^{j'} |j'\rangle ((t,\cdot ),(t',\cdot ))\nonumber \\&\quad = \delta \nu G^j \Delta ^{\rightarrow 0} G^{j'}((t,\cdot ),(t',\cdot )+\cdots \end{aligned}$$

  (8.24)

  has an extra $2^{-(j'-j)}$-prefactor due to a reduced volume of integration in time. If $t-t'\approx 2^{j''}\gg 2^{j'}$, then the leading term in the series vanishes, so that

  $$\begin{aligned}&A^j \langle j| \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B^{j'} |j'\rangle ((t,\cdot ),(t',\cdot )) \nonumber \\&\quad =\left( (\delta \nu )^2 G^j \Delta ^{\rightarrow 0}G^{\rightarrow 1}\Delta ^{\rightarrow 0}G^{j'}+\ldots \right) ((t,\cdot ),(t',\cdot )), \end{aligned}$$

  (8.25)

  where the middle propagator $G^{\rightarrow 1}$ has scale $j''+O(1)$, leading for the same reason to an extra $2^{-(j''-j)}2^{-(j''-j')}$-prefactor. Gradients $\nabla ^{\kappa '},\nabla ^{\kappa ''}$ are easily turned into prefactors by using elementary heat kernel estimates as in Lemma 3.4 (i).

• (iii) Let us now bound

  $$\begin{aligned} D:= & {} A \Big [ (1-\delta \nu B\Delta ^{\rightarrow 0}A)^{-1}-(1-\delta \nu B^{\rightarrow 1}\Delta ^{\rightarrow 0} A^{\rightarrow 1})^{-1} \Big ] B \nonumber \\= & {} A \Big [ (1-\delta \nu B\Delta ^{\rightarrow 0}A)^{-1} \ \cdot \ \delta \nu \left( B\Delta ^{\rightarrow 0}A-B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1}\right) \nonumber \\&(1-\delta \nu B^{\rightarrow 1}\Delta ^{\rightarrow 0} A^{\rightarrow 1})^{-1} \Big ] B\nonumber \\= & {} A \Big [ (1-\delta \nu B\Delta ^{\rightarrow 0}A)^{-1}\ \cdot \ \delta \nu \left( B^0|0\rangle \, \Delta ^{\rightarrow 0} A +B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^0 \langle 0| \, \right) \nonumber \\&(1-\delta \nu B^{\rightarrow 1}\Delta ^{\rightarrow 0} A^{\rightarrow 1})^{-1} \Big ] B. \nonumber \\ \end{aligned}$$

  (8.26)

  Thus

  $$\begin{aligned} D= & {} \delta \nu \Big (G^0 + \delta \nu \, G_{1,eff}\, \Delta ^{\rightarrow 0}G^0\Big ) \Delta ^{\rightarrow 0} \Big ( A(1-\delta \nu \, B^{\rightarrow 1}\Delta ^{\rightarrow 0}A^{\rightarrow 1})^{-1} B \Big ) \nonumber \\&+ \delta \nu \, \tilde{G}_{1,eff} \Delta ^{\rightarrow 0}G^0 \end{aligned}$$

  (8.27)

  where the kernel

  $$\begin{aligned} \tilde{G}_{1,eff}(\cdot ,\cdot )&=A(1-\delta \nu \, B\Delta ^{\rightarrow 0}A)^{-1}B^{\rightarrow 1}(\cdot ,\cdot )=AB^{\rightarrow 1}(\cdot ,\cdot )\nonumber \\&\quad +\delta \nu AB\Delta ^{\rightarrow 0}AB^{\rightarrow 1}(\cdot ,\cdot )+\ldots \nonumber \\&= G^{\rightarrow 1}(\cdot ,\cdot )+\delta \nu \, G_{1,eff}\Delta ^{\rightarrow 0}AB^{\rightarrow 1}(\cdot ,\cdot ) \end{aligned}$$

  (8.28)

  is bounded (using again and again the commutation trick) by $O(1)\, G_{\nu ^{(0)}+O(\lambda ^2)}(\cdot ,\cdot )$. Hence

  $$\begin{aligned}&|D((t,x),(t',x'))| \lesssim (t-t')^{-1} G_{\nu ^{(0)}+O(\lambda ^2)}((t,x),(t',x')). \end{aligned}$$

  (8.29)

• (iv) Finally,

  $$\begin{aligned}&G_{1,eff}-\tilde{G}_{eff}-D\nonumber \\&\qquad =A \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B - A^{\rightarrow 1} \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B^{\rightarrow 1} \nonumber \\&\qquad = A^0 \langle 0|\, \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B + A^{\rightarrow 1} \left( 1-\delta \nu B^{\rightarrow 1} \Delta ^{\rightarrow 0} A^{\rightarrow 1} \right) ^{-1} B^0 |0\rangle \nonumber \\&\qquad =G^0 \end{aligned}$$

  (8.30)

  and $G^0((\varepsilon ^{-1}t,\varepsilon ^{-1/2}x),(\varepsilon ^{-1}t',\varepsilon ^{-1/2}x'))=0$ for $\varepsilon$ small enough.

$\square$

Remark

Using a suitably chosen cut-off $\chi ^0$ with vanishing first momenta (obtained e.g. by subtracting the beginning of the Taylor expansion of its Fourier transform near zero), i.e. such that $\int dx\, x_{i_1}\cdots x_{i_p} \chi ^0(x)=0$ for $1\le i_1,\ldots ,i_p\le d$ and $p=2,3,\ldots ,n-1$ one gets $\nabla ^p(\widehat{\chi ^0})(0)=0$, $2\le p\le n-1$, which makes it possible to reduce the prefactor $O(\lambda ^2 \varepsilon ^{1^-})$ in (8.15) to $O(\lambda ^2)$ times an arbitrary large power of $\varepsilon$.