# Renormalization of singular elliptic stochastic PDEs using flow equation

Paweł Duch
Faculty of Mathematics and Computer Science
Adam Mickiewicz University in Poznań
ul. Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
<EMAIL_ADDRESS>
###### Abstract

We develop a solution theory for singular elliptic stochastic PDEs with
fractional Laplacian, additive white noise and cubic non-linearity. The method
covers the whole sub-critical regime. It is based on the Wilsonian
renormalization group theory and the Polchinski flow equation.

###### Contents

  1. 1 Introduction
  2. 2 Effective force
  3. 3 Construction of effective force coefficients
  4. 4 Regularizing kernels
  5. 5 Function spaces for coefficients
  6. 6 Deterministic bounds for irrelevant coefficients
  7. 7 Generalized effective force coefficients
  8. 8 Taylor polynomial and remainder
  9. 9 Deterministic bounds for relevant coefficients
  10. 10 Deterministic bounds for long-range part of coefficients
  11. 11 Deterministic construction of solution
  12. 12 Cumulants of effective force coefficients
  13. 13 Probabilistic analysis
  14. 14 Function spaces for cumulants
  15. 15 Flow equation for cumulants
  16. 16 Uniform bounds for cumulants

##  1 Introduction

A general technique that allows to renormalize and prove universality of
parabolic singular SPDEs with fractional Laplacian, additive noise and
polynomial non-linearity was developed in [4]. The goal of this paper is to
give a different application of this technique. We present a self-contained
construction of solutions of the following non-local singular elliptic SPDEs

$\big{(}1+(-\Delta)^{\sigma/2}\big{)}\varPhi(x)=\xi(x)+\lambda\varPhi(x)^{3}-\infty\varPhi(x),\quad
x\in\mathbb{R}^{\mathsf{d}},$ (1.1)

where $\mathsf{d}\in\\{1,\ldots,6\\}$, $\sigma\in(\mathsf{d}/3,\mathsf{d}/2]$,
$(-\Delta)^{\sigma/2}$ is the fractional Laplacian, $\lambda\in\mathbb{R}$ is
sufficiently small and $\xi$ is the periodization of the white noise on
$\mathbb{R}^{\mathsf{d}}$ with period $2\pi$. Recall that the regularity of
the noise $\xi$ is slightly worse than $-\mathsf{d}/2$ and the expected
regularity of the solution is slightly worse than $\sigma-\mathsf{d}/2$. For
$\sigma>\mathsf{d}/2$ the above equation is not singular and can be solved
using classical PDE theory. For $\sigma\leq\mathsf{d}/2$ the solution is not a
function but only a distribution. As a result, the cubic term is ill-defined
and has to be renormalized by subtracting appropriate mass counterterm (for
$\mathsf{d}>6$ other counterterms are needed even if one takes into account
all the symmetries of the equation). The renormalization problem is tractable
only if the equation is sub-critical (super-renormalizable). This is the case
if the expected regularity of the renormalized non-linearity is better than
the regularity of $\xi$, i.e. $3(\sigma-\mathsf{d}/2)>-\mathsf{d}/2$. Let us
remark that for $\mathsf{d}=5$ and $\sigma=2$ the above equation is sometimes
called the elliptic quantization equation of the $\varPhi^{4}_{3}$ model
(provided $\xi$ is replaced by the white noise on $\mathbb{R}^{\mathsf{d}}$).

Let $G\in L^{1}(\mathbb{R}^{\mathsf{d}})$ be the fundamental solution for the
pseudo-differential operator $\mathbf{Q}:=1+(-\Delta)^{\sigma/2}$ and let
$G_{\kappa}$ be the smooth approximation of $G$ with a spatial UV cutoff of
order $[\kappa]:=\kappa^{1/\sigma}$ introduced in Def. 2.1. We rewrite the
above singular SPDE in the following regularized mild form

$\varPhi=G_{\kappa}\ast F_{\kappa}[\varPhi],\qquad\kappa\in(0,1/2].$ (1.2)

The functional $F_{\kappa}[\varphi]$, called the force, is defined by

$F_{\kappa}[\varphi](x):=\xi(x)+\lambda\,\varphi^{3}(x)+\sum_{i=1}^{i_{\sharp}}\lambda^{i}\,c_{\kappa}^{[i]}\varphi(x),$
(1.3)

where $i_{\sharp}=\lfloor\sigma/(3\sigma-\mathsf{d})\rfloor$ and the
parameters $c_{\kappa}^{[i]}\in\mathbb{R}$ depending on the UV cutoff $\kappa$
are called the counterterms. Let us state our main result.

######  Theorem 1.1.

There exists a choice of counterterms and a random variable $\lambda_{0}$ such
that $\mathbb{E}(\lambda_{0}^{-n})<\infty$ for every $n\in\mathbb{N}_{+}$ and
for every random variable $\lambda\in[-\lambda_{0},\lambda_{0}]$ and
$\kappa\in(0,1/2]$ Eq. (1.2) has a periodic solution $\varPhi_{\kappa}\in
C^{\infty}(\mathbb{R}^{\mathsf{d}})$ and for any $\beta<\sigma-\mathsf{d}/2$
the limit $\lim_{\kappa\searrow 0}\varPhi_{\kappa}$ exists almost surely in
the Besov space $\mathscr{C}^{\beta}(\mathbb{R}^{\mathsf{d}})$.

###### Proof.

We first establish bounds for cumulants of the enhanced noise introduced in
Def. 7.1. The bounds are stated in Theorem 16.1 and hold true for an
appropriate choice of the counterterms. Using these bounds and a Kolmogorov
type argument we deduce bounds for the enhanced noise stated in Theorem 13.1.
This together with the deterministic result of Theorem 11.4 implies the
statement. ∎

######  Remark 1.2.

There exists a general technique developed in [5, 3, 2, 1] based on the theory
of regularity structures that allows to systematically renormalize virtually
all sub-critical singular SPDEs with local differential operators. However,
due to the slow decay of the kernel $G$ at infinity, this technique does not
cover Eq. (1.2).

##  2 Effective force

The basic object of the flow equation approach is the effective force
functional

$\mathscr{S}(\mathbb{M})\ni\varphi\mapsto
F_{\kappa,\mu}[\varphi]\in\mathscr{S}^{\prime}(\mathbb{M})$ (2.1)

depending on the UV cutoff $\kappa\in(0,1/2]$ and the flow parameter
$\mu\in[0,1]$, where $\mathscr{S}(\mathbb{M})$ and
$\mathscr{S}^{\prime}(\mathbb{M})$ denote the space of Schwartz functions and
Schwartz distributions on $\mathbb{M}=\mathbb{R}^{\mathsf{d}}$, respectively.

######  Definition 2.1.

Fix $\chi\in C^{\infty}(\mathbb{R})$ such that $\chi(r)=0$ for $|r|\leq 1$ and
$\chi(r)=1$ for $|r|>2$ and let $\chi_{\mu}(r):=\chi(r(1-\mu)/\mu)$ for
$\mu\in(0,1]$. Let $G\in L^{1}(\mathbb{M})$ be the fundamental solution for
the pseudo-differential operator $\mathbf{Q}:=1+(-\Delta)^{\sigma/2}$. For
$\kappa\in(0,1/2]$ and $\mu\in(0,1]$ the smooth kernels
$G_{\kappa},G_{\kappa,\mu}\in L^{1}(\mathbb{M})$ are defined by

$G_{\kappa}(x):=\chi_{\kappa}(|x|^{\sigma})\,G(x),\qquad
G_{\kappa,\mu}(x):=\chi_{\kappa}(|x|^{\sigma})\,\chi_{\mu}(|x|^{\sigma})\,G(x).$
(2.2)

######  Remark 2.2.

Since $\chi_{\kappa}(r)\chi_{\mu}(r)=\chi_{\kappa}(r)$ for $\mu\leq\kappa/2$
it holds $G_{\kappa,\mu}=G_{\kappa}$ for $\mu\leq\kappa/2$ and
$G_{\kappa,\mu}=G_{\mu}$ for $\mu\geq 2\kappa$. Moreover,
$\lim_{\kappa\searrow 0}G_{\kappa}=G$ in $L^{1}(\mathbb{M})$ and
$G_{\kappa,1}=0$ for all $\kappa\in(0,1/2]$.

By definition, the effective force satisfies the following flow equation

$\langle\partial_{\mu}F_{\kappa,\mu}[\varphi],\psi\rangle=-\langle\mathrm{D}F_{\kappa,\mu}[\varphi,\partial_{\mu}G_{\kappa,\mu}\ast
F_{\kappa,\mu}[\varphi]],\psi\rangle$ (2.3)

with the boundary condition $F_{\kappa,0}[\varphi]=F_{\kappa}[\varphi]$, where
$\varphi,\psi\in\mathscr{S}(\mathbb{M})$ and the force $F_{\kappa}[\varphi]$
is defined by Eq. (1.3). The paring between a distribution $V$ and a test
function $\psi$ is denoted by $\langle V,\psi\rangle$ and
$\langle\mathrm{D}V[\varphi,\zeta],\psi\rangle$ is the derivative of the
functional $\mathscr{S}(\mathbb{M})\ni\varphi\mapsto\langle
V[\varphi],\psi\rangle\in\mathbb{R}$ in the direction
$\zeta\in\mathscr{S}(\mathbb{M})$. In contrast to the force
$F_{\kappa}[\varphi]$ the effective force $F_{\kappa,\mu}[\varphi]$ is
generically a non-local functional of $\varphi$.

We claim that $\varPhi_{\kappa}:=G_{\kappa}\ast F_{\kappa,1}[0]$ is a solution
of Eq. (2.4). The above statement is a consequence of the equalities
$G_{\kappa,\mu}=G_{\kappa}$ and $F_{\kappa,\mu}[\varphi]=F_{\kappa}[\varphi]$
that hold for all $\mu\in[0,\kappa/2]$ and the identity

$F_{\kappa,1}[0]=F_{\kappa,\mu}[G_{\kappa,\mu}\ast F_{\kappa,1}[0]]$ (2.4)

that holds for all $\mu\in[0,1]$. In order to prove the last identity we use
the flow equation to show that the difference between the LHS and RHS of Eq.
(2.4), denoted by $g_{\kappa,\mu}$, satisfies the following linear ODE

$\partial_{\mu}g_{\kappa,\mu}=-\mathrm{D}F_{\kappa,\mu}[G_{\kappa,\mu}\ast
F_{\kappa,1}[0],\partial_{\mu}G_{\kappa,\mu}\ast g_{\kappa,\mu}]$ (2.5)

with the boundary condition $g_{\kappa,1}=0$. This implies that
$g_{\kappa,\mu}=0$ for all $\mu\in[0,1]$.

######  Remark 2.3.

The effective force plays also a central role in the approach to singular
SPDEs proposed earlier by Kupiainen [6, 7] and applied to the dynamical
$\varPhi^{4}_{3}$ model and the KPZ equation. The method developed by
Kupiainen is based on the Wilsonian discrete renormalization group theory
[11]. In this approach one uses the fact that for $\mu\geq\eta$ the effective
force satisfies the following equation

$F_{\kappa,\mu}[\varphi]=F_{\kappa,\eta}[(G_{\kappa,\eta}-G_{\kappa,\mu})\ast
F_{\kappa,\mu}[\varphi]+\varphi].$ (2.6)

Given $F_{\kappa,\eta}[\varphi]$ the above equation is viewed as an equation
for $F_{\kappa,\mu}[\varphi]$ that can be solved using the Banach fixed point
theorem. One defines recursively the effective force $F_{\kappa,\mu}[\varphi]$
for $\kappa=2L^{-N}$ and $\mu\in\\{L^{-N},\ldots,1\\}$, where $L>1$, starting
with $F_{2L^{-N}}[\varphi]=F_{2L^{-N},L^{-N}}[\varphi]$ and finishing with
$F_{2L^{-N},1}[\varphi]$. In order to prove uniform bounds for
$F_{2L^{-N},1}[\varphi]$ one has to appropriately adjust the counterterms
which are the coefficients of the force $F_{2L^{-N}}[\varphi]$. The fine-
tuning problem becomes exceedingly difficult for equations close to
criticality. The flow equation provides a different, more efficient and
simpler method of constructing the effective force that allows to treat
equations arbitrarily close to criticality.

##  3 Construction of effective force coefficients

The starting point of the construction of the effective force is the formal
ansatz

$\langle
F_{\kappa,\mu}[\varphi],\psi\rangle:=\sum_{i=0}^{\infty}\sum_{m=0}^{\infty}\lambda^{i}\,\langle
F^{i,m}_{\kappa,\mu},\psi\otimes\varphi^{\otimes m}\rangle,$ (3.1)

where $\lambda$ is the prefactor of the cubic non-linearity in the original
equation. The distributions
$F^{i,m}_{\kappa,\mu}\in\mathscr{S}^{\prime}(\mathbb{M}^{1+m})$,
$i,m\in\mathbb{N}_{0}$, are called the effective force coefficients. By
definition the expression $\langle
F^{i,m}_{\kappa,\mu},\psi\otimes\varphi_{1}\otimes\ldots\otimes\varphi_{m}\rangle$
is invariant under permutations of the test functions
$\varphi_{1},\ldots,\varphi_{m}\in\ \mathscr{S}(\mathbb{M})$. The coefficients
$F^{i,m}_{\kappa}$ of the force $F_{\kappa}[\varphi]$ are defined by an
equality analogous to Eq. (3.1).

######  Remark 3.1.

Let us list the non-vanishing force coefficients $F^{i,m}_{\kappa}$:

$\displaystyle F^{0,0}_{\kappa}(x)=\xi(x),\qquad
F^{1,3}_{\kappa}(x;x_{1},x_{2},x_{3})=\delta_{\mathbb{M}}(x-x_{1})\delta_{\mathbb{M}}(x-x_{2})\delta_{\mathbb{M}}(x-x_{3}),$
(3.2) $\displaystyle
F^{i,1}_{\kappa}(x;x_{1})=c_{\kappa}^{[i]}\delta_{\mathbb{M}}(x-x_{1}),\quad
i\in\\{0,\ldots,i_{\sharp}\\}.$ (3.3)

The flow equation (2.3) for the effective force $F_{\kappa,\mu}[\varphi]$
formally implies that the effective force coefficients $F^{i,m}_{\kappa,\mu}$
satisfy the following flow equation

$\langle\partial_{\mu}^{\phantom{i}}F^{i,m}_{\kappa,\mu}\,,\,\psi\otimes\varphi^{\otimes
m}\rangle\\\
=-\sum_{j=0}^{i}\sum_{k=0}^{m}\,(1+k)\,\big{\langle}F^{j,1+k}_{\kappa,\mu}\otimes
F^{i-j,m-k}_{\kappa,\mu},\psi\otimes\varphi^{\otimes
k}\otimes\mathbf{V}\partial_{\mu}G_{\kappa,\mu}\otimes\varphi^{\otimes(m-k)}\big{\rangle}$
(3.4)

with the boundary condition $F^{i,m}_{\kappa,0}=F^{i,m}_{\kappa}$, where
$\mathbf{V}\partial_{\mu}G_{\kappa,\mu}\in
C^{\infty}(\mathbb{M}\times\mathbb{M})$ is defined by
$\mathbf{V}\partial_{\mu}G_{\kappa,\mu}(x,y):=\partial_{\mu}G_{\kappa,\mu}(x-y)$.

The basic idea behind the flow equation approach is a recursive construction
of the effective force coefficients $F^{i,m}_{\kappa,\mu}$:

  1. (0)

We set $F^{0,0}_{\kappa,\mu}=\xi$ and $F^{i,m}_{\kappa,\mu}=0$ if $m>3i$.

  2. (I)

Assuming that all $F^{i,m}_{\kappa,\mu}$ with $i<i_{\circ}$, or $i=i_{\circ}$
and $m>m_{\circ}$ were constructed we define
$\partial_{\mu}F^{i,m}_{\kappa,\mu}$ with $i=i_{\circ}$ and $m=m_{\circ}$ with
the use of Eq. (3.4).

  3. (II)

Subsequently, $F^{i,m}_{\kappa,\mu}$ is defined by
$F^{i,m}_{\kappa,\mu}=F^{i,m}_{\kappa}+\int_{0}^{\mu}\partial_{\eta}F^{i,m}_{\kappa,\eta}\,\mathrm{d}\eta$.

Using this procedure we construct all the effective force coefficients
$F^{i,m}_{\kappa,\mu}$ for arbitrary $\kappa\in(0,1/2]$, $\mu\in[0,1]$.

######  Remark 3.2.

One easily shows that the effective force coefficients $F^{i,m}_{\kappa,\mu}$
actually vanish if $i=0$ and $m>0$, or $i>0$ and $m>2(i-1)+3$. The only non-
zero coefficients $F^{i,m}_{\kappa,\mu}$ which are independent of the value of
the flow parameter $\mu$ are $F^{0,0}_{\kappa,\mu}(x)=\xi(x)$ and
$F^{1,3}_{\kappa,\mu}(x;x_{1},x_{2},x_{3})=F^{1,3}_{\kappa}(x;x_{1},x_{2},x_{3})$.
These coefficients happen to be independent of the UV cutoff $\kappa$. Let us
give some further examples:

$\begin{gathered}F^{1,2}_{\kappa,\mu}(x;x_{1},x_{2})=3\varPsi_{\kappa,\mu}(x)\,\delta_{\mathbb{M}}(x-x_{1})\delta_{\mathbb{M}}(x-x_{2}),\\\
F^{1,1}_{\kappa,\mu}(x;x_{1})=(3\varPsi_{\kappa,\mu}^{2}(x)+c_{\kappa}^{[1]})\,\delta_{\mathbb{M}}(x-x_{1}),\qquad
F^{1,0}_{\kappa,\mu}(x)=\varPsi_{\kappa,\mu}^{3}(x)+c_{\kappa}^{[1]}\,\varPsi_{\kappa,\mu}(x),\end{gathered}$
(3.5)

where $\varPsi_{\kappa,\mu}:=G_{\kappa\shortparallel\mu}\ast\xi$ and
$G_{\kappa\shortparallel\mu}:=G_{\kappa}-G_{\kappa,\mu}$ is the so-called
fluctuation propagator. The coefficient
$F^{2,5}_{\kappa,\mu}(x;x_{1},\ldots,x_{5})$ is obtained from the distribution
$\delta_{\mathbb{M}}(x-x_{1})\delta_{\mathbb{M}}(x-x_{2})\,G_{\kappa\shortparallel\mu}(x-x_{3})\,\delta_{\mathbb{M}}(x_{3}-x_{4})\delta_{\mathbb{M}}(x_{3}-x_{5})$
by symmetrization in variables $x_{1},\ldots,x_{5}$. We also have

$F^{2,1}_{\kappa,\mu}(x;x_{1})=(3\varPsi_{\kappa,\mu}^{2}(x)+c_{\kappa}^{[1]})\,G_{\kappa\shortparallel\mu}(x-x_{1})\,(3\varPsi_{\kappa,\mu}^{2}(x_{1})+c_{\kappa}^{[1]})\\\
+\left(\int_{\mathbb{M}}6\varPsi_{\kappa,\mu}(x)\,G_{\kappa\shortparallel\mu}(x-x_{2})\,(\varPsi_{\kappa,\mu}^{3}(x_{2})+c_{\kappa}^{[1]}\varPsi_{\kappa,\mu}(x_{2}))\,\mathrm{d}x_{2}+c_{\kappa}^{[2]}\right)\delta_{\mathbb{M}}(x-x_{1}),$
(3.6)
$F^{2,0}_{\kappa,\mu}(x)=(3\varPsi_{\kappa,\mu}^{2}(x)+c_{\kappa}^{[1]})\,G_{\kappa\shortparallel\mu}(x-x_{1})\,(\varPsi_{\kappa,\mu}^{3}(x_{1})+c_{\kappa}^{[1]}\varPsi_{\kappa,\mu}(x_{1}))+c_{\kappa}^{[2]}\,\varPsi_{\kappa,\mu}(x).$
(3.7)

For the sake of brevity, we did not give expressions for the coefficients
$F^{2,4}_{\kappa,\mu}$, $F^{2,3}_{\kappa,\mu}$, $F^{2,2}_{\kappa,\mu}$, which
should be constructed after $F^{2,5}_{\kappa,\mu}$ and before
$F^{2,1}_{\kappa,\mu}$.

######  Remark 3.3.

The effective force is an analog of the effective potential in QFT. A
recursive construction of the effective potential coefficients based on the
flow equation is the backbone of a very simple proof of perturbative
renormalizability of QFT models proposed by Polchinski [10] (see [8] for a
review).

The solution of Eq. (1.2) is formally given by the following sum

$\varPhi_{\kappa}=G_{\kappa}\ast
F_{\kappa,1}[0]:=G_{\kappa}\ast\sum_{i=0}^{\infty}\lambda^{i}F^{i,0}_{\kappa,1}.$
(3.8)

Assuming that $\lambda$ is sufficiently small in Sec. 11 we prove that the
above series converges absolutely and $\varPhi_{\kappa}$, as defined above,
solves Eq. (1.2) and converges almost surely as $\kappa\searrow 0$ in the
Besov space $\mathscr{C}^{\beta}(\mathbb{M})$ for every
$\beta<\sigma-\mathsf{d}/2$. To this end, we will establish certain bounds for
$\partial_{\kappa}^{r}\partial_{\mu}^{s}F_{\kappa,\mu}^{i,m}$ which are stated
in Sec. 6. The bounds involve a regularizing kernel $K_{\mu}$, which is
introduced in Sec. 4, and a norm
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{\mathcal{V}^{m}}$, which is introduced in Sec. 5.

##  4 Regularizing kernels

######  Definition 4.1.

For $n\in\mathbb{N}_{+}$ let
$\mathcal{K}^{n}\subset\mathscr{S}^{\prime}(\mathbb{M}^{n})$ be the space of
signed measures $K$ on $\mathbb{M}^{n}$ with finite total variation $|K|$. We
set
$\|K\|_{\mathcal{K}^{n}}=\int_{\mathbb{M}^{n}}|K(\mathrm{d}x_{1}\ldots\mathrm{d}x_{n})|$.
If $n=1$, then we write
$\mathcal{K}^{1}=\mathcal{K}\subset\mathscr{S}^{\prime}(\mathbb{M})$.

######  Remark 4.2.

It holds $\|K\|_{\mathcal{K}^{n}}=\|K\|_{L^{1}(\mathbb{M}^{n})}$ for any $K\in
L^{1}(\mathbb{M}^{n})\subset\mathcal{K}^{n}$. Note also that
$\delta_{\mathbb{M}^{n}}\in\mathcal{K}^{n}$, where $\delta_{\mathbb{M}^{n}}$
is the Dirac delta in $\mathbb{M}^{n}$.

######  Definition 4.3.

For $\mu\geq 0$ the kernel $K_{\mu}\in\mathcal{K}$ is the unique solution of
the equation $\mathbf{P}_{\mu}K_{\mu}=\delta_{\mathbb{M}}$, where
$\mathbf{P}_{\mu}:=1-[\mu]^{2}\Delta_{x}$ and $[\mu]:=\mu^{1/\sigma}$. We set
$K_{\mu}^{\ast 0}:=\delta_{\mathbb{M}}$, $\mathbf{P}^{0}_{\mu}:=1$ and
$K^{\ast({\mathtt{g}}+1)}_{\mu}:=K_{\mu}\ast K^{\ast{\mathtt{g}}}_{\mu}$,
$\mathbf{P}^{{\mathtt{g}}+1}_{\mu}:=\mathbf{P}_{\mu}\,\mathbf{P}^{{\mathtt{g}}}_{\mu}$
for ${\mathtt{g}}\in\mathbb{N}_{0}$. We omit the index $\mu$ if $\mu=1$.

######  Remark 4.4.

It holds $K_{0}=\delta_{\mathbb{M}}$ and $K_{\mu}\in L^{1}(\mathbb{M})$ for
$\mu>0$. For ${\mathtt{g}}\in\mathbb{N}_{0}$ and $\mu\geq 0$ the kernel
$K_{\mu}^{\ast{\mathtt{g}}}$ is a positive measure with total mass
$\|K_{\mu}^{\ast{\mathtt{g}}}\|_{\mathcal{K}}=1$. We have
$\mathbf{P}^{\mathtt{g}}_{\mu}K^{\ast{\mathtt{g}}}_{\mu}=\delta_{\mathbb{M}}$.
The fact that the regularizing kernel $K_{\mu}^{\ast{\mathtt{g}}}$ is an
inverse of a differential operator simplifies the analysis in Sec. 8.

######  Lemma 4.5.

For any $\mu>0$ and $\eta\geq 0$ there exists $K_{\mu,\eta}\in\mathcal{K}$
such that

$\begin{gathered}K_{\mu}=K_{\mu,\eta}\ast
K_{\eta},\qquad\|K_{\mu,\eta}\|_{\mathcal{K}}=1\vee(2[\eta/\mu]^{2}-1).\end{gathered}$
(4.1)

In particular, if $\mu\geq\eta$, then $\|K_{\mu,\eta}\|_{\mathcal{K}}=1$.

###### Proof.

Note that
$K_{\mu,\eta}=[\eta/\mu]^{2}\delta_{\mathbb{M}}+(1-[\eta/\mu]^{2})K_{\mu}$ and
$\|\delta_{\mathbb{M}}\|_{\mathcal{K}}=\|K_{\mu}\|_{\mathcal{K}}=1$. ∎

######  Definition 4.6.

Let $\mathbb{T}=\mathbb{M}/(2\pi\mathbb{Z})^{\mathsf{d}}$. For $V\in
L^{1}(\mathbb{M})$ we define $\mathbf{T}V\in L^{1}(\mathbb{T})$ by

$\mathbf{T}V(x):=\sum_{y\in(2\pi\mathbb{Z})^{\mathsf{d}}}V(x+y).$ (4.2)

######  Remark 4.7.

For $K\in L^{1}(\mathbb{M})$ and periodic $f\in C(\mathbb{T})$ it holds $K\ast
f=\mathbf{T}K\star f$, where $\ast$ and $\star$ are the convolutions in
$\mathbb{M}$ and $\mathbb{T}$, respectively.

######  Lemma 4.8.

Let ${\mathtt{g}}\in\mathbb{N}_{0}$, $a\in\mathbb{N}_{0}^{\mathsf{d}}$ and
$n\in[1,\infty]$. The following is true:

  1. (A)

If $|a|\leq{\mathtt{g}}$, then
$\|\partial^{a}K^{\ast{\mathtt{g}}}_{\mu}\|_{\mathcal{K}}\lesssim[\mu]^{-|a|}$
uniformly in $\mu>0$.

  2. (B)

$\|\mathbf{T}K^{\ast\mathsf{d}}_{\mu}\|_{L^{n}(\mathbb{T})}\lesssim[\mu]^{-\mathsf{d}(n-1)/n}$
uniformly in $\mu\in(0,1]$.

  3. (C)

It holds $\|\partial_{\mu}K_{\mu}\|_{\mathcal{K}}\lesssim[\mu]^{-\sigma}$
uniformly in $\mu>0$.

######  Remark 4.9.

Let $f\in C(\mathbb{M})$, ${\mathtt{g}}\in\mathbb{N}_{0}$ and
$a\in\mathbb{N}_{0}^{\mathsf{d}}$ be such that $|a|\leq{\mathtt{g}}$. Then
$K^{\ast{\mathtt{g}}}_{\mu}\ast f\in C^{\mathtt{g}}(\mathbb{M})$ and
$\|\partial^{a}K^{\ast{\mathtt{g}}}_{\mu}\ast
f\|_{L^{\infty}(\mathbb{M})}\lesssim[\mu]^{-|a|}\|f\|_{L^{\infty}(\mathbb{M})}$
uniformly over $\mu>0$ and $f\in C(\mathbb{M})$.

##  5 Function spaces for coefficients

######  Definition 5.1.

For $\alpha<0$ and $\phi\in C^{\infty}_{\mathrm{b}}(\mathbb{M})$ we define

$\|\phi\|_{\mathscr{C}^{\alpha}(\mathbb{M})}:=\sup_{\mu\in(0,1]}[\mu]^{-\alpha}\,\|K_{\mu}^{\ast{\mathtt{g}}}\ast\phi\|_{L^{\infty}(\mathbb{M})},\qquad{\mathtt{g}}=\lceil-\alpha\rceil\in\mathbb{N}_{+}.$
(5.1)

The space $\mathscr{C}^{\alpha}(\mathbb{M})$ is the completion of
$C^{\infty}_{\mathrm{b}}(\mathbb{M})$ equipped with the norm
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{\mathscr{C}^{\alpha}(\mathbb{M})}$.

######  Definition 5.2.

Let $m\in\mathbb{N}_{0}$. The vector space $\mathcal{V}^{m}$ consists of $V\in
C(\mathbb{M}^{1+m})$ such that

$\|V\|_{\mathcal{V}^{m}}:=\\\
\sup_{x\in\mathbb{M}}\int_{\mathbb{M}^{m}}|V(x;y_{1},\ldots,y_{m})|\,\mathrm{d}y_{1}\ldots\mathrm{d}y_{m}$
(5.2)

is finite and the function $x\mapsto V(x;y_{1}+x,\ldots,y_{n}+x)$ is $2\pi$
periodic for every $y_{1},\ldots,y_{m}\in\mathbb{M}$. For
${\mathtt{g}}\in\mathbb{N}_{0}$ the space $\mathcal{D}^{m}_{\mathtt{g}}$
consists of $V\in\mathscr{S}^{\prime}(\mathbb{M}^{1+m})$ such that it holds
$K^{\ast{\mathtt{g}},\otimes(1+m)}\ast V\in\mathcal{V}^{m}$. The space
$\mathcal{D}^{m}$ is the union of the spaces $\mathcal{D}^{m}_{\mathtt{g}}$,
${\mathtt{g}}\in\mathbb{N}_{0}$. We also set $\mathcal{V}^{0}=\mathcal{V}$,
$\mathcal{D}^{0}_{\mathtt{g}}=\mathcal{D}_{\mathtt{g}}$ and
$\mathcal{D}^{0}=\mathcal{D}$.

######  Remark 5.3.

Observe that $\mathcal{V}=\mathcal{D}_{0}=C(\mathbb{T})$. Moreover, we have
$\|v\|_{\mathcal{V}}=\|v\|_{L^{\infty}(\mathbb{M})}$ for any
$v\in\mathcal{V}$. Since $K$ is the inverse of a differential operator
$\mathcal{D}=\mathscr{S}^{\prime}(\mathbb{T})$.

######  Definition 5.4.

The permutation group of the set $\\{1,\ldots,n\\}$ is denoted by
$\mathcal{P}_{n}$. For $m\in\mathbb{N}_{0}$ and $V\in\mathcal{D}^{m}$ and
$\pi\in\mathcal{P}_{m}$ we define $\mathbf{Y}_{\pi}V\in\mathcal{D}^{m}$ by

$\big{\langle}\mathbf{Y}_{\pi}V,\psi\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{m}\varphi_{q}\big{\rangle}:=\big{\langle}V,\psi\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{m}\varphi_{\pi(q)}\rangle,$
(5.3)

where $\psi,\varphi_{1},\ldots,\varphi_{m}\in\ \mathscr{S}(\mathbb{M})$.

######  Remark 5.5.

The map $\mathbf{Y}_{\pi}:\mathcal{V}^{m}\to\mathcal{V}^{m}$ is well defined
and has norm one.

######  Definition 5.6.

Let $m\in\mathbb{N}_{0}$, $k\in\\{0,\ldots,m\\}$. We define the trilinear map
$\mathbf{B}\,:\,\mathscr{S}(\mathbb{M})\times\mathcal{V}^{1+k}\times\mathcal{V}^{m-k}\to\mathcal{V}^{m}$
by

$\mathbf{B}(G,W,U)(x;y_{1},\ldots,y_{m})\\\
:=\int_{\mathbb{M}^{2}}W(x;y_{0},\ldots,y_{k})\,G(y_{0}-z)\,U(z;y_{1+k},\ldots,y_{m})\,\mathrm{d}y_{0}\mathrm{d}z.$
(5.4)

######  Remark 5.7.

The RHS of Eq. (3.4) can be written compactly using the map $\mathbf{B}$.

######  Lemma 5.8.

It holds

$\|\mathbf{B}(G,W,U)\|_{\mathcal{V}^{m}}\leq\|G\|_{\mathcal{K}}\,\|W\|_{\mathcal{V}^{1+k}}\|U\|_{\mathcal{V}^{m-k}}.$
(5.5)

###### Proof.

We have

$\|\mathbf{B}(G,W,U)\|_{\mathcal{V}^{m}}=\sup_{x\in\mathbb{M}}\int_{\mathbb{M}^{m}}|\mathbf{B}(G,W,U)(x;y_{1},\ldots,y_{m})|\,\mathrm{d}y_{1}\ldots\mathrm{d}y_{m}\\\
\leq\sup_{x\in\mathbb{M}}\int_{\mathbb{M}^{2+m}}|W(x;y_{0},\ldots,y_{k})|\,|G(y_{0}-z)|\,|U(z;y_{1+k},\ldots,y_{m})|\,\mathrm{d}z\mathrm{d}y_{0}\ldots\mathrm{d}y_{m}.$
(5.6)

It is easy to see that the last line is bounded by the RHS of (5.5). ∎

######  Remark 5.9.

The fact that
$P_{\mu}^{\mathtt{g}}K_{\mu}^{\ast{\mathtt{g}}}=\delta_{\mathbb{M}}$ implies
that for any $\mu>0$ it holds

$\begin{gathered}K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\mathbf{B}(G,W,U)=\mathbf{B}\big{(}\mathbf{P}^{2{\mathtt{g}}}_{\mu}G,K_{\mu}^{\ast{\mathtt{g}},\otimes(2+k)}\ast
W,K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m-k)}\ast U\big{)}.\end{gathered}$ (5.7)

This allows to define $\mathbf{B}(G,W,U)\in\mathcal{D}^{m}$ for all
$W\in\mathcal{D}^{1+k}$, $U\in\mathcal{D}^{m-k}$.

##  6 Deterministic bounds for irrelevant coefficients

######  Definition 6.1.

Recall that $\sigma\in(\mathsf{d}/3,\mathsf{d}/2]$, $[\mu]=\mu^{1/\sigma}$ and
set

$\dim(\xi):=\mathsf{d}/2>0,\qquad\dim(\varPhi):=\mathsf{d}/2-\sigma\geq
0,\qquad\dim(\lambda):=3\sigma-\mathsf{d}>0.$ (6.1)

######  Definition 6.2.

For $\varepsilon\geq 0$ and $i,m\in\mathbb{N}_{0}$ we define

$\varrho_{\varepsilon}(i,m):=-\dim(\Xi)-\varepsilon+m\,(\dim(\Phi)+2\varepsilon)+i\,(\dim(\lambda)-6\varepsilon)\in\mathbb{R}.$
(6.2)

We omit $\varepsilon$ if $\varepsilon=0$. The effective force coefficients
$F^{i,m}_{\kappa,\mu}$ such that $\varrho(i,m)\leq 0$ are called relevant. The
remaining coefficients are called irrelevant.

######  Remark 6.3.

The number of relevant coefficients $F^{i,m}_{\kappa,\mu}$ such that $m\leq
3i$ is always finite (recall that $F^{i,m}_{\kappa,\mu}=0$ if $m>3i$). For
example, in the case $\mathsf{d}=5$ and $\sigma=2$ the relevant coefficients
are $F^{0,0}_{\kappa,\mu}$, $F^{1,0}_{\kappa,\mu}$, $F^{1,1}_{\kappa,\mu}$,
$F^{1,2}_{\kappa,\mu}$, $F^{1,3}_{\kappa,\mu}$, $F^{2,0}_{\kappa,\mu}$,
$F^{2,1}_{\kappa,\mu}$. For explicit expressions for these coefficients see
Remark 3.2.

######  Remark 6.4.

For arbitrary $\varepsilon>0$ and $i,m\in\mathbb{N}_{0}$ such that $m\leq 3i$
it holds $\varrho_{\varepsilon}(i,m)<\varrho(i,m)$. Let $i_{\diamond}$ be the
smallest integer such that $\varrho(i_{\diamond}+1,0)>0$. It holds
$i_{\diamond}\in\mathbb{N}_{+}$. Moreover, let $\varrho_{\diamond}>0$ be the
minimum of $\varrho(i,m)+\mathtt{l}$ for $i\in\\{0,\ldots,i_{\diamond}\\}$,
$m\in\\{0,\ldots,3i\\}$, $\mathtt{l}\in\mathbb{N}_{+}$ such that
$\varrho(i,m)+\mathtt{l}>0$. Define
$\varepsilon_{\diamond}:=\dim(\xi)/3\wedge\dim(\lambda)/9\wedge\varrho_{\diamond}/(7+6i_{\diamond})\wedge\sigma$.
We claim that for all $\varepsilon\in(0,\varepsilon_{\diamond})$ and all
$i,m,\mathtt{l}\in\mathbb{N}_{0}$ it holds
$\varrho_{\varepsilon}(i,m)+\mathtt{l}>0$ if $\varrho(i,m)+\mathtt{l}>0$. In
what follows, we fix some $\varepsilon\in(0,\varepsilon_{\diamond}/3)$.

######  Lemma 6.5.

For all ${\mathtt{g}}\in\mathbb{N}_{0}$, $r\in\mathbb{N}_{0}$ it holds
$\partial_{\kappa}^{r}\partial_{\mu}G_{\kappa,\mu}\in
C^{\infty}_{\mathrm{c}}(\mathbb{M})$ and

$\|\mathbf{P}^{{\mathtt{g}}}_{\mu}\partial_{\kappa}^{r}\partial_{\mu}G_{\kappa,\mu}\|_{\mathcal{K}}\lesssim[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{-\varepsilon
r}$ (6.3)

uniformly in $\kappa\in(0,1/2]$, $\mu\in(0,1]$, where $G_{\kappa,\mu}$ was
introduced in Def. 2.1.

###### Proof.

Let $a\in\mathbb{N}_{0}^{\mathsf{d}}$. It holds
$|\partial^{a}G(x)|\lesssim|x|^{\sigma-\mathsf{d}-|a|}$ uniformly for $|x|\leq
2$. If $\sigma\in 2\mathbb{N}_{+}$, then $\partial^{a}G$ decays fast at
infinity. In general, $|\partial^{a}G(x)|\lesssim|x|^{-\sigma-\mathsf{d}-|a|}$
uniformly for $|x|>1$. Moreover, we have

$|\partial^{a}\partial_{\kappa}^{r}\chi_{\kappa}(|x|^{\sigma})|\lesssim[\kappa]^{(\varepsilon-\sigma)r}|x|^{(\sigma-\varepsilon)r-|a|},\qquad|\partial^{a}\partial_{\mu}\chi_{\mu}(|x|^{\sigma})|\lesssim|x|^{\sigma-|a|}/\mu^{2}$
(6.4)

uniformly in $\kappa\in(0,1/2]$, $\mu\in(0,1]$ and $x\in\mathbb{M}$.
Furthermore, $|\partial^{a}\partial_{\mu}\chi_{\mu}(|x|^{\sigma})|$ vanishes
unless $\mu<(1-\mu)|x|^{\sigma}\leq 2\mu$. Using the above properties and
considering separately $\mu\in(0,1/2]$ and $\mu\in(1/2,1]$ we obtain
$\|\partial^{a}\partial_{\kappa}^{r}\partial_{\mu}G_{\kappa,\mu}\|_{\mathcal{K}}\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{-\varepsilon
r-|a|}$. This implies the lemma since
$\mathbf{P}_{\mu}^{\mathtt{g}}=(1-[\mu]^{2}\Delta_{x})^{\mathtt{g}}$. ∎

######  Remark 6.6.

Given ${\mathtt{g}}\in\mathbb{N}_{0}$ there exists $\tilde{R}>0$ such that

$\frac{(1+m)\,\sigma}{\varrho_{\varepsilon}(i,m)}\leq\tilde{R}^{1/2},\qquad\|\mathbf{P}^{2{\mathtt{g}}}_{\mu}\partial_{\kappa}^{r}\partial_{\mu}G_{\kappa,\mu}\|_{\mathcal{K}}\leq
1/6\,\tilde{R}^{1/2}\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{-\varepsilon}$
(6.5)

for all $i,m\in\mathbb{N}_{0}$ such that $\varrho(i,m)>0$ and all
$r\in\\{0,1\\}$, $\kappa\in(0,1/2]$, $\mu\in(0,1]$. The first of the above
bounds implies, in particular, that
$1/(\dim(\varPhi)+2\varepsilon)\leq\tilde{R}^{1/2}$.

######  Theorem 6.7.

Fix ${\mathtt{g}}\in\mathbb{N}_{0}$. Let $\tilde{R}>1$ as in Remark 6.6.
Assume that for $r\in\\{0,1\\}$, $s=0$, all $i,m\in\mathbb{N}_{0}$ such that
$\varrho(i,m)\leq 0$ and all $\kappa\in(0,1/2]$, $\mu\in(0,1]$ the following
bound holds

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\mu}^{s}F_{\kappa,\mu}^{i,m}\|_{\mathcal{V}^{m}}\leq\frac{\tilde{R}^{1-s/2+2(3i-m)}}{4(1+i)^{2}\,4(1+m)^{2-s}}[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{\varrho_{\varepsilon}(i,m)-\sigma
s}.$ (6.6)

Then the above bound holds for all $r,s\in\\{0,1\\}$, $i,m\in\mathbb{N}_{0}$,
$\kappa\in(0,1/2]$, $\mu\in(0,1]$.

######  Remark 6.8.

By Theorems 16.1, 13.1 9.3 10.1 (applied in this order), there exists a choice
of counterterms such that the assumption of the above theorem is satisfied for
some random $\tilde{R}>0$ such that $\mathbb{E}\tilde{R}^{n}<\infty$ for any
$n\in\mathbb{N}_{+}$.

######  Remark 6.9.

If the bound (6.6) holds for ${\mathtt{g}}=\mathtt{f}$, then it holds for all
${\mathtt{g}}>\mathtt{f}$.

######  Remark 6.10.

Let us comment on the assumption in two simple cases. For $i=0$, $m=0$ the
bound (6.6) says that
$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\xi\|_{\mathcal{V}}\leq\tilde{R}/4^{2}\,[\mu]^{-\dim(\xi)-\varepsilon}$,
which is known to be true for ${\mathtt{g}}=\mathsf{d}$ and some random $R>0$
such that $\mathbb{E}\tilde{R}^{n}<\infty$ for all $n\in\mathbb{N}_{+}$. For
$i=1$, $m=3$ the bound (6.6) says that $\|K_{\mu}^{\ast{\mathtt{g}},\otimes
4}\ast F^{1,3}_{\kappa,\mu}\|_{\mathcal{V}^{3}}\leq\tilde{R}/4^{5}$. It is
easy to see that the above bound is satisfied for $\tilde{R}=4^{5}$ using the
fact that
$F^{1,3}_{\kappa,\mu}(x;x_{1},x_{2},x_{3})=\delta_{\mathbb{M}}(x-x_{1})\delta_{\mathbb{M}}(x-x_{2})\delta_{\mathbb{M}}(x-x_{3})$.

######  Remark 6.11.

For $\lambda\leq\tilde{R}^{-6}$ the bounds (6.6) imply convergence of the
series (3.8) defining $\varPhi_{\kappa}$, which is our candidate for the
solution of Eq. (1.2). In fact, one easily proves that
$\|K^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}\varPhi_{\kappa}\|_{\mathcal{V}}\leq
R\,[\kappa]^{(\varepsilon-\sigma)r}$ for $r\in\\{0,1\\}$ and all
$\kappa\in(0,1]$. This implies the existence of the limit
$\lim_{\kappa\searrow 0}\varPhi_{\kappa}$ in
$\mathscr{S}^{\prime}(\mathbb{M})$. In Sec. 11 we will prove that
$\varPhi_{\kappa}$ solves Eq. (3.8) and $\lim_{\kappa\searrow
0}\varPhi_{\kappa}$ exists in
$\mathscr{C}^{-\dim(\varPhi)-2\varepsilon}(\mathbb{M})$.

###### Proof.

Fix some $i_{\circ}\in\mathbb{N}_{+}$, $m_{\circ}\in\mathbb{N}_{0}$ and assume
that the statement with $s=0$ is true for all $i,m\in\mathbb{N}_{+}$ such that
either $i<i_{\circ}$ or $i=i_{\circ}$ and $m>m_{\circ}$. We shall prove the
statement for $i=i_{\circ}$, $m=m_{\circ}$. We first consider the case $s=1$.
Using the flow equation (3.4) and the notation introduced in Sec. 5 we obtain

$\partial_{\mu}\partial_{\kappa}^{r}F^{i,m}_{\kappa,\mu}=-\frac{1}{m!}\sum_{\pi\in\mathcal{P}_{m}}\sum_{j=0}^{i}\sum_{k=0}^{m}\sum_{u+v+w=r}(1+k)\,\\\
\times\mathbf{Y}_{\pi}\mathbf{B}\big{(}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu},\partial_{\kappa}^{v}F^{j,1+k}_{\kappa,\mu},\partial_{\kappa}^{w}F^{i-j,m-k}_{\kappa,\mu}\big{)},$
(6.7)

where $r,u,v,w\in\\{0,1\\}$. By Remark 5.5, Lemma 5.8 and Remark 5.9 we get

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\mu}\partial_{\kappa}^{r}F^{i,m}_{\kappa,\mu}\|_{\mathcal{V}^{m}}\leq(1+m)~{}\sum_{u+v+w=r}\|\mathbf{P}_{\mu}^{\mathtt{g}}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu}\|_{\mathcal{K}}\\\
\times\sum_{j=0}^{i}\sum_{k=0}^{m}\|K_{\mu}^{\ast{\mathtt{g}},\otimes(2+k)}\ast\partial_{\kappa}^{v}F^{j,1+k}_{\kappa,\mu}\|_{\mathcal{V}^{1+k}}~{}\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m-k)}\partial_{\kappa}^{w}F^{i-j,m-k}_{\kappa,\mu}\|_{\mathcal{V}^{m-k}}.$
(6.8)

The statement of the theorem with $s=1$ follows now from the induction
hypothesis, Remark 6.6, the equality

$\varrho_{\varepsilon}(i,m)-\sigma=\varrho_{\varepsilon}(j,1+k)+\varrho_{\varepsilon}(i-j,m-k)-\varepsilon$
(6.9)

and the bound

$\sum_{j=0}^{i}\sum_{k=0}^{m}\frac{\tilde{R}^{1+2(3j-k-1)}}{4(1+j)^{2}\,4(2+k)^{2}}\frac{\tilde{R}^{1+2(3i-3j-m+k)}}{4(1+i-j)^{2}\,4(1+m-k)^{2}}\leq\frac{\tilde{R}^{2(3i-m)}}{4(1+i)^{2}\,4(1+m)^{2}},$
(6.10)

which is a consequence of the following inequality

$\sum_{j=0}^{i}\frac{1}{4(1+j)^{2}\,4(1+i-j)^{2}}\leq\frac{1}{4(1+i)^{2}}.$
(6.11)

In order to prove the statement of the theorem for $s=0$ and
$i,m\in\mathbb{N}_{0}$ such that $\varrho(i,m)>0$ we use the identity

$F^{i,m}_{\kappa,\mu}=F^{i,m}_{\kappa}+\int_{0}^{\mu}\partial_{\eta}F^{i,m}_{\kappa,\eta}\,\mathrm{d}\eta.$
(6.12)

We first observe that $F^{i,m}_{\kappa,0}=F^{i,m}_{\kappa}=0$ if
$\varrho(i,m)>0$. Next, we note that

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m}\|_{\mathcal{V}^{m}}\leq\|K_{\eta}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m}\|_{\mathcal{V}^{m}}$
(6.13)

for $\eta\leq\mu$ by Lemma 4.5. The statement of the theorem with $s=0$
follows now from the statement with $s=1$ and the bounds

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}F_{\kappa,\mu}^{i,m}\|_{\mathcal{V}^{m}}\\\
\leq\int_{0}^{\mu}\|K_{\eta}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m}\|_{\mathcal{V}^{m}}\,\mathrm{d}\eta$
(6.14)

and

$\int_{0}^{\mu}[\eta]^{\varrho_{\varepsilon}(i,m)-\sigma}\,\mathrm{d}\eta\leq\sigma/\varrho_{\varepsilon}(i,m)~{}[\mu]^{\varrho_{\varepsilon}(i,m)}\leq\tilde{R}^{1/2}/(1+m)\,[\mu]^{\varrho_{\varepsilon}(i,m)}$
(6.15)

We stress that the first inequality in the last bound is valid only if
$\varrho_{\varepsilon}(i,m)>0$, which holds provided $\varrho(i,m)>0$ by
Remark 6.4. ∎

##  7 Generalized effective force coefficients

In order to prove the bound (6.6) for all relevant effective force
coefficients we will use generalized coefficients introduced in this section.

######  Definition 7.1.

We denote the set of multi-indices by
$\mathfrak{M}=\mathbb{N}_{0}^{\mathsf{d}}$. For for $m\in\mathbb{N}_{+}$ and
$a=(a_{1},\ldots,a_{m})\in\mathfrak{M}^{m}$ we define
$|a|:=|a_{1}|+\ldots+|a_{m}|$. We also set $\mathfrak{M}^{0}\equiv\\{0\\}$.
For $a\in\mathfrak{M}$ we define $\mathcal{X}^{a}\in C^{\infty}(\mathbb{M})$
by $\mathcal{X}^{a}(x):=\frac{1}{a!}x^{a}$ and for $a\in\mathfrak{M}^{m}$ we
define $\mathcal{X}^{m,a}\in C^{\infty}(\mathbb{M}^{1+m})$ by

$\mathcal{X}^{m,a}(x;y_{1},\ldots,y_{m}):=\frac{1}{a!}\,(x-y_{1})^{a_{1}}\ldots(x-y_{m})^{a_{m}}.$
(7.1)

For $i,m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$ we define
$F^{i,m,a}_{\kappa,\mu}\in\mathscr{S}^{\prime}(\mathbb{M}^{1+m})$ and
$f^{i,m,a}_{\kappa,\mu}\in\mathscr{S}^{\prime}(\mathbb{M})$ by

$F^{i,m,a}_{\kappa,\mu}:=\mathcal{X}^{m,a}F^{i,m}_{\kappa,\mu},\qquad\langle
f^{i,m,a}_{\kappa,\mu},\psi\rangle:=\langle F^{i,m,a}_{\kappa,\mu},\psi\otimes
1_{\mathbb{M}}^{\otimes m}\rangle,$ (7.2)

where $\psi\in\mathscr{S}(\mathbb{M})$ and $1_{\mathbb{M}}(x)=1$ for all
$x\in\mathbb{M}$. The force coefficients $F^{i,m,a}_{\kappa}$ and
$f^{i,m,a}_{\kappa}$ are defined analogously. The effective force coefficients
$F^{i,m,a}_{\kappa,\mu}$ or $f^{i,m,a}_{\kappa,\mu}$ such that
$\varrho(i,m)+|a|\leq 0$ are called relevant. The remaining coefficients are
called irrelevant. The finite collection of all relevant coefficients
$f^{i,m,a}_{\kappa,\mu}$ such that $m\leq 3i$ is called the enhanced noise
(recall that $f^{i,m,a}_{\kappa,\mu}=0$ if $m>3i$).

######  Remark 7.2.

Let us list the non-zero force coefficients
$f^{i,m,a}_{\kappa\phantom{0}}=f^{i,m,a}_{\kappa,0}\in\mathscr{S}^{\prime}(\mathbb{M})$:

$f^{0,0,0}_{\kappa}=\xi(x),\quad f^{1,3,0}_{\kappa}=1,\quad
f^{i,1,0}_{\kappa}=c^{[i]}_{\kappa},\quad i\in\\{1,\ldots,i_{\sharp}\\}.$
(7.3)

######  Remark 7.3.

For example, in the case $\mathsf{d}=5$ and $\sigma=2$ the enhanced noise
consists of $f^{0,0,0}_{\kappa,\mu}=\xi$, $f^{1,0,0}_{\kappa,\mu}$,
$f^{1,1,0}_{\kappa,\mu}$, $f^{1,2,0}_{\kappa,\mu}$,
$f^{1,3,0}_{\kappa,\mu}=1$, $f^{2,0,0}_{\kappa,\mu}$,
$f^{2,1,0}_{\kappa,\mu}$. Explicit expressions for all these coefficients can
be easily obtained using Remark 3.2.

######  Remark 7.4.

Given $i,m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$ such that
$\varrho(i,m)+|a|\in(-\mathtt{l},1-\mathtt{l}]$ for some
$\mathtt{l}\in\mathbb{N}_{+}$ the relevant coefficient
$F^{i,m,a}_{\kappa,\mu}$ can be expressed in terms of irrelevant coefficients
$F^{i,m,b}_{\kappa,\mu}$, $|b|=\mathtt{l}$, and relevant coefficients
$f^{i,m,b}_{\kappa,\mu}$, $|b|<\mathtt{l}$. The above fact plays a crucial
role in the proof of bounds for relevant coefficients given in Sec. 9. In Sec.
8 we show that it is a consequence of the Taylor theorem.

######  Remark 7.5.

For a list of multi-indices $a=(a_{1},\ldots,a_{m})\in\mathfrak{M}^{m}$ and a
permutation $\pi\in\mathcal{P}_{n}$ we set
$\pi(a):=(a_{\pi(1)},\ldots,a_{\pi(m)})$. Note that it holds

$\mathcal{X}^{m,a}(x;y_{1},\ldots,y_{m})\\\
=\mathcal{X}^{m,\pi(a)}(x;y_{\pi(1)},\ldots,y_{\pi(m)})$ (7.4)

We claim that

$\mathcal{X}^{m,a}(x;y_{1},\ldots,y_{m})=\sum_{b,c,d}\mathcal{X}^{1+k,(b_{1+k}+\ldots+b_{m},b_{1},\ldots,b_{k})}(x;y_{0},y_{1},\ldots,y_{k})\\\
\mathcal{X}^{1,c_{1+k}+\ldots+c_{m}}(y_{0};z)\,\mathcal{X}^{m-k,(d_{1+k},\ldots,d_{m})}(z;y_{1+k},\ldots,y_{m}),$
(7.5)

where the sum is over all $b_{p},c_{p},d_{p}$ with $p\in\\{1,\ldots,m\\}$ such
that $b_{p}=a_{p}$, $c_{p}=0$, $d_{p}=0$ for $p\in\\{1,\ldots,k\\}$ and
$b_{p}+c_{p}+d_{p}=a_{p}$ for $p\in\\{1+k,\ldots,m\\}$. Throughout the paper
we use the following schematic notation for the sums of the above type

$\mathcal{X}^{m,a}(x;y_{1},\ldots,y_{m})\\\
=\sum_{b+c+d=a}\mathcal{X}^{1+k,b}(x;y_{0},y_{1},\ldots,y_{k})\mathcal{X}^{c}(y_{0}-z)\mathcal{X}^{m-k,d}(z;y_{1+k},\ldots,y_{m}).$
(7.6)

Let us note that the formula (7.5) above was obtained by writing
$(y_{p}-x)^{a_{p}}$, $p\in\\{1+k,\ldots,m\\}$, as

$(y_{p}-x)^{a_{p}}=\sum_{\begin{subarray}{c}b_{p},c_{p},d_{p}\\\
b_{p}+c_{p}+d_{p}=a_{p}\end{subarray}}\frac{a_{p}!}{b_{p}!c_{p}!d_{p}!}~{}(x-y_{0})^{b_{p}}(y_{0}-z)^{c_{p}}(z-y_{p})^{d_{p}}.$
(7.7)

Using the flow equation (3.4) and the identity (7.5) we show that the
effective force coefficients $F^{i,m,a}_{\kappa,\mu}$ satisfy the following
flow equation

$\partial_{\mu}^{\phantom{i}}\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\mu}=-\frac{1}{m!}\sum_{\pi\in\mathcal{P}_{m}}\sum_{j=0}^{i}\sum_{k=0}^{m}\sum_{u+v+w=r}\sum_{b+c+d=\pi(a)}(1+k)\,\\\
\times\mathbf{Y}_{\pi}\mathbf{B}\big{(}\mathcal{X}^{c}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu},\partial_{\kappa}^{v}F^{j,1+k,b}_{\kappa,\mu},\partial_{\kappa}^{w}F^{i-j,m-k,d}_{\kappa,\mu}\big{)},$
(7.8)

where $r\in\mathbb{N}_{0}$, $u,v,w\in\\{0,\ldots,r\\}$ and the sum over multi-
indices $b,c,d$ is restricted to the set specified below Eq. (7.5).

##  8 Taylor polynomial and remainder

######  Definition 8.1.

For $a\in\mathfrak{M}^{m}$, $V\in C^{\infty}(\mathbb{M}^{1+m})$ we define
$\partial^{a}V\in C^{\infty}(\mathbb{M}^{1+m})$ by
$\partial^{a}V(x;x_{1},\ldots,x_{m}):=\partial^{a_{1}}_{x_{1}}\ldots\partial^{a_{m}}_{x_{m}}V(x;x_{1},\ldots,x_{m})$.
The above map extends in an obvious way to
$\mathscr{S}^{\prime}(\mathbb{M}^{1+m})\supset\mathcal{D}^{m}$.

######  Definition 8.2.

For $v\in\mathcal{D}$ we define $\mathbf{L}^{m}v\in\mathcal{D}^{m}$ by the
equality

$\langle\,\mathbf{L}^{m}v\,,\,\psi\otimes\varphi_{1}\otimes\ldots\otimes\varphi_{m}\,\rangle=\langle
v\,,\,\psi\varphi_{1}\ldots\varphi_{m}\rangle,$ (8.1)

where $\psi,\varphi_{1},\ldots,\varphi_{m}\in\mathscr{S}(\mathbb{M})$ are
arbitrary.

######  Definition 8.3.

For $\tau>0$ and $V\in C(\mathbb{M}^{1+m})$ we define $\mathbf{Z}_{\tau}V\in
C(\mathbb{M}^{1+m})$ by the equality

$\mathbf{Z}_{\tau}V(x;y_{1},\ldots,y_{m}):=\tau^{-\mathsf{d}m}\,V(x;x+(y_{1}-x)/\tau,\ldots,x+(y_{m}-x)/\tau).$
(8.2)

The above map extends in an obvious way to
$\mathscr{S}^{\prime}(\mathbb{M}^{1+m})\supset\mathcal{D}^{m}$.

######  Definition 8.4.

The linear map $\mathbf{I}\,:\,\mathcal{V}^{m}\to\mathcal{V}$ is defined by

$\mathbf{I}V(x):=\int_{\mathbb{M}^{m}}V(x;y_{1},\ldots,y_{m})\,\mathrm{d}y_{1}\ldots\mathrm{d}y_{m}.$
(8.3)

The map $\mathbf{I}$ is extended to $V\in\mathcal{D}^{m}_{\mathtt{g}}$ by the
formula

$\langle\mathbf{I}V,\psi\rangle:=\langle K^{\ast{\mathtt{g}},\otimes(1+m)}\ast
V,\mathbf{P}^{\mathtt{g}}\psi\otimes 1^{\otimes m}_{\mathbb{M}}\rangle,$ (8.4)

where $\psi\in\mathscr{S}(\mathbb{M})$.

######  Lemma 8.5.

The map $\mathbf{I}$ is well defined and has the following properties.

  1. (A)

$\mathbf{I}\big{(}K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast
V\big{)}=K^{\ast{\mathtt{g}}}_{\mu}\ast\mathbf{I}V$ for any
$V\in\mathcal{V}^{m}$, ${\mathtt{g}}\in\mathbb{N}_{0}$ and $\mu>0$.

  2. (B)

$\|\mathbf{I}V\|_{\mathcal{V}}\leq\|V\|_{\mathcal{V}^{m}}$ for
$V\in\mathcal{V}^{m}$.

###### Proof.

The well definedness and Part (A) follow from
$\int_{\mathbb{M}}K_{\mu}(x)\,\mathrm{d}x=1$. Part (B) is a consequence of
Def. 5.2 of the norm $\mathcal{V}^{m}$. ∎

######  Definition 8.6.

For $\mathtt{l}\in\mathbb{N}_{+}$, $a\in\mathfrak{M}^{m}$ such that
$|a|<\mathtt{l}$ and two collections of distributions: $v^{b}\in\mathcal{D}$,
$b\in\mathfrak{M}^{m}$, $|b|<\mathtt{l}$, and $V^{b}\in\mathcal{D}^{m}$,
$b\in\mathfrak{M}^{m}$, $|b|=\mathtt{l}$, the distribution
$\mathbf{X}_{\mathtt{l}}^{a}(v^{b},V^{b})\in\mathcal{D}^{m}$ is defined by the
equality

$\mathbf{X}_{\mathtt{l}}^{a}(v^{b},V^{b}):=\sum_{|a+b|<\mathtt{l}}\,(-1)^{|b|}\,\tbinom{a+b}{a}\partial^{b}\mathbf{L}^{m}(v^{a+b})\\\
+\sum_{|a+b|=\mathtt{l}}|b|\,(-1)^{|b|}\,\tbinom{a+b}{a}\int_{0}^{1}(1-\tau)^{|b|-1}\,\partial^{b}\mathbf{Z}_{\tau}(V^{a+b})\,\mathrm{d}\tau,$
(8.5)

where the sums above are over $b\in\mathfrak{M}^{m}$.

######  Theorem 8.7.

Let $\mathtt{l}\in\mathbb{N}_{+}$ and ${\mathtt{g}}\in\mathbb{N}_{0}$. There
exists a constant $c>0$ such that the following statement is true. Let
$V^{b}\in\mathcal{V}^{m}$, $v^{b}\in\mathcal{V}$ be as in Def. 8.6 and
$\mu\in(0,1]$. Assume that there exists a constant $C>0$ such that for
$b\in\mathfrak{M}$

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast V^{b}\|_{\mathcal{V}^{m}}\leq
C\,[\mu]^{[b]},\quad|b|=\mathtt{l},\quad\quad\|K_{\mu}^{\ast{\mathtt{g}}}\ast
v^{b}\|_{\mathcal{V}}\leq C\,[\mu]^{[b]}.\quad|b|<\mathtt{l},$ (8.6)

Then for $a\in\mathfrak{M}$ such that $|a|<\mathtt{l}$ it holds

$\|K_{\mu}^{\ast(6{\mathtt{g}}+\mathtt{l}),\otimes(1+m)}\ast\mathbf{X}_{\mathtt{l}}^{a}(v^{b},V^{b})\|_{\mathcal{V}^{m}}\leq
c\,C\,[\mu]^{|a|}.$ (8.7)

###### Proof.

The theorem follows from Def. 8.6 of the map $\mathbf{X}^{a}_{\mathtt{l}}$,
Lemma 8.9 and the bound
$\|\partial^{c}K^{\ast\mathtt{l}}_{\mu}\|_{\mathcal{K}}\lesssim|\mu|^{-|c|}$,
$c\in\mathfrak{M}$, $|c|\leq\mathtt{l}$, proved in Lemma 4.8 (A). ∎

######  Theorem 8.8.

Let $\mathtt{l}\in\mathbb{N}_{+}$ and $V\in\mathcal{D}^{m}$ such that
$\mathcal{X}^{m,b}V\in\mathcal{D}^{m}$ for any $b\in\mathfrak{M}^{m}$,
$|b|\leq\mathtt{l}$. Then
$\mathcal{X}^{m,a}V=\mathbf{X}_{\mathtt{l}}^{a}(\mathbf{I}(\mathcal{X}^{m,b}V),\mathcal{X}^{m,b}V)$
for all $a\in\mathfrak{M}^{m}$, $|a|<\mathtt{l}$.

###### Proof.

First recall that for any $\varphi\in C^{\infty}(\mathbb{R}^{N})$,
$N\in\mathbb{N}_{+}$, it holds

$\varphi(y)=\sum_{|b|<\mathtt{l}}\frac{1}{b!}(y-x)^{b}\,(\partial^{b}\varphi)(x)\\\
+\sum_{|b|=\mathtt{l}}\frac{1}{b!}\,(y-x)^{b}\int_{0}^{1}(1-\tau)^{\mathtt{l}-1}\,(\partial^{b}\varphi)(x+\tau(y-x))\,\mathrm{d}\tau$
(8.8)

by the Taylor theorem, where $b\in\mathbb{N}_{0}^{N}$. Consequently, for any
$\varphi\in\mathscr{S}(\mathbb{M}^{1+m})$ we have

$(\mathcal{X}^{m,a}\varphi)(x;y_{1},\ldots,y_{m})=\sum_{|a+b|<\mathtt{l}}\tbinom{a+b}{b}\,\mathcal{X}^{a+b}(x;y_{1},\ldots,y_{m})\,(\partial^{b}\varphi)(x;x,\ldots,x)\\\
+\sum_{|a+b|=\mathtt{l}}\tbinom{a+b}{b}\,\mathcal{X}^{a+b}(x;y_{1},\ldots,y_{m})\int_{0}^{1}(1-\tau)^{\mathtt{l}-1}\,(\mathbf{Z}_{\tau}^{\dagger}\partial^{b}\varphi)(x;y_{1},\ldots,y_{m})\,\mathrm{d}\tau,$
(8.9)

where the map
$\mathbf{Z}_{\tau}^{\dagger}:=\tau^{\mathsf{d}m}\mathbf{Z}_{1/\tau}$ is a
formal dual to the map $\mathbf{Z}_{\tau}$ and the sums are over
$b\in\mathfrak{M}$. To complete the proof we test $\mathcal{X}^{m,a}V$ with
$\varphi\in\mathscr{S}(\mathbb{M}^{1+m})$, apply the above formula and use the
definitions of the maps $\mathbf{X}_{\mathtt{l}}^{a}$ and $\mathbf{I}$. ∎

######  Lemma 8.9.

Let $\mathtt{h}\in\mathbb{N}_{0}$. The following bounds:

  1. (A)

$\|K^{\ast\mathtt{h},\otimes(1+m)}_{\mu}\ast\mathbf{L}^{m}v\|_{\mathcal{V}^{m}}\lesssim\|K^{\ast\mathtt{h}}_{\mu}\ast
v\|_{\mathcal{V}}$,

  2. (B)

$\|K^{\ast
2\mathtt{h},\otimes(1+m)}_{\mu}\ast\mathbf{Z}_{\tau}V\|_{\mathcal{V}^{m}}\lesssim\|K^{\ast\mathtt{h},\otimes(1+m)}_{\mu}\ast
V\|_{\mathcal{V}^{m}}$

hold uniformly in $\tau,\mu\in(0,1]$ and $v\in\mathcal{V}$,
$V\in\mathcal{V}^{m}$.

###### Proof.

By the exact scaling of the norms and kernels it is enough to prove the lemma
for $\mu=1$. Using
$\mathbf{P}^{\mathtt{h}}K^{\ast\mathtt{h}}=\delta_{\mathbb{M}}$ we obtain

$(K^{\ast\mathtt{h},\otimes(1+m)}\ast\mathbf{L}^{m}v)(x;y_{1},\ldots,y_{m})\\\
=\int_{\mathbb{M}}(K^{\ast\mathtt{h}}\ast
v)(z)~{}\mathbf{P}^{\mathtt{h}}(\partial_{z})K^{\ast\mathtt{h}}(x-z)K^{\ast\mathtt{h}}(y_{1}-z)\ldots
K^{\ast\mathtt{h}}(y_{m}-z)\,\mathrm{d}z\\\
=\int_{\mathbb{M}}(K^{\ast\mathtt{h}}\ast
v)(z)~{}H^{\ast\mathtt{h}}_{0}(x-z,y_{1}-z,\ldots,y_{m}-z)\,\mathrm{d}z,$
(8.10)

where $H_{0}\in\mathcal{K}^{1+m}$ is defined in the lemma below and
$\mathbf{P}^{\mathtt{h}}(r)=(1-r^{2})^{\mathtt{h}}$. This proves the bound (A)
since
$K^{\ast\mathtt{h},\otimes(1+m)}\ast\mathbf{L}^{m}v=H_{0}^{\ast\mathtt{h}}\ast\mathbf{L}^{m}(K^{\ast\mathtt{h}}\ast
v).$ The bound (B) follows from the identities

$\begin{gathered}K^{\otimes(1+m)}=(K\otimes\check{K}^{\otimes
m}_{\tau})\ast\mathbf{Z}_{\tau}(\delta_{\mathbb{M}}\otimes K^{\otimes m}),\\\
K^{\otimes(1+m)}=H_{\tau}\ast\mathbf{Z}_{\tau}(K\otimes\delta_{\mathbb{M}}^{\otimes
m}),\end{gathered}$ (8.11)

where the kernels $\check{K}_{\tau}$ and $\check{H}_{\tau}$ are defined in the
lemma below. Indeed, applying the above equalities recursively and using
$\mathbf{Z}_{\tau}H\ast\mathbf{Z}_{\tau}V=\mathbf{Z}_{\tau}(H\ast V)$ we
arrive at

$K^{\ast
2\mathtt{h},\otimes(1+m)}\ast\mathbf{Z}_{\tau}V=\check{H}^{\ast\mathtt{h}}_{\tau}\ast\mathbf{Z}_{\tau}(K^{\ast\mathtt{h},\otimes(1+m)}\ast
V),\quad\check{H}_{\tau}:=(K\otimes\check{K}^{\otimes m}_{\tau})\ast
H_{\tau},$ (8.12)

which implies Part (B) since $\|\check{H}_{\tau}\|_{\mathcal{K}^{1+m}}\lesssim
1$ uniformly in $\tau\in(0,1]$ by the lemma below. It remains to establish the
identities (8.11). The first one follows from the fact that
$\mathbf{Z}_{\tau}(\delta_{\mathbb{M}}\otimes K^{\otimes
m})=\delta_{\mathbb{M}}\otimes\breve{K}_{\tau}^{\otimes m}$, where
$\breve{K}_{\tau}(x):=\tau^{-1-\mathsf{d}}K(x/\tau)$. To show the second one
we observe that by the definition of $\check{H}_{\tau}$ given in the lemma
below

$\mathbf{Z}_{1/\tau}(H_{\tau})(x;y_{1},\ldots,y_{m})=\tau^{\mathsf{d}m}\,\mathbf{P}(\partial_{x})\,K(x)K(x+\tau(y_{1}-x))\ldots
K(x+\tau(y_{1}-x)).$ (8.13)

After convolving both sides with the kernel
$K\otimes\delta_{\mathbb{M}}^{\otimes m}$ we obtain

$\mathbf{Z}_{1/\tau}(H_{\tau})\ast\big{(}K\otimes\delta_{\mathbb{M}}^{\otimes
m}\big{)}=\mathbf{Z}_{1/\tau}(K^{\otimes(1+m)})$ (8.14)

The second of the identities (8.11) follows after applying the map
$\mathbf{Z}_{\tau}$ to both sides of the above equality. ∎

######  Lemma 8.10.

The distributions $\check{K}_{\tau}\in\mathscr{S}^{\prime}(\mathbb{M})$ and
$H_{\tau}\in\mathscr{S}^{\prime}(\mathbb{M}^{1+m})$ defined by

$\begin{gathered}\check{K}_{\tau}(x):=\mathbf{P}(\tau\partial_{x})K(x)=\tau^{2}\delta_{\mathbb{M}}+(1-\tau^{2})K,\\\
H_{\tau}(x;y_{1},\ldots,y_{m}):=\mathbf{P}(\partial_{x}+(1-\tau)\partial_{y})\,K(x)K(y_{1})\ldots
K(y_{m}),\end{gathered}$ (8.15)

where $\partial_{y}:=\partial_{y_{1}}+\ldots+\partial_{y_{m}}$ and
$\mathbf{P}(r)=1-r^{2}$, are polynomials in $\tau>0$ of degree $2$ whose
coefficients belong to $\mathcal{K}$ and $\mathcal{K}^{1+m}$, respectively.

##  9 Deterministic bounds for relevant coefficients

######  Lemma 9.1.

For every ${\mathtt{g}}\in\mathbb{N}_{0}$, $a\in\mathfrak{M}$ and
$r\in\mathbb{N}_{0}$ the following bound

$\|\mathbf{P}^{{\mathtt{g}}}_{\mu}\mathcal{X}^{a}\partial_{\kappa}^{r}\partial_{\mu}G_{\kappa,\mu}\|_{\mathcal{K}}\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{|a|-\varepsilon
r}$ (9.1)

holds uniformly in $\kappa,\mu\in(0,1/2]$.

######  Remark 9.2.

For $\sigma\notin 2\mathbb{N}_{+}$ the bound stated in the lemma does not hold
uniformly for all $\mu\in(0,1]$ because of the slow decay of the kernel $G(x)$
at infinity. The above lemma can be easily proved using the method of the
proof of Lemma 6.5.

######  Theorem 9.3.

Fix $R>1$. Assume that for all $i,m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$
such that $\varrho(i,m)+|a|\leq 0$ there exists
${\mathtt{g}}\in\mathbb{N}_{0}$ such that

$\|K^{\ast{\mathtt{g}}}_{\mu}\ast\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}\|_{\mathcal{V}}\leq
R\,[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{\varepsilon}(i,m)+|a|}$
(9.2)

for all $r\in\\{0,1\\}$ and $\kappa,\mu\in(0,1/2]$. Then for all
$i,m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$ there exists
${\mathtt{g}}\in\mathbb{N}_{0}$ such that

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\mu}^{s}F_{\kappa,\mu}^{i,m,a}\|_{\mathcal{V}^{m}}\lesssim
R^{1+3i-m}\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{\varrho_{\varepsilon}(i,m)+|a|-\sigma
s}$ (9.3)

for all $r,s\in\\{0,1\\}$ uniformly in $\kappa,\mu\in(0,1/2]$. The constants
of proportionality in the above bounds depend only on $i,m\in\mathbb{N}_{+}$,
$a\in\mathfrak{M}^{m}$ and are otherwise universal.

######  Remark 9.4.

Let us remind the reader that the coefficients $F^{i,m,a}_{\kappa,\mu}$ and
$f^{i,m,a}_{\kappa,\mu}$ such that $\varrho(i,m)+|a|\leq 0$ are called
relevant and $F^{i,m,a}_{\kappa,\mu}$ and $f^{i,m,a}_{\kappa,\mu}$ vanish
unless $m\leq 3i$. There only finitely many non-zero relevant coefficients
$F^{i,m,a}_{\kappa,\mu}$ or $f^{i,m,a}_{\kappa,\mu}$. Recall that the
collection of the relevant coefficients $f^{i,m,a}_{\kappa,\mu}$ such that
$m\leq 3i$ is called the enhanced noise.

######  Remark 9.5.

The above theorem together with Theorem 10.1 imply that under the assumption
of the above theorem there exists ${\mathtt{g}}\in\mathbb{N}_{0}$ and a
universal constant $c>1$ such that the assumption of Theorem 6.7 is satisfied
with $\tilde{R}=cR$.

###### Proof.

The base case: We observe that
$F^{0,0,0}_{\kappa,\mu}=f^{0,0,0}_{\kappa,\mu}=\xi$. Hence, for $i=0$, $m=0$
the statement follows from the assumption.

The inductive step: Fix some $i_{\circ}\in\mathbb{N}_{+}$,
$m_{\circ}\in\mathbb{N}_{0}$ and assume that the statement with $s=0$ is true
for all $i,m\in\mathbb{N}_{+}$ such that either $i<i_{\circ}$ or $i=i_{\circ}$
and $m>m_{\circ}$. We shall prove the statement for $i=i_{\circ}$,
$m=m_{\circ}$. As in the proof of Theorem 6.7 the induction hypothesis, the
flow equation (7.8), and the identities

$\varrho_{\varepsilon}(i,m)+|a|-\sigma=\varrho_{\varepsilon}(j,1+k)+|b|+|c|+\varrho_{\varepsilon}(i-j,m-k)+|d|-\varepsilon,$
(9.4) $R^{1+3j-k-1}R^{1+3(i-j)-m+k}=R^{1+3i-m}$ (9.5)

imply the statement for $s=1$.

In order to prove the statement for $s=0$ let us first assume that
$a\in\mathfrak{M}^{m}$ is such that $\varrho(i,m)+|a|>0$. Then
$F^{i,m,a}_{\kappa,0}=F^{i,m,a}_{\kappa}=0$. Consequently, the analogs of Eq.
(6.12) and the bound (6.13) imply that

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}F_{\kappa,\mu}^{i,m,a}\|_{\mathcal{V}^{m}}\\\
\leq\int_{0}^{\mu}\|K_{\eta}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m,a}\|_{\mathcal{V}^{m}}\,\mathrm{d}\eta.$
(9.6)

The statement follows now from the bound for
$\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m,a}$ and the fact
that $\varrho_{\varepsilon}(i,m)+|a|-\varepsilon>0$ if $\varrho(i,m)+|a|>0$ by
Remark 6.4.

Now let us suppose that $a\in\mathfrak{M}^{m}$ is such that
$\varrho(i,m)+|a|\leq 0$. Then our bound for
$\|K_{\eta}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m,a}\|_{\mathcal{V}^{m}}$
is not integrable at $\eta=0$. Consequently, the strategy used in the previous
paragraph does not work. If $m=0$, then $a=0$ and
$\partial_{\kappa}^{r}F^{i,0,0}_{\kappa,\mu}=\partial_{\kappa}^{r}f^{i,0,0}_{\kappa,\mu}$.
As a result, the statement follows from the assumption. If
$m\in\mathbb{N}_{+}$, then we use the identity

$\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\mu}=\mathbf{X}^{a}_{\mathtt{l}}(\partial_{\kappa}^{r}f^{i,m,b}_{\kappa,\mu},\partial_{\kappa}^{r}F^{i,m,b}_{\kappa,\mu}),$
(9.7)

which follows from Theorem 8.8. We choose $\mathtt{l}\in\mathbb{N}_{+}$ to be
the smallest positive integer such that $\varrho(i,m)+\mathtt{l}>0$. With this
choice of $\mathtt{l}$ the RHS of the above equality involves only the
coefficients $f^{i,m,b}_{\kappa,\mu}$ such that $\varrho(i,m,b)\leq 0$, which
satisfy the assumed bound, and the coefficients $F^{i,m,b}_{\kappa,\mu}$ such
that $\varrho(i,m,b)>0$, for which the statement of the theorem has already
been established. To finish the proof of the inductive step we apply Theorem
8.7 with
$C\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{\varepsilon}(i,m)-\varepsilon
r}$. ∎

##  10 Deterministic bounds for long-range part of coefficients

######  Theorem 10.1.

Fix $R>1$. Assume that for all $i,m\in\mathbb{N}_{0}$ there exists
${\mathtt{g}}\in\mathbb{N}_{0}$ such that

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\mu}^{s}F_{\kappa,\mu}^{i,m}\|_{\mathcal{V}^{m}}\lesssim
R^{1+3i-m}\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{\varrho_{\varepsilon}(i,m)-\sigma
s}$ (10.1)

for $r,s\in\\{0,1\\}$ uniformly in $\kappa,\mu\in(0,1/2]$. Then for all
$i,m\in\mathbb{N}_{0}$ there exists ${\mathtt{g}}\in\mathbb{N}_{0}$ such that
the above bound is true for $r,s\in\\{0,1\\}$ uniformly in $\kappa\in(0,1/2]$,
$\mu\in(0,1]$. The constants of proportionality in the above bounds depend
only on $i,m\in\mathbb{N}_{+}$ and are otherwise universal.

###### Proof.

The base case: We observe that
$F^{0,0}_{\kappa,\mu}=f^{0,0,0}_{\kappa,\mu}=\xi$ is independent of
$\kappa,\mu$. The assumed bound and Lemma 4.5 imply that

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\xi\|_{\mathcal{V}}\leq\|K_{\mu/2}^{\ast{\mathtt{g}}}\ast\xi\|_{\mathcal{V}}\lesssim[\mu]^{\varrho_{\varepsilon}(0,0)}$
(10.2)

uniformly in $\mu\in(0,1]$. This proves the statement for $i=0$ and $m=0$.

The induction step: Fix some $i_{\circ}\in\mathbb{N}_{+}$,
$m_{\circ}\in\mathbb{N}_{0}$ and assume that the statement with $s=0$ is true
for all $i,m\in\mathbb{N}_{+}$ such that either $i<i_{\circ}$ or $i=i_{\circ}$
and $m>m_{\circ}$. We shall prove the statement for $i=i_{\circ}$,
$m=m_{\circ}$. The induction hypothesis and the flow equation imply the
statement for $s=1$. In order to prove the statement for $s=0$ we use the
identity

$F^{i,m}_{\kappa,\mu}=F^{i,m}_{\kappa,1/2}+\int_{1/2}^{\mu}\partial_{\eta}F^{i,m}_{\kappa,\eta}\,\mathrm{d}\eta,$
(10.3)

which by Lemma 4.5 implies that

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}F_{\kappa,\mu}^{i,m}\|_{\mathcal{V}^{m}}\\\
\leq\|K_{1/2}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}F_{\kappa,1/2}^{i,m}\|_{\mathcal{V}^{m}}+\int_{1/2}^{\mu}\|K_{\eta}^{\ast{\mathtt{g}},\otimes(1+m)}\ast\partial_{\kappa}^{r}\partial_{\eta}F_{\kappa,\eta}^{i,m}\|_{\mathcal{V}^{m}}\mathrm{d}\eta.$
(10.4)

This completes the proof of inductive step. ∎

##  11 Deterministic construction of solution

Assuming the bound (6.6) for all $r,s\in\\{0,1\\}$, $i,m\in\mathbb{N}_{0}$,
$\kappa\in(0,1/2]$, $\mu\in(0,1]$ with fixed ${\mathtt{g}}\in\mathbb{N}_{0}$,
$\tilde{R}>1$ we show how to construct a solution of Eq. (1.2).

######  Definition 11.1.

The families of sets $\mathcal{V}_{\mu}$, $\mathcal{B}_{\mu}$, $\mu\in(0,1]$,
are defined as follows

$\displaystyle\mathcal{V}_{\mu}$
$\displaystyle:=\\{K^{\ast{\mathtt{g}}}_{\mu}\ast\varphi\in\mathcal{V}\,|\,\varphi\in\mathcal{V}\\},$
(11.1) $\displaystyle\mathcal{B}_{\mu}$
$\displaystyle:=\\{K^{\ast{\mathtt{g}}}_{\mu}\ast\varphi\in\mathcal{V}\,|\,\varphi\in\mathcal{V},\,\|\varphi\|_{\mathcal{V}}<\tilde{R}^{2}\,[\mu]^{-\dim(\varPhi)-2\varepsilon}\\}.$

By Lemma 4.5, if $\varphi\in\mathcal{B}_{\mu}$, then
$\varphi\in\mathcal{B}_{\eta}$ for any $\eta$ in a sufficiently small
neighbourhood of $\mu$. For $\kappa\in(0,1]$, $\mu\in[0,1]$ and
$\lambda\leq\tilde{R}^{-6}$ we define the functionals

$F_{\kappa,\mu}\,:\,\mathcal{B}_{\mu}\to\mathcal{D},\qquad\mathrm{D}F_{\kappa,\mu}\,:\,\mathcal{B}_{\mu}\times\mathcal{V}_{\mu}\to\mathcal{D}$
(11.2)

by the following formulas

$\langle
F_{\kappa,\mu}[\varphi],\psi\rangle:=\sum_{i=0}^{\infty}\sum_{m=0}^{\infty}\lambda^{i}\,\langle
F^{i,m}_{\kappa,\mu},\psi\otimes\varphi^{\otimes m}\rangle,$ (11.3)
$\langle\mathrm{D}F_{\kappa,\mu}[\varphi,\zeta],\psi\rangle:=\sum_{i=0}^{\infty}\sum_{m=0}^{\infty}\lambda^{i}\,(1+m)\,\langle
F^{i,1+m}_{\kappa,\mu},\psi\otimes\zeta\otimes\varphi^{\otimes m}\rangle,$
(11.4)

where $\psi\in\mathscr{S}(\mathbb{M})$, $\varphi\in\mathcal{B}_{\mu}$ and
$\zeta\in\mathcal{V}_{\mu}$.

######  Lemma 11.2.

For $\lambda\leq\tilde{R}^{-6}$ the functionals $F_{\kappa,\mu}$,
$\mathrm{D}F_{\kappa,\mu}$ are well defined and satisfy the following bounds

$\begin{gathered}\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}F_{\kappa,\mu}[\varphi]\|_{\mathcal{V}}\leq\tilde{R}\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{-\dim(\xi)-\varepsilon}\\\
\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}\mathrm{D}F_{\kappa,\mu}[\varphi,\zeta]\|_{\mathcal{V}}\leq\tilde{R}\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{-\dim(\xi)-\varepsilon}\,\|\mathbf{P}^{\mathtt{g}}_{\mu}\zeta\|_{\mathcal{V}}\end{gathered}$
(11.5)

for $r\in\\{0,1\\}$ and $\kappa\in(0,1/2]$, $\mu\in(0,1]$ and
$\varphi\in\mathcal{B}_{\mu}$, $\zeta\in\mathcal{V}_{\mu}$. The functional
$\mathrm{D}F_{\kappa,\mu}$ is the directional derivative of $F_{\kappa,\mu}$.
Moreover, given $\eta\in(0,1]$ and $\varphi\in\mathcal{B}_{\eta}$ the flow
equation (2.3) holds for all $\mu$ in a sufficiently small neighbourhood of
$\eta$.

###### Proof.

The proof is straightforward. For example, using Def. 5.2 of the space
$\mathcal{V}_{m}$ and the bounds for the effective force coefficients we
obtain

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast
F_{\kappa,\mu}[K_{\mu}^{\ast{\mathtt{g}}}\ast\varphi]\|\\\
\leq\sum_{i=0}^{\infty}\sum_{m=0}^{\infty}\frac{\lambda^{i}\tilde{R}^{1+2(3i-m)}}{4(1+i)^{2}\,4(1+m)^{2}}\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{\varrho_{\varepsilon}(i,m)}~{}\tilde{R}^{2m}\,[\mu]^{-m(\dim(\varPhi)+2\varepsilon)}$
(11.6)

for $\varphi\in\mathcal{V}$ such that
$\|\varphi\|_{\mathcal{V}}\leq\tilde{R}^{2}\,[\mu]^{-\dim(\varPhi)-2\varepsilon}$,
which implies the desired bound. The flow equation (2.3) follows from the flow
equation (3.4). ∎

######  Lemma 11.3.

For $\lambda\leq\tilde{R}^{-6}$ and $\kappa\in(0,1/2]$, $\mu\in(0,1]$ it holds

  * (A)

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}\leq\tilde{R}\,[\kappa]^{r(\varepsilon-\sigma)}[\mu]^{-\dim(\xi)-\varepsilon}$,

  * (B)

$\|\mathbf{P}_{\mu}^{\mathtt{g}}G_{\kappa,\mu}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}\leq
1/3\,\tilde{R}^{2}\,[\kappa]^{r(\varepsilon-\sigma)}[\mu]^{-\dim(\varPhi)-2\varepsilon}$,

  * (C)

$F_{\kappa,1}[0]=F_{\kappa,\mu}[G_{\kappa,\mu}\ast F_{\kappa,1}[0]]$,

  * (D)

$\partial_{\kappa}F_{\kappa,1}[0]=(\partial_{\kappa}F_{\kappa,\mu})[G_{\kappa,\mu}\ast
F_{\kappa,1}[0]]+\mathrm{D}F_{\kappa,\mu}[G_{\kappa,\mu}\ast
F_{\kappa,1}[0],\partial_{\kappa}(G_{\kappa,\mu}\ast F_{\kappa,1}[0])]$.

###### Proof.

By Lemma 11.2 and the equality $G_{\kappa,1}=0$ the statement holds true for
$\mu=1$. Chose $\nu\in(0,1]$ and assume that the statement holds true for all
$\mu\in[\nu,1]$. Then by Lemma 4.5 we have

$\|K_{\eta}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}\leq
2\,\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}\leq
2\tilde{R}\,[\kappa]^{r(\varepsilon-\sigma)}[\mu]^{-\dim(\xi)-\varepsilon}$
(11.7)

for all $\eta\in[\tau\mu,1]$, $\mu\in[\nu,1]$ for some $\tau\in(0,1)$
depending only on $\tilde{R}$. As a result,

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}\leq
2\tilde{R}\,[\kappa]^{r(\varepsilon-\sigma)}[\mu]^{-\dim(\xi)-\varepsilon}$
(11.8)

for all $\mu\in[\tau\nu,1]$. This together with Remark 6.6 and the estimate

$\|\mathbf{P}_{\mu}^{\mathtt{g}}G_{\kappa,\mu}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}\leq\int_{\mu}^{1}\|\mathbf{P}_{\eta}^{2{\mathtt{g}}}\partial_{\eta}G_{\kappa,\eta}\|_{\mathcal{K}}\,\|K_{\eta}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}F_{\kappa,1}[0]\|_{\mathcal{V}}$
(11.9)

imply Part (B) for all $\mu\in[\tau\nu,1]$ and shows that $G_{\kappa,\mu}\ast
F_{\kappa,1}[0]\in\mathcal{B}_{\mu}$. Using Lemma 11.2 and the reasoning
presented in Sec. 2 below Eq. (2.4) we prove that Parts (C), (D) hold for all
$\mu\in[\tau\kappa,1]$. Part (A) for all $\mu\in[\tau\kappa,1]$ follows now
from Lemma 11.2. To complete the proof of the theorem it is enough to apply
the above reasoning recursively with $\nu=\tau^{n}$, $n\in\mathbb{N}_{0}$. ∎

######  Theorem 11.4.

Fix $R>1$. Assume that for all $i,m\in\mathbb{N}_{+}$, $a\in\mathfrak{M}^{m}$
such that $\varrho(i,m)+|a|\leq 0$ there exists
${\mathtt{g}}\in\mathbb{N}_{0}$ such that

$\|K^{\ast{\mathtt{g}}}_{\mu}\ast\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}\|_{\mathcal{V}}\leq
R\,[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{\varepsilon}(i,m)+|a|}$
(11.10)

for all $r\in\\{0,1\\}$ and $\kappa\in(0,1/2]$, $\mu\in(0,1]$. There exists
${\mathtt{g}}\in\mathbb{N}_{0}$ and a universal constant $c>1$ such that for
$\tilde{R}=c\,R$, $\lambda<\tilde{R}^{-6}$ and $\kappa\in(0,1/2]$ the function
$\varPhi_{\kappa}:=G_{\kappa}\ast F_{\kappa,\mu}[0]$ is well defined, solves
Eq. (1.2) and satisfies the bound

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}\varPhi_{\kappa}\|_{\mathcal{V}}\leq\tilde{R}^{2}\,[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{-\dim(\varPhi)-2\varepsilon}$
(11.11)

for all $\kappa,\mu\in(0,1]$.

###### Proof.

By Theorems 6.7, 9.3 and 10.1 the assumption implies that there exists
${\mathtt{g}}\in\mathbb{N}_{0}$ and a universal constant $c>1$ such that the
bound (6.6) holds true for all $r,s\in\\{0,1\\}$, $i,m\in\mathbb{N}_{0}$ and
$\kappa\in(0,1/2]$, $\mu\in(0,1]$ with $\tilde{R}=c\,R$. The function
$\varPhi_{\kappa}$ is well-defined by Lemma 11.2. To conclude, we observe that

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}\varPhi_{\kappa}\|_{\mathcal{V}}\leq\sum_{u+v=r}\int_{0}^{1}\|\mathbf{P}^{\mathtt{g}}_{\eta}\partial_{\kappa}^{u}\partial_{\eta}G_{\kappa,\eta}\|_{\mathcal{K}}\,\|K_{\eta}^{\ast{\mathtt{g}}}\ast
K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{v}F_{\kappa,1}[0]\|_{\mathcal{V}}\,\mathrm{d}\eta,$
(11.12)

use Remark 6.6 and apply Lemma 11.3 with $\mu$ replaced by $\eta\vee\mu$. ∎

##  12 Cumulants of effective force coefficients

In the remaining part of the paper we show that the assumption of Theorem 11.4
is satisfied for some random $R$ such that $\mathbb{E}R^{n}<\infty$ for all
$n\in\mathbb{N}_{+}$. This will follow from the bounds for the joint cumulants
of the effective force coefficients $F^{i,m,a}_{\kappa,\mu}$ proved in Sec. 16
and the probabilistic estimates proved in Sec. 13. The bounds for the joint
cumulants of $F^{i,m,a}_{\kappa,\mu}$ involve the norm
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{\mathcal{V}^{\mathsf{m}}}$ introduced in Sec. 14. We prove
the above-mentioned bounds using a certain flow equation given in Sec. 15. We
consider joint cumulants of $F^{i,m,a}_{\kappa,\mu}$ instead of
$f^{i,m,a}_{\kappa,\mu}$ because there is no flow equation for the joint
cumulants of $f^{i,m,a}_{\kappa,\mu}$.

######  Definition 12.1.

Let $p\in\mathbb{N}_{+}$, $I=\\{1,\ldots,p\\}$ and $\zeta_{q}$, $q\in I$, be
random variables. The joint cumulant of $(\zeta_{q})_{q\in I}$ is defined by
the formula

$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1},\ldots,\zeta_{p}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\equiv\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}=(-\mathrm{i})^{p}\partial_{t_{1}}\ldots\partial_{t_{p}}\log\mathbb{E}\exp(\mathrm{i}t_{1}\zeta_{1}+\ldots+\mathrm{i}t_{p}\zeta_{p})\big{|}_{t_{1}=\ldots=t_{p}=0}.$
(12.1)

In particular,
$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\equiv\mathbb{E}\zeta$,
$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1},\zeta_{2}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}=\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1}\zeta_{2}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}-\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{2}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}$.
The joint cumulant
$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}$
does not depend on the order of elements in the list $(\zeta_{q})_{q\in I}$.

######  Lemma 12.2.

Let $p\in\mathbb{N}_{+}$, $I=\\{1,\ldots,p\\}$ and
$\zeta_{1},\ldots,\zeta_{p},\Phi,\Psi$ be random variables. It holds

$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1}\ldots\zeta_{p}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}=\sum_{r=1}^{p}\sum_{\begin{subarray}{c}I_{1},\ldots,I_{r}\subset
I,\\\ I_{1}\cup\ldots\cup I_{r}=I\\\
I_{1},\ldots,I_{r}\neq\emptyset\end{subarray}}\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I_{1}}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\ldots\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I_{r}}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}},$
(12.2)
$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I},\Phi\Psi\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}=\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I},\Phi,\Psi\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}+\sum_{\begin{subarray}{c}I_{1},I_{2}\subset
I\\\ I_{1}\cup
I_{2}=I\end{subarray}}\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I_{1}},\Phi\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}~{}\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I_{2}},\Psi\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}.$
(12.3)

######  Remark 12.3.

For the proof of the above lemma see Proposition 3.2.1 in [9].

######  Definition 12.4.

Given $n\in\mathbb{N}_{+}$, $I=\\{1,\ldots,n\\}$,
$m_{1},\ldots,m_{n}\in\mathbb{N}_{0}$ and random distributions
$\zeta_{q}\in\mathscr{S}^{\prime}(\mathbb{M}^{1+m}_{q})$, $q\in I$, we define
the deterministic distribution
$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\equiv\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1},\ldots,\zeta_{n}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\in\mathscr{S}^{\prime}(\mathbb{M}^{n}\times\mathbb{M}^{m_{1}+\ldots+m_{n}})$
by the equality

$\langle\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\zeta_{1},\ldots,\zeta_{n}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}},\psi_{1}\otimes\ldots\otimes\psi_{n}\otimes\varphi_{1}\otimes\ldots\otimes\varphi_{n}\rangle:=\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\langle\zeta_{1},\psi_{1}\otimes\varphi_{1}\rangle,\ldots,\langle\zeta_{n},\psi_{n}\otimes\varphi_{n}\rangle\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}},$
(12.4)

where $\psi_{q}\in\mathscr{S}(\mathbb{M})$,
$\varphi_{q}\in\mathscr{S}(\mathbb{M}^{m_{q}})$, $q\in I$, are arbitrary.

######  Definition 12.5.

A list $(i,m,a,s,r)$, where $i,m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$ and
$s\in\\{0,1\\}$, $r\in\\{0,1,2\\}$ is called an index. Let
$n\in\mathbb{N}_{+}$ and

$\mathtt{I}\equiv((i_{1},m_{1},a_{1},s_{1},r_{1}),\ldots,(i_{n},m_{n},a_{n},s_{n},r_{n}))$
(12.5)

be a list of indices. We set $n(\mathtt{I}):=n$,
$i(\mathtt{I}):=i_{1}+\ldots+i_{n}$,
$\mathsf{m}(\mathtt{I}):=(m_{1},\ldots,m_{n})$,
$m(\mathtt{I}):=m_{1}+\ldots+m_{n}$, $a(\mathtt{I}):=|a_{1}|+\ldots+|a_{n}|$,
$s(\mathtt{I}):=s_{1}+\ldots+s_{n}$ and $r(\mathtt{I}):=r_{1}+\ldots+r_{n}$.
We use the following notation for the joint cumulants of the effective force
coefficients

$E^{\mathtt{I}}_{\kappa,\mu}:=\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\partial_{\mu}^{s_{1}}\partial_{\kappa}^{r_{1}}F^{i_{1},m_{1},a_{1}}_{\kappa,\mu},\ldots,\partial_{\mu}^{s_{n}}\partial_{\kappa}^{r_{n}}F^{i_{n},m_{n},a_{n}}_{\kappa,\mu}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\in\mathscr{S}^{\prime}(\mathbb{M}^{n(\mathtt{I})}\times\mathbb{M}^{m(\mathtt{I})}).$
(12.6)

######  Definition 12.6.

For $\varepsilon\geq 0$ and a list of indices $\mathtt{I}$ of the form (12.5)
we define

$\varrho_{\varepsilon}(\mathtt{I}):=\varrho_{\varepsilon}(i_{1},m_{1})+|a_{1}|+\ldots+\varrho_{\varepsilon}(i_{n},m_{n})+|a_{n}|\in\mathbb{R}.$
(12.7)

We also set $\varrho(\mathtt{I}):=\varrho_{0}(\mathtt{I})$. The cumulant
$E^{\mathtt{I}}_{\kappa,\mu}$ such that
$\varrho(\mathtt{I})+(n-1)\mathsf{d}\leq 0$ are called relevant. The remaining
cumulants are called irrelevant.

######  Remark 12.7.

If $\mathrm{i}(\mathtt{I})=0$, then either $E^{\mathtt{I}}_{\kappa,\mu}=0$ or
$n(\mathtt{I})=2$, $m(\mathtt{I})=0$, $a(\mathtt{I})=0$, $s(\mathtt{I})=0$,
$r(\mathtt{I})=0$. In the later case $E^{\mathtt{I}}_{\kappa,\mu}$ is relevant
and coincides with the covariance of the white noise. For
$\mathrm{i}(\mathtt{I})>0$ the only relevant cumulants are the expectations of
the relevant force coefficients.

######  Remark 12.8.

For $\varepsilon>0$ and any list of indices $\mathtt{I}$ such that
$m(\mathtt{I})\leq 3i(\mathtt{I})$ it holds
$\varrho_{\varepsilon}(\mathtt{I})<\varrho(\mathtt{I})$. Moreover,
$\varrho_{\varepsilon}(\mathtt{I})+(n(\mathtt{I})-1)\mathsf{d}>0$ for
$\varepsilon\in(0,\varepsilon_{\diamond})$ and lists of indices $\mathtt{I}$
such that $\varrho(\mathtt{I})+(n(\mathtt{I})-1)\mathsf{d}>0$, where
$\varepsilon_{\diamond}$ was introduced in Remark 6.4. Recall that
$\varepsilon\in(0,\varepsilon_{\diamond}/3)$ is fixed (see Remark 6.4).

##  13 Probabilistic analysis

######  Theorem 13.1.

Fix $n\in 2\mathbb{N}_{+}$ such that $\mathsf{d}/n<\varepsilon$ and
$i,m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$ such that $\varrho(i,m)+|a|\leq
0$. For $s\in\\{0,1\\}$, $r\in\\{0,1,2\\}$ we define the list of indices
$\mathtt{I}=\mathtt{I}(s,r)=((i,m,a,s,r),\ldots,(i,m,a,s,r))$,
$n(\mathtt{I})=n$. Assume that there exists ${\mathtt{g}}\in\mathbb{N}_{0}$
such that for $s\in\\{0,1\\}$, $r\in\\{0,1,2\\}$ the following bound

$\int_{\mathbb{T}^{n-1}}[K^{\ast{\mathtt{g}},\otimes(n+nm)}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu}](x_{1},\ldots,x_{n})\,\mathrm{d}x_{2}\ldots\mathrm{d}x_{n}\\\
\lesssim[\kappa]^{(\varepsilon-\sigma)r(\mathtt{I})}[\mu]^{\varrho_{3\varepsilon}(\mathtt{I})-\sigma
s(\mathtt{I})+(n-1)\mathsf{d}}$ (13.1)

holds uniformly in $x_{1}\in\mathbb{M}$, $\kappa,\mu\in(0,1/2]$, where

$[K^{\ast{\mathtt{g}},\otimes(n+nm)}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu}](x_{1},\ldots,x_{n})\\\
:=\int_{\mathbb{M}^{nm}}|(K^{\ast{\mathtt{g}},\otimes(n+nm)}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu})(x_{1},\ldots,x_{n};\mathrm{y}_{1},\ldots,\mathrm{y}_{n})|\,\mathrm{d}\mathrm{y}_{1}\ldots\mathrm{d}\mathrm{y}_{n}.$
(13.2)

Then there exists ${\mathtt{g}}\in\mathbb{N}_{0}$ and a random variable $R>0$
such that $\mathbb{E}R^{n}<\infty$ and for $r\in\\{0,1\\}$ the following bound

$\|K_{\mu}^{\ast{\mathtt{g}}}\ast\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}\|_{\mathcal{V}}\leq
R\,[\kappa]^{(\varepsilon-\sigma)r}\,[\mu]^{\varrho_{\varepsilon}(i,m)+|a|}.$
(13.3)

holds uniformly in $\kappa,\mu\in(0,1/2]$.

######  Remark 13.2.

The function $[K^{\ast{\mathtt{g}},\otimes(n+nm)}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu}]$ is $2\pi$ periodic in all variables and the
integral on the LHS of the bound (13.1) is an integral over one period.

######  Remark 13.3.

The assumption of the above theorem is verified in Theorem 16.1.

###### Proof.

By Remark 4.7 and Lemma 4.8 (B), which says that
$\|\mathbf{T}K^{\ast\mathsf{d}}_{\mu}\|_{\mathcal{V}}\lesssim[\mu]^{-\mathsf{d}}$,
the assumption of the theorem implies that for
$\mathtt{f}={\mathtt{g}}+\mathsf{d}$ the following bound

$[K^{\ast\mathtt{f},\otimes(n+nm)}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu}](x_{1},\ldots,x_{n})\lesssim[\kappa]^{(\varepsilon-\sigma)r(\mathtt{I})}[\mu]^{\varrho_{3\varepsilon}(\mathtt{I})-\sigma
s(\mathtt{I})}$ (13.4)

holds uniformly in $x_{1},\ldots,x_{n}\in\mathbb{M}$, $\kappa,\mu\in(0,1/2]$.
Using the above bound with $x_{1}=\ldots=x_{n}=x$, the relation between the
expectation of a product of random variables and their joint cumulants given
by Eq. (12.2), the equalities
$\varrho_{3\varepsilon}(\mathtt{I})=n(\varrho_{3\varepsilon}(i,m)+|a|)$,
$s(\mathtt{I})=ns$, $r(\mathtt{I})=nr$ and

$(K^{\ast\mathtt{f}}_{\mu}\ast\partial_{\mu}^{s}\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu})(x)=\int_{\mathbb{M}^{m}}\\!(K^{\ast\mathtt{f},\otimes(1+m)}_{\mu}\ast\partial_{\mu}^{s}\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\mu})(x;y_{1},\ldots,y_{m})\,\mathrm{d}y_{1}\ldots\mathrm{d}y_{m}$
(13.5)

we obtain for $r\in\\{0,1,2\\}$, $s\in\\{0,1\\}$ the following bound

$\sup_{x\in\mathbb{M}}\mathbb{E}(K^{\ast\mathtt{f}}_{\mu}\ast\partial_{\mu}^{s}\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}(x))^{n}\lesssim[\kappa]^{n(\varepsilon-\sigma)r}\,[\mu]^{n(\varrho_{3\varepsilon}(i,m)+|a|-\sigma
s)}$ (13.6)

uniform in $x\in\mathbb{M}$, $\kappa,\mu\in(0,1/2]$. Lemma 13.5 and Lemma 4.8
(C) imply

$\mathbb{E}\|\partial_{\mu}^{s}\partial_{\kappa}^{r}(K^{\ast\mathtt{h}}_{\mu}\ast
f^{i,m,a}_{\kappa,\mu})\|^{n}_{\mathcal{V}}\lesssim[\kappa]^{n(\varepsilon-\sigma)r}\,[\mu]^{n(\varrho_{3\varepsilon}(i,m)+|a|-\sigma
s-\varepsilon)}$ (13.7)

with $\mathtt{h}=\mathtt{f}+\mathsf{d}$. The theorem follows now from Lemma
13.6 applied with

$\zeta_{2\kappa,2\mu}=K^{\ast\mathtt{h}}_{\mu}\ast\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu},\qquad
r\in\\{0,1\\},$ (13.8)

and $-\omega=(\varepsilon-\sigma)r$ and
$-\rho=\varrho_{3\varepsilon}(i,m)+|a|-2\varepsilon\geq\varrho_{\varepsilon}(i,m)+|a|$,
where the last inequality holds for $m\leq 3i$. ∎

######  Remark 13.4.

The advantage of the bound of the form (13.1) over (13.4) is that the former
bound contains the extra factor $[\mu]^{(n-1)\mathsf{d}}$ on the RHS, and
consequently can be more easily proved by induction using the flow equation
for cumulants stated in Sec. 15 and the equality
$E^{\mathtt{I}}_{\kappa,\mu}=E^{\mathtt{I}}_{\kappa,0}+\int_{0}^{\mu}\partial_{\eta}E^{\mathtt{I}}_{\kappa,\eta}\,\mathrm{d}\eta$.

######  Lemma 13.5.

Let $n\in 2\mathbb{N}_{+}$. There exists a constant $C>0$ such that for all
random fields $\zeta\in L^{n}(\mathbb{T})$ and $\mu\in(0,1/2]$ it holds

$\mathbb{E}\|K_{\mu}^{\ast\mathsf{d}}\ast\zeta\|^{n}_{L^{\infty}(\mathbb{T})}\leq
C\,[\mu]^{-\mathsf{d}}\,\mathbb{E}\|\zeta\|^{n}_{L^{n}(\mathbb{T})}.$ (13.9)

###### Proof.

Note that
$K_{\mu}^{\ast\mathsf{d}}\ast\zeta=\mathbf{T}K_{\mu}^{\ast\mathsf{d}}\star\zeta$,
where $\star$ is the convolution in $\mathbb{T}$ and
$\mathbf{T}K_{\mu}^{\ast\mathsf{d}}$ is the periodization of
$K_{\mu}^{\ast\mathsf{d}}$ (see Def. 4.6). Using the Young inequality for
convolutions we obtain

$\mathbb{E}\|K_{\mu}^{\ast\mathsf{d}}\ast\zeta\|^{n}_{L^{\infty}(\mathbb{T})}\leq\|\mathbf{T}K_{\mu}^{\ast\mathsf{d}}\|_{L^{n/(n-1)}(\mathbb{T})}^{n}\,\mathbb{E}\|\zeta\|^{n}_{L^{n}(\mathbb{T})}.$
(13.10)

The lemma follows now from Lemma 4.8 (B). ∎

######  Lemma 13.6.

Fix $n\in\mathbb{N}_{+}$. There exists a universal constant $c>0$ such that if

$\mathbb{E}\|\partial_{\mu}^{s}\partial_{\kappa}^{u}\zeta_{\kappa,\mu}^{\phantom{l}}\|^{n}_{L^{\infty}(\mathbb{T})}\\\
\leq C\,[\kappa]^{-n(\omega+\sigma u-\varepsilon u)}[\mu]^{-n(\rho+\sigma
s-\varepsilon s)}$ (13.11)

for some differentiable random function $\zeta:(0,1]^{2}\to
L^{\infty}(\mathbb{T})$, some $C>0$, $\omega,\rho\leq 0$ and all
$u,s\in\\{0,1\\}$ and $\kappa,\mu\in(0,1]$, then

$\mathbb{E}\Big{(}\sup_{\kappa,\mu\in(0,1]}[\kappa]^{n\omega}[\mu]^{n\rho}\,\|\zeta_{\kappa,\mu}\|^{n}_{L^{\infty}(\mathbb{T})}\Big{)}\leq
c\,C.$ (13.12)

###### Proof.

For all $\kappa,\mu\in(0,1]$ we have

$[\kappa]^{\omega}[\mu]^{\rho}\,\|\zeta_{\kappa,\mu}\|_{L^{\infty}(\mathbb{T})}\leq\|\zeta_{1,1}\|_{L^{\infty}(\mathbb{T})}\\\
+\int_{\mu}^{1}[\eta]^{\rho}\,\|\partial_{\eta}\zeta_{1,\eta}\|_{L^{\infty}(\mathbb{T})}\,\mathrm{d}\eta+\int_{\mu}^{1}\int_{\kappa}^{1}[\nu]^{\omega}[\eta]^{\rho}\,\|\partial_{\eta}\partial_{\nu}\zeta_{\nu,\eta}\|_{L^{\infty}(\mathbb{T})}\,\mathrm{d}\nu\mathrm{d}\eta$
(13.13)

By the Minkowski inequality we get the bound

$\mathbb{E}\Big{(}\sup_{\kappa,\mu\in(0,1]}[\kappa]^{n\omega}[\mu]^{n\rho}\,\|\zeta_{\kappa,\mu}\|^{n}_{L^{\infty}(\mathbb{T})}\Big{)}^{\\!\frac{1}{n}}\leq\mathbb{E}\Big{(}\|\zeta_{1,1}\|^{n}_{L^{\infty}(\mathbb{T})}\Big{)}^{\\!\frac{1}{n}}\\\
+\int_{0}^{1}[\mu]^{\rho}\,\mathbb{E}\Big{(}\|\partial_{\mu}\zeta_{1,\mu}\|^{n}_{L^{\infty}(\mathbb{T})}\Big{)}^{\\!\frac{1}{n}}\,\mathrm{d}\mu+\int_{0}^{1}\int_{0}^{1}[\kappa]^{\omega}[\mu]^{\rho}\,\mathbb{E}\Big{(}\|\partial_{\mu}\partial_{\kappa}\zeta_{\kappa,\mu}\|^{n}_{L^{\infty}(\mathbb{T})}\Big{)}^{\\!\frac{1}{n}}\,\mathrm{d}\kappa\mathrm{d}\mu,$
(13.14)

which implies the statement. ∎

##  14 Function spaces for cumulants

######  Definition 14.1.

Let $n\in\mathbb{N}_{+}$,
$\mathsf{m}=(m_{1},\ldots,m_{n})\in\mathbb{N}_{0}^{n}$ and
$m=m_{1}+\ldots+m_{n}$. The vector space $\mathcal{V}^{\mathsf{m}}$ consists
of functions $V\in C(\mathbb{M}^{n}\times\mathbb{M}^{m})$ such that

$\|V\|_{\mathcal{V}^{\mathsf{m}}}:=\\\
\sup_{x_{1}\in\mathbb{M}}\int_{\mathbb{T}^{n-1}}[V](x_{1},\ldots,x_{n})\,\mathrm{d}x_{2}\ldots\mathrm{d}x_{n},$
(14.1)

where

$[V](x_{1},\ldots,x_{n}):=\int|V(x_{1},\ldots,x_{n};y_{1},\ldots,y_{m})|\,\mathrm{d}y_{1}\ldots\mathrm{d}y_{m},$
(14.2)

is finite and the function $\mathbf{U}^{\mathsf{m}}V\in
C(\mathbb{M}^{n}\times\mathbb{M}^{m})$ defined by

$(\mathbf{U}^{\mathsf{m}}V)(x_{1},\ldots,x_{n};y_{1},\ldots,y_{m}):=V(x_{1},\ldots,x_{n};\mathrm{y}_{1}+x_{1},\ldots,\mathrm{y}_{n}+x_{n})$
(14.3)

is $2\pi$ periodic in variables $x_{1},\ldots,x_{n}$ for every

$(y_{1},\ldots,y_{m})=(\mathrm{y}_{1},\ldots,\mathrm{y}_{n})\in\mathbb{M}^{m_{1}}\times\ldots\times\mathbb{M}^{m_{n}}=\mathbb{M}^{m},$
(14.4)

where for arbitrary $x\in\mathbb{M}$ and
$\mathrm{y}=(y_{1},\ldots,y_{m})\in\mathbb{M}^{m}$, $m\in\mathbb{N}_{0}$, we
use the following notation
$\mathrm{y}+x:=(y_{1}+x,\ldots,y_{n}+x)\in\mathbb{M}^{m}$. For
${\mathtt{g}}\in\mathbb{N}_{0}$ the space
$\mathcal{D}^{\mathsf{m}}_{\mathtt{g}}$ consists of distributions
$V\in\mathscr{S}^{\prime}(\mathbb{M}^{n}\times\mathbb{M}^{m})$ such that
$K^{\ast{\mathtt{g}},\otimes(n+m)}_{\mu}\ast V\in\mathcal{V}^{\mathsf{m}}$.
The space $\mathcal{D}^{\mathsf{m}}$ is the union of the spaces
$\mathcal{D}^{\mathsf{m}}_{\mathtt{g}}$, ${\mathtt{g}}\in\mathbb{N}_{0}$.

######  Remark 14.2.

For $V\in\mathcal{V}^{\mathsf{m}}$, the function $[V]$ is $2\pi$ periodic.
Note also that for $n=1$ and $\mathsf{m}=m$ we have
$\mathcal{V}^{\mathsf{m}}=\mathcal{V}^{m}$,
$\mathcal{D}^{\mathsf{m}}=\mathcal{D}^{m}$,
$\mathcal{D}^{\mathsf{m}}_{\mathtt{g}}=\mathcal{D}^{m}_{\mathtt{g}}$, where
the spaces $\mathcal{V}^{m}$, $\mathcal{D}^{m}$,
$\mathcal{D}^{m}_{\mathtt{g}}$ were introduced in Def. 5.2.

######  Definition 14.3.

For $V\in\mathcal{D}^{\mathsf{m}}$ and $\pi\in\mathcal{P}_{m_{1}}$ we define
$\mathbf{Y}_{\pi}V\in\mathcal{D}^{\mathsf{m}}$ by

$\big{\langle}\mathbf{Y}_{\pi}V,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{n}\psi_{q}\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{m}\varphi_{q}\big{\rangle}:=\big{\langle}V,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{n}\psi_{q}\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{m_{1}}\varphi_{\pi(q)}\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=m_{1}+1}^{m}\varphi_{q}\big{\rangle},$
(14.5)

where $\psi_{1},\ldots,\psi_{n},\varphi_{1},\ldots,\varphi_{m}\in\
\mathscr{S}(\mathbb{M})$. For $V\in\mathcal{D}^{\mathsf{m}}$ and
$\omega\in\mathcal{P}_{n}$ we define
$\mathbf{Y}^{\omega}V\in\mathcal{D}^{\omega(\mathsf{m})}$, where
$\omega(\mathsf{m}):=(m_{\omega(1)},\ldots,m_{\omega(n)})$, by

$\big{\langle}\mathbf{Y}^{\omega}V,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{n}\psi_{q}\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{n}\varphi_{q}\big{\rangle}:=\big{\langle}V,{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{n}\psi_{\omega(q)}\otimes{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\displaystyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\textstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptstyle\otimes$}}}}}{\mathbin{\vbox{\hbox{\scalebox{1.5}{$\scriptscriptstyle\otimes$}}}}}}_{q=1}^{n}\varphi_{\omega(q)}\big{\rangle},$
(14.6)

where and $\psi_{q}\in\ \mathscr{S}(\mathbb{M})$,
$\varphi_{q}\in\mathscr{S}(\mathbb{M}^{m_{q}})$ for $q\in\\{1,\ldots,n\\}$.

######  Remark 14.4.

Recall that $\mathcal{P}_{n}$ is the permutation group. The map
$\mathbf{Y}_{\pi}:\mathcal{V}^{\mathsf{m}}\to\mathcal{V}^{\mathsf{m}}$ is well
defined and bounded with norm one. The map
$\mathbf{Y}^{\omega}:\mathcal{V}^{\mathsf{m}}\to\mathcal{V}^{\mathsf{m}}$ is
also well-defined but in general not bounded because of the distinguished role
of the variable $x_{1}$ in Def. 14.1 of the norm
$\|{\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}\|_{\mathcal{V}^{\mathsf{m}}}$.

######  Definition 14.5.

Let $n\in\mathbb{N}_{+}$ and $V\in\mathscr{S}^{\prime}(\mathbb{M}^{n})$. We
say that $V$ is translationally invariant iff $\langle
V,\psi_{1}\otimes\ldots\otimes\psi_{n}\rangle=\langle
V,\psi_{1}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip
1.42262pt}}}}}}+x)\otimes\ldots\otimes\psi_{n}({\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\displaystyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\textstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptstyle\bullet$\hskip
1.42262pt}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.6}{\hskip
1.42262pt$\scriptscriptstyle\bullet$\hskip 1.42262pt}}}}}}+x)\rangle$ for any
$x\in\mathbb{M}$ and $\psi_{1},\ldots,\psi_{n}\in\mathscr{S}(\mathbb{M})$.

######  Remark 14.6.

$\|\mathbf{Y}^{\omega}V\|_{\mathcal{V}^{\mathsf{m}}}\leq\|V\|_{\mathcal{V}^{\mathsf{m}}}$
for translationally-invariant $V\in\mathcal{V}^{\mathsf{m}}$ since

$\|V\|_{\mathcal{V}^{\mathsf{m}}}=(2\pi)^{-\mathsf{d}}\int_{\mathbb{T}^{n}}[V](x_{1},\ldots,x_{n})\,\mathrm{d}x_{1}\ldots\mathrm{d}x_{n}.$
(14.7)

######  Definition 14.7.

Fix $n\in\mathbb{N}_{0}$, $\hat{n}\in\\{1,\ldots,n\\}$,
$m_{1},\ldots,m_{n+1}\in\mathbb{N}_{0}$ and

$\begin{gathered}\mathsf{m}=(m_{1}+m_{n+1},m_{2},\ldots,m_{n})\in\mathbb{N}_{0}^{n},\qquad\tilde{\mathsf{m}}=(m_{1}+1,m_{2},\ldots,m_{n+1})\in\mathbb{N}_{0}^{n+1},\\\
\hat{\mathsf{m}}=(m_{1}+1,m_{2},\ldots,m_{\hat{n}})\in\mathbb{N}_{0}^{\hat{n}},\qquad\check{\mathsf{m}}=(m_{\hat{n}+1},\ldots,m_{n+1})\in\mathbb{N}_{0}^{n-\hat{n}+1}.\end{gathered}$
(14.8)

The bilinear map
$\mathbf{A}\,:\,\mathscr{S}(\mathbb{M})\times\mathcal{V}^{\tilde{\mathsf{m}}}\to\mathcal{V}^{\mathsf{m}}$
is defined by

$\mathbf{A}(G,V)(x_{1},\ldots,x_{n};\mathrm{y}_{1},\mathrm{y}_{n+1},\mathrm{y}_{2},\ldots,\mathrm{y}_{n})\\\
:=\int_{\mathbb{M}^{2}}V(x_{1},\ldots,x_{n+1};y,\mathrm{y}_{1}\ldots\mathrm{y}_{n+1})\,G(y-x_{n+1})\,\mathrm{d}y\mathrm{d}x_{n+1}.$
(14.9)

The trilinear map
$\mathbf{B}\,:\,\mathscr{S}(\mathbb{M})\times\mathcal{V}^{\hat{\mathsf{m}}}\times\mathcal{V}^{\check{\mathsf{m}}}\to\mathcal{V}^{\mathsf{m}}$
is defined by

$\mathbf{B}(G,W,U)(x_{1},\ldots,x_{n};\mathrm{y}_{1},\mathrm{y}_{n+1},\mathrm{y}_{2},\ldots,\mathrm{y}_{n}):=\int_{\mathbb{M}^{2}}W(x_{1},\ldots,x_{\hat{n}};y,\mathrm{y}_{1},\ldots,\mathrm{y}_{\hat{n}})\\\
\times\,G(y-x_{n+1})\,U(x_{\hat{n}+1},\ldots,x_{n+1};\mathrm{y}_{\hat{n}+1},\ldots,\mathrm{y}_{n+1})\,\mathrm{d}y\mathrm{d}x_{n+1}.$
(14.10)

In the above equations $\mathrm{y}_{j}\in\mathbb{M}^{m_{j}}$,
$j\in\\{1,\ldots,n+1\\}$.

######  Remark 14.8.

The map $\mathbf{B}$ is a generalization of the map $\mathbf{B}$ introduced in
Def. 5.6. The maps $\mathbf{A}$ and $\mathbf{B}$ appear on the RHS of the flow
equation for the cumulants of the effective force coefficients introduced in
Sec. 15.

######  Lemma 14.9.

If $V$, $W$ and $U$ are translationally invariant, then $\mathbf{A}(G,V)$ and
$\mathbf{B}(G,W,U)$ are also translationally invariant and it holds

$\|\mathbf{A}(G,V)\|_{\mathcal{V}^{\mathsf{m}}}\leq\|\mathbf{T}G\|_{\mathcal{V}}\,\|V\|_{\mathcal{V}^{\tilde{\mathsf{m}}}},$
(14.11)
$\|\mathbf{B}(G,W,U)\|_{\mathcal{V}^{\mathsf{m}}}\leq\|G\|_{\mathcal{K}}\,\|W\|_{\mathcal{V}^{\check{\mathsf{m}}}}\,\|U\|_{\mathcal{V}^{\hat{\mathsf{m}}}},$
(14.12)

where $\mathbf{T}G$ is the periodization of $G$ introduced in Def. 4.6.

###### Proof.

The translational invariance follows easily from the definition of the maps
$\mathbf{A}$ and $\mathbf{B}$. Using Def. 14.1, Def. 4.6 and Remark 4.7 we
obtain:

$\begin{gathered}(x_{1},\ldots,x_{n})\leq\|\mathbf{T}G\|_{L^{\infty}(\mathbb{M})}\,\,\int_{\mathbb{T}}[V](x_{1},\ldots,x_{n},x)\,\mathrm{d}x,\\\
[\mathbf{B}(G,W,U)](x_{1},\ldots,x_{n})\leq\|G\|_{L^{1}(\mathbb{M})}\,[W](x_{1},\ldots,x_{\hat{n}})\,\sup_{x\in\mathbb{M}}\,[U](x_{\hat{n}+1},\ldots,x_{n},x).\end{gathered}$
(14.13)

The statement follows now from Remark 14.6. ∎

######  Remark 14.10.

The fact that
$P_{\mu}^{\mathtt{g}}K_{\mu}^{\ast{\mathtt{g}}}=\delta_{\mathbb{M}}$ implies
that for any $\mu>0$ it holds

$\begin{gathered}K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m)}\ast\mathbf{A}(G,V)=\mathbf{A}\big{(}\mathbf{P}^{2{\mathtt{g}}}_{\mu}G,K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m+2)}\ast
V\big{)},\\\
K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m)}\ast\mathbf{B}(G,W,U)=\mathbf{B}\big{(}\mathbf{P}^{2{\mathtt{g}}}_{\mu}G,K_{\mu}^{\ast{\mathtt{g}},\otimes(\hat{n}+\hat{m}+1)}\ast
W,K_{\mu}^{\ast{\mathtt{g}},\otimes(n-\hat{n}+\check{m}+1)}\ast
U\big{)},\end{gathered}$ (14.14)

where $m=m_{1}+\ldots+m_{n+1}$, $\hat{m}=m_{1}+\ldots+m_{\hat{n}}$ and
$\check{m}=m_{\hat{n}+1}+\ldots+m_{n+1}$. Note also that by Lemma 4.8 (B)

$\|\mathbf{T}\mathbf{P}^{2{\mathtt{g}}}_{\mu}G\|_{\mathcal{V}}\leq\|\mathbf{T}K_{\mu}^{\ast\mathsf{d}}\|_{\mathcal{V}}\|\mathbf{P}^{2{\mathtt{g}}+\mathsf{d}}_{\mu}G\|_{\mathcal{K}}\lesssim[\mu]^{-\mathsf{d}}\,\|\mathbf{P}^{2{\mathtt{g}}+\mathsf{d}}_{\mu}G\|_{\mathcal{K}}$
(14.15)

uniformly in $\mu\in(0,1]$.

##  15 Flow equation for cumulants

######  Lemma 15.1.

Let $n\in\mathbb{N}_{+}$, $i_{1}\in\mathbb{N}_{0}$,
$m_{1},\ldots,m_{n}\in\mathbb{N}_{0}$, $a_{1}\in\mathfrak{M}^{m_{1}}$,
$r_{1}\in\\{0,1,2\\}$ and $I\equiv\\{2,\ldots,n\\}$. For any random
distributions $\zeta_{q}\in\mathcal{D}^{m_{q}}$, $q\in I$, the following flow
equation holds

$\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\partial_{\mu}\partial_{\kappa}^{r_{1}}F^{i_{1},m_{1},a_{1}}_{\kappa,\mu},(\zeta_{q})_{q\in
I}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}=-\frac{1}{m_{1}!}\sum_{\pi\in\mathcal{P}_{m_{1}}}\sum_{j=0}^{i_{1}}\sum_{k=0}^{m_{1}}\sum_{u+v+w=r_{1}}\sum_{b+c+d=\pi(a_{1})}(1+k)\,\\\
\times\Bigg{(}\sum_{\begin{subarray}{c}I_{1},I_{2}\subset I\\\ I_{1}\cup
I_{2}=I\end{subarray}}\mathbf{Y}_{\pi}\mathbf{B}\big{(}\mathcal{X}^{c}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu},\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\partial_{\kappa}^{v}F^{j,1+k,b}_{\kappa,\mu},(\zeta_{q})_{q\in
I_{1}}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}},\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}(\zeta_{q})_{q\in
I_{2}},\partial_{\kappa}^{w}F^{i_{1}-j,m_{1}-k,d}_{\kappa,\mu}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\big{)}\\\
+\mathbf{Y}_{\pi}\mathbf{A}\big{(}\mathcal{X}^{c}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu},\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}\partial_{\kappa}^{v}F^{j,1+k,b}_{\kappa,\mu},(\zeta_{q})_{q\in
I},\partial_{\kappa}^{w}F^{i_{1}-j,m_{1}-k,d}_{\kappa,\mu}\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\big{)}\Bigg{)},$
(15.1)

where we used the notation introduced in Def. 12.4.

###### Proof.

The statement follows immediately from Eqs. (7.8), (12.3). ∎

######  Lemma 15.2.

Let
$\mathtt{J}\equiv(\mathtt{J}_{1},\ldots,\mathtt{J}_{n})=((i_{1},m_{1},a_{1},s_{1},r_{1}),\ldots,(i_{n},m_{n},a_{n},s_{n},r_{n}))$
be a list of indices such that $s_{l}=1$ for some $l\in\\{1,\ldots,n\\}$.

  1. (A)

The distribution $E^{\mathtt{J}}_{\kappa,\mu}$ can be expressed as a linear
combination of distributions of the form

$\mathbf{Y}^{\omega}\mathbf{Y}_{\pi}\mathbf{A}\big{(}\mathcal{X}^{c}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu},E^{\mathtt{K}}_{\kappa,\mu}\big{)}\qquad\textrm{or}\qquad\mathbf{Y}^{\omega}\mathbf{Y}_{\pi}\mathbf{B}\big{(}\mathcal{X}^{c}\partial_{\kappa}^{u}\partial_{\mu}G_{\kappa,\mu},E^{\mathtt{L}}_{\kappa,\mu},E^{\mathtt{M}}_{\kappa,\mu}\big{)},$
(15.2)

where $u\in\\{0,\ldots,r_{l}\\}$, $c\in\mathfrak{M}$ is some multi-index,
$\mathtt{K}$, $\mathtt{L}$, $\mathtt{M}$ are some lists of indices and
$\omega\in\mathcal{P}_{n}$, $\pi\in\mathcal{P}_{m_{l}}$ are some permutation.

  2. (B)

The lists of indices $\mathtt{K}$, $\mathtt{L}$, $\mathtt{M}$ satisfy the
following conditions

$\begin{split}&n(\mathtt{K})=n(\mathtt{J})+1,\\\
&i(\mathtt{K})=i(\mathtt{J}),\\\ &m(\mathtt{K})=m(\mathtt{J})+1,\\\
&a(\mathtt{K})+|c|=a(\mathtt{J}),\\\ &s(\mathtt{K})=s(\mathtt{J})-1,\\\
&r(\mathtt{K})=r(\mathtt{J})-r,\\\
&\varrho_{3\varepsilon}(\mathtt{J})-\sigma=\varrho_{3\varepsilon}(\mathtt{K})-\mathsf{d}-3\varepsilon,\end{split}~{}\textrm{or}\qquad\begin{split}&n(\mathtt{L})+n(\mathtt{M})=n(\mathtt{J})+1,\\\
&i(\mathtt{L})+i(\mathtt{M})=i(\mathtt{J}),\\\
&m(\mathtt{L})+m(\mathtt{M})=m(\mathtt{J})+1,\\\
&a(\mathtt{L})+a(\mathtt{M})+|c|=a(\mathtt{J}),\\\
&s(\mathtt{L})+s(\mathtt{M})=s(\mathtt{J})-1,\\\
&r(\mathtt{L})+r(\mathtt{M})=r(\mathtt{J})-r,\\\
&\varrho_{3\varepsilon}(\mathtt{J})-\sigma=\varrho_{3\varepsilon}(\mathtt{L})+\varrho_{3\varepsilon}(\mathtt{M})-3\varepsilon.\end{split}$
(15.3)

  3. (C)

Fix some ${\mathtt{g}}\in\mathbb{N}_{0}$. Suppose that the bound

$\big{\|}K^{\ast{\mathtt{g}},\otimes(n(\mathtt{I})+m(\mathtt{I}))}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu}\big{\|}_{\mathcal{V}^{\mathsf{m}(\mathtt{I})}}\\\
\lesssim[\kappa]^{(\varepsilon-\sigma)r(\mathtt{I})}[\mu]^{\varrho_{3\varepsilon}(\mathtt{I})-\sigma
s(\mathtt{I})+(n(\mathtt{I})-1)\mathsf{d}}$ (15.4)

holds uniformly in $\kappa,\mu\in(0,1/2]$ for all lists of indices
$\mathtt{I}\in\\{\mathtt{K},\mathtt{L},\mathtt{M}\\}$, where
$\mathtt{K},\mathtt{L},\mathtt{M}$ are arbitrary lists of indices satisfying
the conditions specified in Part (B) given the list of indices $\mathtt{J}$.
Then the above bound holds uniformly in $\kappa,\mu\in(0,1/2]$ for
$\mathtt{I}=\mathtt{J}$.

###### Proof.

Without loss of generality we can assume that $l=1$. Part (A) of the theorem
follows immediately from Lemma 15.1 applied with

$\zeta_{q}\equiv\partial_{\mu}^{s_{q}}\partial_{\kappa}^{r_{q}}F^{i_{q},m_{q},a_{q}}_{\kappa,\mu},\qquad
q\in\\{2,\ldots,n\\}.$ (15.5)

The multi-index $c\in\mathfrak{M}$, $u\in\\{0,\ldots,r_{1}\\}$ and the
permutation $\pi\in\mathcal{P}_{m_{1}}$ in the statement coincide with the
respective objects in Eq. (15.1). The permutation $\omega\in\mathcal{P}_{n}$
is trivial because of the assumption $l=1$. Moreover, it holds

$\begin{gathered}\mathtt{K}=((j,1+k,b,0,v),\,\mathtt{J}_{2},\ldots,\mathtt{J}_{n},\,(i_{1}-j,m_{1}-k,d,0,w))\\\
\mathtt{L}\sqcup\mathtt{M}=((j,1+k,b,0,v),\,\mathtt{J}_{2},\ldots\mathtt{J}_{n},\,(i_{1}-j,m_{1}-k,d,0,w)),\end{gathered}$
(15.6)

where $\mathtt{L}\sqcup\mathtt{M}$ denotes the concatenation of the lists
$\mathtt{L}$, $\mathtt{M}$ and $j\in\\{1,\ldots,i_{1}\\}$,
$k\in\\{0,\ldots,m_{1}\\}$, $b\in\mathfrak{M}^{1+k}$,
$d\in\mathfrak{M}^{m-k}$, $v,w\in\\{0,\ldots,r_{1}\\}$ coincide with the
respective objects in Eq. (15.1). This implies that the lists $\mathtt{K}$,
$\mathtt{L}$, $\mathtt{M}$ satisfy the conditions given in Part (B). The last
condition follows from Def. (12.6) and
$\dim(\Xi)+\dim(\Phi)=\mathsf{d}-\sigma$. To prove Part (C) we use Parts (A),
(B), Lemma 14.9, Remark 14.10 and Lemma 6.5 applied with $r\in\\{0,1,2\\}$. ∎

##  16 Uniform bounds for cumulants

######  Theorem 16.1.

There exists a unique choice of the counterterms $c^{[i]}_{\kappa}$ in Eq.
(1.3) such that: $\mathbb{E}f^{1,3,0}_{\kappa,1/2}=1$,
$\mathbb{E}f^{i,1,0}_{\kappa,1/2}=\mathfrak{f}^{[i]}$,
$i\in\\{1,\ldots,i_{\sharp}\\}$, and $\mathbb{E}f^{i,m,a}_{\kappa,1/2}=0$ for
all other $i\in\mathbb{N}_{+}$, $m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$
such that $\varrho(i,m)+|a|\leq 0$. Fix arbitrary
$\mathfrak{f}^{[i]}\in\mathbb{R}$, $i\in\\{1,\ldots,i_{\sharp}\\}$. For all
list of indices $\mathtt{I}$ there exists ${\mathtt{g}}\in\mathbb{N}_{0}$ such
that the bound

$\|K^{\ast{\mathtt{g}},\otimes(n+m)}_{\mu}\ast
E^{\mathtt{I}}_{\kappa,\mu}\|_{\mathcal{V}^{\mathsf{m}}}\lesssim[\kappa]^{(\varepsilon-\sigma)r(\mathtt{I})}[\mu]^{\varrho_{3\varepsilon}(\mathtt{I})-\sigma
s(\mathtt{I})+(n-1)\mathsf{d}}$ (16.1)

holds uniformly in $\kappa,\mu\in(0,1/2]$, where $n=n(\mathtt{I})$,
$\mathsf{m}=\mathsf{m}(\mathtt{I})$, $m=m(\mathtt{I})$. Moreover, the
following condition is satisfied

$E^{\mathtt{I}}_{\kappa,\mu}(x_{1},\ldots,x_{n};\mathrm{y}_{1},\ldots,\mathrm{y}_{m})=(-1)^{a(\mathtt{I})}E^{\mathtt{I}}_{\kappa,\mu}(-x_{1},\ldots,-x_{n};-\mathrm{y}_{1},\ldots,-\mathrm{y}_{m}),$
(16.2)

where $x_{j}\in\mathbb{M}$, $\mathrm{y}_{j}\in\mathbb{M}^{m_{j}}$ for
$j\in\\{1,\ldots,n\\}$ and $E^{\mathtt{I}}_{\kappa,\mu}=0$ unless $m+n\in
2\mathbb{N}_{0}$.

######  Remark 16.2.

By stationarity $\mathbb{E}f^{i,m,a}_{\kappa,\mu}(x)$ is a constant. Since
$\partial_{\mu}F^{1,3}_{\kappa,\mu}=0$ it holds
$\mathbb{E}f^{1,3,0}_{\kappa,1/2}=f^{1,3,0}_{\kappa,0}=1$. There is no
distinguished value of $\mathbb{E}f^{i,1,0}_{\kappa,1/2}=\mathfrak{f}^{[i]}$,
$i\in\\{1,\ldots,i_{\sharp}\\}$, ultimately, because there is no distinguished
function $\chi$ in Def. 2.1 of the cutoff propagator $G_{\kappa}$. The
vanishing of $\mathbb{E}f^{i,m,a}_{\kappa,1/2}$ for all other
$i\in\mathbb{N}_{+}$, $m\in\mathbb{N}_{0}$, $a\in\mathfrak{M}^{m}$ such that
$\varrho(i,m)+|a|\leq 0$ is enforced by the properties of the cumulants given
in the last sentence of the theorem (here we use the assumption
$\mathsf{d}\in\\{1,\ldots,6\\}$). Observe that these properties are
consequences of the following symmetries of Eq. (1.2):
$\varPhi(x)\mapsto-\varPhi(x)$, $\xi(x)\mapsto-\xi(x)$ and
$\varPhi(x)\mapsto\varPhi(-x)$, $\xi(x)\mapsto\xi(-x)$, which in particular
preserve the law of $\xi$. The counterterms $c^{[i]}_{\kappa}$ are related to
the renormalization parameters $\mathfrak{f}^{[i]}$ by the following formula

$c^{[i]}_{\kappa}:=f^{i,1,0}_{\kappa}=f^{i,1,0}_{\kappa,0}=\mathfrak{f}^{[i]}-\int_{0}^{1/2}\mathbb{E}\,\partial_{\mu}f^{i,1,0}_{\kappa,\mu}\,\mathrm{d}\mu,\qquad
i\in\\{1,\ldots,i_{\sharp}\\}.$ (16.3)

The constants $\mathfrak{f}^{[i]}\in\mathbb{R}$,
$i\in\\{1,\ldots,i_{\sharp}\\}$ parameterize the class of solutions of Eq.
(1.2) constructed in the paper (generically this is an over-parametrization).

###### Proof.

The base case: We first prove the theorem for list of indices $\mathtt{I}$
such that $i(\mathtt{I})=0$. In this case the cumulants
$E^{\mathtt{I}}_{\kappa,\mu}$ coincide with the cumulants of the white noise
$\xi$. The only non-vanishing cumulant is the covariance which corresponds to
$n(\mathtt{I})=2$, $\mathsf{m}(\mathtt{I})=(0,0)$, $m(\mathtt{I})=0$,
$a(\mathtt{I})=0$, $r(\mathtt{I})=0$ and $s(\mathtt{I})=0$. It holds

$\|\mathopen{\hbox{${\langle}$}\kern-2.33336pt\leavevmode\hbox{${\langle}$}}K_{\mu}^{\ast{\mathtt{g}}}\ast\xi,K_{\mu}^{\ast{\mathtt{g}}}\ast\xi\mathclose{\hbox{${\rangle}$}\kern-2.33336pt\leavevmode\hbox{${\rangle}$}}\|_{\mathcal{V}^{\mathsf{m}}}\leq\sup_{x_{2}\in\mathbb{T}}\int_{\mathbb{T}}|\mathbb{E}(\xi(\mathrm{d}x_{1})\xi(x_{2}))|=1.$
(16.4)

This finishes the proof of the base case.

Induction step: Fix $i\in\mathbb{N}_{+}$ and $m\in\mathbb{N}_{0}$. Assume that
the theorem is true for all lists of indices $\mathtt{I}$ such that either
$i(\mathtt{I})<i$, or $i(\mathtt{I})=i$ and $m(\mathtt{I})>m$. We shall prove
the theorem for all $\mathtt{I}$ such that $i(\mathtt{I})=i$ and
$m(\mathtt{I})=m$.

Consider the case $s(\mathtt{I})>0$. In this case by Lemma 15.2 (A), (B) the
cumulants $E^{\mathtt{I}}_{\kappa,\mu}$ can be expressed in terms of the
cumulants for which the statement of the theorem has already been established.
As a result, the bound (16.1) with $s(\mathtt{I})>0$ follows from the
inductive assumption and Lemma 15.2 (C).

Now consider
$\mathtt{I}=((i_{1},m_{1},a_{1},0,r_{1}),\ldots,(i_{n},m_{n},a_{n},0,r_{n}))$,
$s(\mathtt{I})=0$. It follows from Def. 12.5 of the cumulants
$E^{\mathtt{I}}_{\kappa,\mu}$ that

$E^{\mathtt{I}}_{\kappa,\mu}=E^{\mathtt{I}}_{\kappa,0}+\sum_{q=1}^{n}\int_{0}^{\mu}E^{\mathtt{I}_{q}}_{\kappa,\eta}\,\mathrm{d}\eta,\quad
E^{\mathtt{I}}_{\kappa,\mu}=E^{\mathtt{I}}_{\kappa,1/2}-\sum_{q=1}^{n}\int_{\mu}^{1/2}E^{\mathtt{I}_{q}}_{\kappa,\eta}\,\mathrm{d}\eta,$
(16.5)

where
$\mathtt{I}_{q}=((i_{1},m_{1},a_{1},0,r_{1}),\ldots,(i_{q},m_{q},a_{q},1,r_{q}),\ldots,(i_{n},m_{n},a_{n},0,r_{n}))$.
Note that $s(\mathtt{I}_{q})=1$, hence the bound (16.1) has already been
established for $E^{\mathtt{I}_{q}}_{\kappa,\eta}$. We will use the first of
Eqs. (16.5) to bound the irrelevant cumulants $E^{\mathtt{I}}_{\kappa,\mu}$,
i.e. those with $\mathtt{I}$ such that
$\varrho(\mathtt{I})+(n(\mathtt{I})-1)\mathsf{d}>0$. The second equality will
be used to bound certain contributions to the relevant cumulants
$E^{\mathtt{I}}_{\kappa,\mu}$, i.e. those with $\mathtt{I}$ such that
$\varrho(\mathtt{I})+(n(\mathtt{I})-1)\mathsf{d}\leq 0$.

First, let us analyse the irrelevant contributions. If $n(\mathtt{I})>1$, then
$E^{\mathtt{I}}_{\kappa,0}$ is a joint cumulant of a list of at least two
random distributions. Since $i(\mathtt{I})=i>0$ one of the elements of this
list is a deterministic distribution of the form
$\partial^{r}_{\kappa}F^{i,m,a}_{\kappa,0}$. Hence, the cumulant vanishes. If
$n(\mathtt{I})=1$ and $\varrho(\mathtt{I})>0$, then
$E^{\mathtt{I}}_{\kappa,0}$ coincides with
$\partial^{r}_{\kappa}F^{i,m,a}_{\kappa,0}=\partial^{r}_{\kappa}F^{i,m,a}_{\kappa\phantom{0}}$
for some $i,m,a$ such that $\varrho(i,m)+|a|>0$ and consequently vanishes. To
prove the bound for $E^{\mathtt{I}}_{\kappa,0}$ we use the first of Eqs.
(16.5). As we argued above, the first term on the RHS of this equation
vanishes. The claim of the theorem is a consequence of the bound

$\|K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m)}\ast
E^{\mathtt{I}}_{\kappa,\mu}\|_{\mathcal{V}^{\mathsf{m}}}\leq\sum_{q=1}^{n}\int_{0}^{\mu}\|K_{\eta}^{\ast{\mathtt{g}},\otimes(n+m)}\ast
E^{\mathtt{I}_{q}}_{\kappa,\eta}\|_{\mathcal{V}^{\mathsf{m}}}\,\mathrm{d}\eta.$
(16.6)

We used the fact $\|K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m)}\ast
E^{\mathtt{I}_{q}}_{\kappa,\eta}\|_{\mathcal{V}^{\mathsf{m}}}\leq\|K_{\eta}^{\ast{\mathtt{g}},\otimes(n+m)}\ast
E^{\mathtt{I}_{q}}_{\kappa,\eta}\|_{\mathcal{V}^{\mathsf{m}}}$ for
$\eta\leq\mu$ which follows from Lemma 4.5.

Next, let us analyse the relevant contributions. We note that the inequality
$\varrho(\mathtt{I})+(n(\mathtt{I})-1)\mathsf{d}\leq 0$ implies
$n(\mathtt{I})=1$. Consequently, $\mathtt{I}=(i,m,a,0,r)$ for some
$r\in\\{0,1,2\\}$ and $a\in\mathfrak{M}$ such that $\varrho(i,m)+|a|\leq 0$.
Hence,
$E^{\mathtt{I}}_{\kappa,\mu}=\mathbb{E}\,\partial^{r}_{\kappa}F^{i,m,a}_{\kappa,\mu}$.
We first study

$\mathbb{E}\,\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}=\mathbf{I}(\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\mu})=\mathbf{I}(K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m)}\ast
E^{\mathtt{I}}_{\kappa,\mu})\in\mathbb{R},$ (16.7)

where the map $\mathbf{I}$ was introduced in Def. 8.4. Note that by the
translational invariance $\mathbb{E}\,f^{i,m,a}_{\kappa,\mu}$ is a constant
function. The application of the map $\mathbf{I}$ to both sides of the second
of Eqs. (16.5) yields

$\mathbb{E}\,\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}=\mathbb{E}\,\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,1/2}-\sum_{q=1}^{n}\int_{\mu}^{1/2}\mathbf{I}(K_{\mu}^{\ast{\mathtt{g}},\otimes(n+m)}\ast
E^{\mathtt{I}_{1}}_{\kappa,\mu})\,\mathrm{d}\eta,$ (16.8)

where
$E^{\mathtt{I}_{1}}_{\kappa,\mu}=\mathbb{E}\,\partial_{\eta}\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\eta}$.
Recalling that $\mathcal{V}^{m}\equiv\mathcal{V}^{\mathsf{m}}$ for $n=1$ and
using Lemma 8.5 we arrive at

$|\mathbb{E}\,\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}|\leq|\mathbb{E}\,\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,1/2}|+\sum_{q=1}^{n}\int_{\mu}^{1/2}\|K^{\ast{\mathtt{g}},\otimes(n+m)}_{\eta}\ast
E^{\mathtt{I}_{1}}_{\kappa,\eta}\|_{\mathcal{V}^{m}}\,\mathrm{d}\eta.$ (16.9)

By assumption, $\mathbb{E}\,f^{i,m,a}_{\kappa,1/2}$ is independent of $\kappa$
and $\mathbb{E}\,\partial_{\kappa}f^{i,m,a}_{\kappa,1/2}=0$. Hence, using the
bound (16.9) and the bound (16.1) applied to $E^{\mathtt{I}_{1}}_{\kappa,\mu}$
we obtain

$|\mathbb{E}\,\partial_{\kappa}^{r}f^{i,m,a}_{\kappa,\mu}|=|\mathbf{I}(\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\mu})|\lesssim
1+\int_{\mu}^{1/2}[\kappa]^{(\varepsilon-\sigma)r}[\eta]^{\varrho_{3\varepsilon}(i,m)+|a|-\sigma-\varepsilon
r}\,\mathrm{d}\eta\\\
\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{3\varepsilon}(i,m)+|a|-\varepsilon
r}.$ (16.10)

If $m=0$, then $a=0$ and
$E^{\mathtt{I}}_{\kappa,\mu}=\mathbb{E}\,\partial^{r}_{\kappa}F^{i,0,0}_{\kappa,\mu}=\mathbb{E}\,\partial_{\kappa}^{r}f^{i,0,0}_{\kappa,\mu}$.
Hence, in this case the statement of the theorem follows from the bound
(16.10). To prove the case $m>0$ we first recall that we have already proved
the following bounds

$\|K_{\mu}^{\ast\mathtt{f},\otimes(1+m)}\ast\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,b}_{\kappa,\mu}\|_{\mathcal{V}^{m}}\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{3\varepsilon}(i,m)+|b|-\varepsilon
r}$ (16.11)

for all irrelevant coefficients $F^{i,m,b}_{\kappa,\mu}$ and the following
bounds

$|\mathbf{I}(\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,b}_{\kappa,\mu})|\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{3\varepsilon}(i,m)+|b|-\varepsilon
r}$ (16.12)

for all relevant coefficients $F^{i,m,b}_{\kappa,\mu}$. Let
$\mathtt{l}\in\mathbb{N}_{+}$ be the smallest positive integer such that
$\varrho(i,m)+\mathtt{l}>0$ and note that by Theorem 8.8 it holds

$\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,a}_{\kappa,\mu}=\mathbf{X}^{a}_{\mathtt{l}}(\mathbf{I}(\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,b}_{\kappa,\mu}),\mathbb{E}\,\partial_{\kappa}^{r}F^{i,m,b}_{\kappa,\mu}).$
(16.13)

The arguments of the map $\mathbf{X}^{a}_{\mathtt{l}}$ above satisfy the
bounds (16.12) and (16.11). This together with Theorem 8.7 applied with
$C\lesssim[\kappa]^{(\varepsilon-\sigma)r}[\mu]^{\varrho_{3\varepsilon}(i,m)-\varepsilon
r}$ implies the statement of the theorem. ∎

## Acknowledgments

I wish to thank Markus Tempelmayr and Pavlos Tsatsoulis for discussions. The
main part of this work was carried out when the author was affiliated at the
Max-Planck Institute for Mathematics in the Sciences in Leipzig. The financial
support of the Max-Planck Society, grant Proj. Bez. M.FE.A.MATN0003 and
partial support from the National Science Centre, Poland, grant ‘Sonata Bis’
2019/34/E/ST1/00053 are gratefully acknowledged.

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