# A critical drift-diffusion equation: intermittent behavior

Felix Otto<EMAIL_ADDRESS>and  Christian Wagner
<EMAIL_ADDRESS>
###### Abstract.

We consider a drift-diffusion process with a time-independent and divergence-
free random drift that is of white-noise character. We are interested in the
critical case of two space dimensions, where one has to impose a small-scale
cut-off for well-posedness, and is interested in the marginally super-
diffusive behavior on large scales.

In the presence of an (artificial) large-scale cut-off at scale L, as a
consequence of standard stochastic homogenization theory, there exist harmonic
coordinates with a stationary gradient $F_{L}$; the merit of these coordinates
being that under their lens, the drift-diffusion process turns into a
martingale.

It has recently been established that the second moments diverge as
$\mathbb{E}|F_{L}|^{2}\sim\sqrt{\ln L}$ for $L\uparrow\infty$. We
quantitatively show that in this limit, and in the regime of small Péclet
number, $|F_{L}|^{2}/\mathbb{E}|F_{L}|^{2}$ is not equi-integrable, and that
$\mathbb{E}|{\rm det}F_{L}|/\mathbb{E}|F_{L}|^{2}$ is small. Hence the
Jacobian matrix of the harmonic coordinates is very peaked and non-conformal.

We establish this asymptotic behavior by characterizing a proxy
$\tilde{F}_{L}$ introduced in previous work as the solution of an Itô SDE w.
r. t. the variable $\ln L$, and which implements the concept of a scale-by-
scale homogenization based on a variance decomposition and admits an efficient
calculus. For this proxy, we establish
$\mathbb{E}|\tilde{F}_{L}|^{4}\gg(\mathbb{E}|\tilde{F}_{L}|^{2})^{2}$ and
$\mathbb{E}({\rm det}\tilde{F}_{L}-1)^{2}\ll 1$. In view of the former
property, we assimilate this phenomenon to intermittency. In fact,
$\tilde{F}_{L}$ behaves like a tensorial stochastic exponential, and as a
field can be assimilated to multiplicative Gaussian chaos.

###### Contents

  1. 1 Introduction
    1. 1.1 Drift-diffusion process $X$, Gaussian ensembles of divergence-free drifts $b$, and critical dimension $d=2$ for decorrelated ensembles
    2. 1.2 UV cut-off, Péclet number $\varepsilon$, annealed second moments, and super-diffusive behavior for $d=2$
    3. 1.3 Stream function $\psi$, elliptic coefficient field $a$, harmonic coordinates, corrector $\phi$
    4. 1.4 IR cut-off $L$, Gaussian free field, asymptotics of $\lambda_{L}$
  2. 2 Statement of result on Jacobian matrix $F={\rm id}+\nabla\phi$ of harmonic coordinates
  3. 3 Strategy of proof
    1. 3.1 Variance decomposition of $\psi$ into a martingale in $\tilde{\lambda}^{2}=1+\varepsilon^{2}\ln L$
    2. 3.2 Definition of the proxy corrector $\tilde{\phi}$ via an Itô SDE in $\tilde{\lambda}^{2}$
    3. 3.3 Intermittency of the proxy Jacobian matrix $\tilde{F}={\rm id}+\nabla\tilde{\phi}$
  4. 4 Proofs
    1. 4.1 Identification of quadratic variations
    2. 4.2 Identification of Itô SDEs
    3. 4.3 Derivation of asymptotics
    4. 4.4 Proof of main result
  5. A Identification of quadratic variation of the drivers
    1. A.1 Quadratic variation of $\nabla d\phi$
    2. A.2 Quadratic variation of $\nabla^{2}d\phi$

##  1\. Introduction

###  1.1. Drift-diffusion process $X$, Gaussian ensembles of divergence-free
drifts $b$, and critical dimension $d=2$ for decorrelated ensembles

We are interested in a drift-diffusion process in $d$-dimensional space,

(1) $\displaystyle dX_{t}=b(X_{t})dt+\sqrt{2}dW_{t}\quad\mbox{and}\quad
X_{t=0}=0$

with a time-independent and divergence-free drift $b$. The factor of
$\sqrt{2}$ is chosen such that the expectation of an observable $u$ evolves
according to $\frac{d}{dt}\mathbb{E}u(X_{t})$
$=\mathbb{E}{\mathcal{L}}u(X_{t})$, where

(2) $\displaystyle{\mathcal{L}}u:=b\cdot\nabla u+\triangle u$

is the generator of the Markov process $X$. The assumption $\nabla\cdot b=0$
ensures that the uniform distribution $\rho=const$ is in the null space of the
adjoint ${\mathcal{L}}^{*}\rho=-\nabla\cdot\rho b+\triangle\rho$. This in
particular rules out traps, i. e. local configurations of $b$ that make it
unlikely for $X$ to escape within moderate time.

We are interested in vector fields $b$ that are randomly chosen from a
stationary and centered Gaussian ensemble. Such an ensemble is characterized
by the (positive semi-definite) Fourier transform111where we use the
physicist’s normalization
$\mathcal{F}c(k)=\int{\frac{dx}{\sqrt{2\pi}^{d}}}\exp(-ik\cdot x)c(x)$
${\mathcal{F}}C(k)$ of the covariance tensor $C(x-y)=\mathbb{E}b(x)\otimes
b(y)$. Loosely speaking, the most decorrelated choice of ensemble is given by

(3) $\displaystyle{\mathcal{F}}C(k)=\varepsilon^{2}({\rm id}-\frac{k\otimes
k}{|k|^{2}}),$

which can be interpreted as the Leray projection222the orthogonal projection
onto the space of a divergence-free fields w. r. t.
$L^{2}(\mathbb{R}^{d})^{d}$, which is the Cameron-Martin space of vectorial
white noise of vectorial white noise, and where we give ourselves the freedom
of choosing some amplitude $\varepsilon\geq 0$.

Does this process share the parabolic scale invariance in law of the Wiener
process $W$, namely $\lambda W_{t}=_{law}W_{\lambda^{2}t}$? We note that we
have333In this context, “annealed law” refers to the expectation w. r. t. both
the environment $b$ and the thermal noise $W$. $\lambda
X_{t}=_{annealed\;law}X_{\lambda^{2}t}$ provided $b$ transforms in law like a
velocity under this parabolic rescaling, that is, $b(\lambda x)$
$=_{law}\frac{\lambda}{\lambda^{2}}b(x)$ $=\lambda^{-1}b(x)$. On the other
hand, it follows from (3) that $b$ scales with the inverse square-root of the
volume444Treating $\varepsilon^{2}$ as non-dimensional (as will soon be
justified), ${\mathcal{F}}C$ is non-dimensional, hence $C$ has units of
inverse volume, and thus $b$ has units of its square-root., that is,
$b(\lambda x)$ $=_{law}\frac{1}{\sqrt{\lambda^{d}}}b(x)$
$=_{law}\lambda^{-d/2}b(x)$. Hence we learn that for the ensemble (3), $d=2$
is a critical dimension.

However, for $d\geq 2$, the ensemble (3) of $b$’s is too rough to give a sense
to the generator (2). Indeed, the above scaling in law indicates that every
realization of $b$ is a Schwartz distribution of regularity no better than
$-\frac{d}{2}$; in fact, by the nature of the Kolmogorov criterion, its
quenched regularity is slightly worse555 like Brownian motion $W$ has a Hölder
regularity slightly worse than $\frac{1}{2}$, as indicated by writing
$-\frac{d}{2}-$. Hence even if the function $u$ were smooth, $b\cdot\nabla u$
would have typical regularity of $-\frac{d}{2}-$. As a consequence, a typical
solution of ${\mathcal{L}}u=0$ would have at best regularity of
$2-\frac{d}{2}-$. Even appealing to $\nabla\cdot b=0$ to rewrite $b\cdot\nabla
u$ $=\nabla\cdot ub$, these regularities would not be sufficient to give a
sense to the product $ub$, since the sum $2-d-$ of the regularities of the
factors, $2-\frac{d}{2}-$ and $-\frac{d}{2}-$, is negative. Also this argument
reveals the criticality of $d=2$.

###  1.2. UV cut-off, Péclet number $\varepsilon$, annealed second moments,
and super-diffusive behavior for $d=2$

In view of the preceding discussion, the ensemble (3) has to be regularized
for (1) to make sense. The simplest way is to introduce an ultra-violet (UV)
cut-off scale. W. l. o. g. we may non-dimensionalize s. t. this scale is
unity, so that (3) is replaced by666In our notation, $I(A)\in\\{0,1\\}$
vanishes unless the statement $A$ holds.

(4) $\displaystyle{\mathcal{F}}C(k)=\varepsilon^{2}I(|k|\leq 1)({\rm
id}-\frac{k\otimes k}{|k|^{2}}).$

As a consequence, every realization $b$ is smooth; more precisely, it varies
little on length scales $\ll 1$ and has a typical amplitude of order
$\varepsilon$. Now that $b$ has acquired a characteristic length scale, we may
associate a characteristic time scale to both diffusion and convection. In
view of the unit diffusivity, and the convection with a drift of order
$\varepsilon$, these characteristic time scales are of the order $1$ and
$\varepsilon^{-1}$, respectively. Hence the parameter $\varepsilon$ plays the
role of the Péclet number, which is defined to be the ratio of both time
scales. The Péclet number is the traditional measure of the relative strength
of convection w. r. t. diffusion.

The presence of the UV cut-off breaks any scale invariance in law. It
introduces non-universal behavior on length scales $\lesssim 1$ so that in
order to see whether $X$ behaves like a diffusion, we should consider $t\gg
1$. The simplest question is to monitor the annealed second moments
$\mathbb{E}(\xi\cdot X_{t})^{2}$ in some direction $\xi\in\mathbb{R}^{d}$ with
$|\xi|=1$; the word “annealed” is to indicate that the expectation is taken
both with respect to the thermal noise $W$ and the environmental noise $b$.
Diffusive behavior means that $\mathbb{E}(\xi\cdot X_{t})^{2}$ scales as
Brownian motion $\mathbb{E}(\xi\cdot W_{t})^{2}=t$.

We digress by giving a heuristic argument that for the regularized ensemble
(4), diffusive behavior of $X$ is (only) to be expected for $d>2$. Arguing by
self-consistency, if $X$ behaves diffusively, we may think of it as a random
walk on a unit lattice, unity being the characteristic length of $b$ discussed
above (and the only characteristic length scale present). Such a random walk
is transient777meaning that it does not return to the same lattice point after
some time iff $d>2$. Since in view of (4), $b$ decorrelates over distances
$\gg 1$, the contribution $b(X_{t})dt$ in (1) thus decorrelates over time.
Hence the drift acts as a (statistically independent) diffusion, increasing
the overall (effective) diffusivity. This indeed can be made rigorous by the
approach of Subsection 1.3, as done in [15, 23].

Let us return to the critical dimension $d=2$ and the asymptotic behavior of
annealed second moments $\mathbb{E}(\xi\cdot X_{t})^{2}$, this time for the
ensemble (4). In the physics literature, cf. [12], the (borderline) super-
diffusive behavior

(5) $\mathbb{E}(\xi\cdot X_{t})^{2}\sim t\sqrt{\ln t}\quad{\rm
as}~{}t\rightarrow\infty$

was informally derived by a renormalization group argument, which is
reproduced in [6, Appendix]. A first rigorous proof of some super-diffusive
behavior was given in [24]. Up to double logarithms and modulo the Laplace
transform, the scaling (5) was rigorously established in [8]. The double
logarithms and the Laplace transform were eliminated in [6]. Moreover, precise
asymptotics were identified in the regime of small Péclet number:

(6) $\frac{\mathbb{E}(\xi\cdot
X_{t})^{2}}{2t\sqrt{1+\frac{1}{2}\varepsilon^{2}\ln t}}\approx 1\quad\mbox{for
all}\;t\geq 1\quad\mbox{provided}\;\varepsilon\ll 1.$

The method of [24, 8] lifts the analysis to the infinite-dimensional Markov
process of the environment as seen from the particle $X$ (as the jargon goes),
which is induced by the generator (2) when replacing spatial derivatives by
what occasionally is called horizontal derivatives. This has the advantage of
naturally extending to a time-dependent Markovian environment [9]. Based on
the Gaussian nature of the underlying (white) noise, the analysis of the
generator relies on the Wiener chaos decomposition of observables. This
functional-analytic approach has also been employed for stochastic PDE that
define a (field-valued) stationary Markov process. Next to the Gaussianity of
the underlying stationary measure on configuration space, it relies on the
control of the skew-symmetric part of the generator by its symmetric one. This
has been carried out for the stochastic Burgers equation999the KPZ equation
formulated in terms of the slope in [14]. It has also been implemented in
scaling-critical cases, like the anisotropic KPZ equation in dimension $d=2$,
see [7]. Very recently, [1] reports a quenched version of (6) valid for all
$\varepsilon$. The method is closer to ours, see Subsection 3.1 for a (short)
discussion.

###  1.3. Stream function $\psi$, elliptic coefficient field $a$, harmonic
coordinates, corrector $\phi$

The method for (6) from [6], on which this paper is based, is much more
specific to the drift-diffusion process (1). It appeals to stochastic
homogenization theory for elliptic equations in divergence form, which we
review now. To the best of our knowledge, [6] was the first work to implement
scale-by-scale homogenization reminiscent of the informal renormalization
group argument in [12] for the critical problem at hand. The condition
$\nabla\cdot b=0$ means that the vector field $b$, seen as a $(d-1)$-form, is
closed. Hence there exists a $(d-2)$-form $\psi$ such that $b$ can be written
as the exterior derivative of $\psi$. As a consequence of (exterior) calculus,
we may represent the generator ${\mathcal{L}}$ as a purely second-order
operator in divergence form:

(7) $\displaystyle{\mathcal{L}}u=\nabla\cdot a\nabla u\quad\mbox{where}\quad
a:={\rm id}+\psi;$

note that (the non-symmetric) $a$ is uniformly positive definite, so that
${\mathcal{L}}$ is elliptic. In the case of $d=2$, $\psi$ can be identified
with a scalar known as stream function, and (7) turns into

(10) $\displaystyle b=J\nabla\psi\quad\mbox{and}\quad a:={\rm id}+\psi
J\quad\mbox{where}\quad J:=\left(\begin{array}[]{rr}0&-1\\\
1&0\end{array}\right).$

It can be easily seen that iff $d>2$, the stationary and centered Gaussian
ensemble (4) of $b$’s is such that there exists a stationary and centered
Gaussian ensemble of skew symmetric matrices $\psi$. Then it follows from
qualitative homogenization theory for stationary and ergodic ensembles of
uniformly elliptic coefficient fields $a$, as initiated in [16, 21], and
adapted to the present case in [20], that $X$ displays diffusive behavior.
More precisely, there exists a $\lambda\in(0,\infty)$ such that for all $\xi$

(11)
$\displaystyle\lambda|\xi|^{2}=\lim_{t\uparrow\infty}\frac{\mathbb{E}(\xi\cdot
X_{t})^{2}}{2t}.$

The statement (11) remains valid almost surely in $b$ when only the
expectation w. r. t. $W$ is taken, as a consequence of what is called quenched
invariance principle.

In order to establish (11), one deforms the affine coordinate
$x\mapsto\xi\cdot x$ into what is called a harmonic coordinate $u$ solving
${\mathcal{L}}u=0$. The merit of $u$ is that $u(X)$ becomes a martingale:

(12) $\displaystyle du(X)=\nabla u(X)\cdot dW,$

which follows from an application of Itô calculus to (1). Making the Ansatz
$u(x)=\xi\cdot x+\phi(x)$, this leads to the PDE $\nabla\cdot
a(\xi+\nabla\phi)=0$ for what is called the corrector $\phi$. By an adaptation
of the Lax-Milgram argument, see [11] for a modern presentation, the above-
mentioned theory establishes that there exists a unique stationary gradient
field $\nabla\phi$ of finite second moments with

(13) $\displaystyle\nabla\cdot
a(\xi+\nabla\phi)=0\quad\mbox{and}\quad\mathbb{E}\nabla\phi=0;$

by stationarity of $\nabla\phi$ we may suppress the argument $x$ in
$\mathbb{E}\nabla\phi=0$. This theory also gives a representation for
$\lambda$ in (11) as the expectation of the (stationary) flux
$a(\xi+\nabla\phi)$

(14) $\displaystyle\lambda\xi=\mathbb{E}a(\xi+\nabla\phi).$

With help of (13), (14) can be reformulated as $\lambda|\xi|^{2}$
$=\mathbb{E}(\xi+\nabla\phi)\cdot a(\xi+\nabla\phi)$. In view of the form (7),
and the second item in (13), this yields

(15)
$\displaystyle\lambda=\mathbb{E}|\xi+\nabla\phi|^{2}=1+\mathbb{E}|\nabla\phi|^{2}\quad\mbox{provided}\;|\xi|=1.$

In particular, a divergence-free drift always increases the diffusivity. The
stochastic homogenization theory for a non-divergence-free drift is much more
subtle, and only established for $d>2$ and $\varepsilon\ll 1$, see [4] for a
somewhat informal and [22] for the rigorous proof.

###  1.4. IR cut-off $L$, Gaussian free field, asymptotics of $\lambda_{L}$

As explained, in $d=2$, the ensemble (4) does not allow for a straightforward
application of stochastic homogenization. In [10], it was suggested to
quantify the super-diffusivity by monitoring the scaling of the effective
diffusivity $\lambda_{L}$ in an artificial large-scale (infra-red) cut-off
$L$. This approach was carried out in [6] in the following way: The infra-red
(IR) cut-off is implemented on the level of the ensemble by replacing (4) by

(16) $\displaystyle{\mathcal{F}}C_{L}(k)=\varepsilon^{2}I(L^{-1}<|k|\leq
1)({\rm id}-\frac{k\otimes k}{|k|^{2}}).$

Since $|k|^{2}{\rm id}-k\otimes k$ $=Jk\otimes Jk$, this ensemble of drifts
$b_{L}$ arises from a (stationary and centered) ensemble of stream functions
$\psi_{L}$ characterized by its (scalar) covariance function
$c_{L}(x-y)=\mathbb{E}\psi_{L}(x)\psi_{L}(y)$ given by its Fourier transform

(17) $\displaystyle{\mathcal{F}}c_{L}(k)=\varepsilon^{2}I(L^{-1}<|k|\leq
1)\frac{1}{|k|^{2}}.$

This ensemble is known as the Gaussian free field, with UV and IR cut-off. The
necessity of both cut-offs for stationarity becomes obvious from the
logarithmic divergence in

(18)
$\displaystyle\mathbb{E}\psi^{2}_{L}=c_{L}(0)=\varepsilon^{2}\int\frac{dk}{2\pi}{\mathcal{F}}c_{L}(k)=\varepsilon^{2}\ln
L.$

In the presence of a stationary ensemble of stream functions, stochastic
homogenization applies, see Subsection 1.3. Setting

(19) $\displaystyle a_{L}:={\rm id}+\psi_{L}J,$

cf. (10), there exists a unique stationary gradient field101010There is no
reason the believe that $\phi_{L}$ itself is stationary. $\nabla\phi_{L}$ of
finite second moments with

(20) $\displaystyle\nabla\cdot
a_{L}(\xi+\nabla\phi_{L})=0\quad\mbox{and}\quad\mathbb{E}\nabla\phi_{L}=0,$

cf. (13). Monitoring

$\displaystyle\lambda_{L}\xi=\mathbb{E}a_{L}(\xi+\nabla\phi_{L}),$

cf. (14), it is established in [6] that

(21) $\displaystyle\lambda_{L}\approx\sqrt{1+\varepsilon^{2}\ln
L}\quad\mbox{provided}\;\varepsilon\ll 1.$

Loosely speaking, in [6], (6) is derived from (21) via approximating
$\mathbb{E}(\xi\cdot X_{t})^{2}$ by $\lambda_{L}t$ for $L=\sqrt{t}$.

##  2\. Statement of result on Jacobian matrix $F={\rm id}+\nabla\phi$ of
harmonic coordinates

Let us denote by $\nabla\phi_{L}^{i}$ the solution of (20) with $\xi=e^{i}$,
where $e^{i}$ denotes the Cartesian vector in direction $i=1,2$. In this
paper, we analyze the harmonic coordinates $x\mapsto
u_{L}(x):=x+(\phi_{L}^{1}(x),\phi_{L}^{2}(x))$ of the cut-off ensemble (16).
More precisely, we study its stationary Jacobian matrix field

(22) $\displaystyle F_{L}:=\nabla u_{L}={\rm
id}+(\nabla\phi_{L}^{1},\nabla\phi_{L}^{2})$

in the limit $L\uparrow\infty$ and regime $\varepsilon\ll 1$. We already know
from (15) and (21) that

(23) $\displaystyle\mathbb{E}|F_{L}|^{2}=2\lambda_{L}\approx
2\sqrt{1+\varepsilon^{2}\ln L}\quad\mbox{for}\;\varepsilon\ll 1,$

where $|F_{L}|^{2}={\rm tr}F_{L}^{*}F_{L}$ denotes the squared Frobenius norm
of $F_{L}$.

Analogous to (12), the harmonic coordinates $x\mapsto u_{L}(x)$ convert the
process $X_{L}$, defined via (1) with $b$ replaced by $b_{L}$, into a two-
component martingale. In view of (12), the quadratic expression
$F_{L}^{*}F_{L}$ of its Jacobian matrix $F_{L}=\nabla u_{L}$ governs the
quadratic variation of this martingale. We thus consider the study of the
asymptotics of $F_{L}$ as a first step towards understanding the finer
structure of the process $X$.

######  Theorem 2.1 (Intermittency).

For any $\alpha<\frac{1}{2}$, provided111111 The regime of $\varepsilon$ and
$L$ in which (25) holds shrinks as $\alpha\uparrow\frac{1}{2}$.
$\varepsilon\ll 1$ and $\lambda_{L}\gg 1$ we have121212Here and in the sequel,
$A\lesssim B$ means $A\leq CB$ for some universal constant $C$, i. e.
independent of the parameters $\varepsilon$ and $L$.

(24) $\displaystyle\mathbb{E}|\det
F_{L}|\lesssim\varepsilon\mathbb{E}|F_{L}|^{2}+1$
$\displaystyle\ll\mathbb{E}|F_{L}|^{2},$ (25)
$\displaystyle\mathbb{E}I(|F_{L}|^{2}\leq\lambda_{L}^{\alpha}\mathbb{E}|F_{L}|^{2})|F_{L}|^{2}$
$\displaystyle\ll\mathbb{E}|F_{L}|^{2}.$

Note that in view of (23), $\lambda_{L}\gg 1$ amounts to $L$ being
exponentially large in $\frac{1}{\varepsilon^{2}}$.

Let us comment upon Theorem 2.1. Its first part (24) amounts to the
observation that typically, the Jacobian matrix $F_{L}(x)$ has two singular
values of very different size. This indicates an extreme non-conformality of
the harmonic coordinates. The second part (25) implies that for typical point
$x$, $|F_{L}(x)|^{2}$ (which is the sum of the squares of the singular values)
is much below its average (which in view of (23) is $2\lambda_{L}$). Hence the
field $|F_{L}|^{2}$ features high peaks, suggesting an intermittent behavior
of the process. In other words, (25) means that the random variable
$|F_{L}|^{2}/\mathbb{E}|F_{L}|^{2}$ has bounded expectation but is not equi-
integrable as $L\uparrow\infty$.

Let us now interpret Theorem 2.1. It is consistent with the following picture
of the process $X$: Since a realization $\psi$ of the Gaussian free field with
UV cut-off has closed stream lines, the process (1) would be trapped in a
finite region if it were not for diffusion. Incidentally, this collection of
loops is an object of active research in view of its emergent conformal
invariance. Diffusion allows the process $X$ to move from one level set to the
next. As $X$ explores larger and larger neighborhoods of its starting point,
it occasionally hits loops of large diameter, which introduces a ballistic
element. In the absence of an IR cut-off, there are loops of larger and larger
diameter, which is at the origin of super-diffusivity. This picture suggests
an intermittent behavior of $X$, with bursts of ballistic motion interrupting
longer portions of more diffusive behavior. The burst-like character is
mirrored by non-equi-integrability (25), the ballistic nature is reflected by
the dominance of one singular value (24). Thus Theorem 2.1 is in contrast (but
not in contradiction) with the quenched invariance principle for $X$ reported
in [1]. Theorem 2.1 seems to capture more of the fine structure of the level
sets of the Gaussian free field.

##  3\. Strategy of proof

###  3.1. Variance decomposition of $\psi$ into a martingale in
$\tilde{\lambda}^{2}=1+\varepsilon^{2}\ln L$

Rather than estimating $F={\rm id}+\nabla\phi_{L}$ directly, we use a
stationary proxy $\tilde{\phi}_{L}$ that is built on the idea of a scale-by-
scale homogenization, see the next Subsection 3.2. In the mathematical
community, the idea of a scale-by-scale homogenization for drift-diffusion
processes with random drift seems to originate in [4]. In an setting of
anomalous diffusion for such processes due to failing homogenization, this
strategy was first implemented in [6] for the problem at hand. Shortly
afterwards, scale-by-scale homogenization has also been employed for the
construction of (time-dependent) drifts that produce anomalous dissipation [2,
5], and very recently has been refined in [1] to a quenched invariance
principle for the problem at hand.

In this paper, we follow the continuum version of the scale-by-scale
homogenization [6] very recently introduced in [19]. This construction relies
on the fact that the family of stationary centered Gaussian fields $\psi_{L}$
characterized by (17) can be coupled to form a process $L\mapsto\psi_{L}$ with
independent increments, i. e.

(26) $\psi_{L_{+}}-\psi_{L}~{}\text{is~{}independent~{}of}~{}\psi_{L}$

for any $L_{+}\geq L$. Note that (26) implies

$\mathbb{E}(\psi_{L_{+}}-\psi_{L})(x)(\psi_{L_{+}}-\psi_{L})(y)=\mathbb{E}\psi_{L_{+}}(x)\psi_{L_{+}}(y)-\mathbb{E}\psi_{L}(x)\psi_{L}(y)$

so that by (17) the increments have covariance characterized by

(27) $\mathcal{F}c_{\psi_{L+}-\psi_{L}}(k)=\varepsilon^{2}I(L_{+}^{-1}<|k|\leq
L^{-1})\frac{1}{|k|^{2}}.$

This amounts to a variance decomposition of $\psi$.

In view of (18), it is natural to replace the independent variable
$L\in[1,\infty)$ by $\tilde{\lambda}^{2}-1\in[0,\infty)$, where

(28) $\tilde{\lambda}^{2}\coloneq 1+\varepsilon^{2}\ln L$

so that $\mathbb{E}\psi^{2}$ $=\tilde{\lambda}^{2}-1$; the notation (28) is
motivated by (21). Hence the martingale $\tilde{\lambda}^{2}-1\mapsto\psi_{L}$
shares features with Brownian motion; we denote its distributional
derivative131313which can also be interpreted as Itô differential by $d\psi$.
Due to the independence of increments and (27), the differential141414That is,
we have  $\int_{\ell_{1}}^{\ell_{2}}[d\psi\,d\psi]\coloneq\lim_{{\rm
fineness}(\\{L_{k}\\})\rightarrow
0}\sum_{k}(\psi_{L_{k}}(x)-\psi_{L_{k-1}}(x))^{2},$ where the limit is taken
over partitions of $[\ell_{1},\ell_{2}]$ with mesh size tending to $0$. It is
well know, that by Gaussianity, the limit is deterministic, and hence by
stationarity also independent of $x$. For a more detailed description, we
refer the reader to the Sections 4.1 and 4.3 in [18], with the warning that a
different notation for the quadratic variation is used within that reference.
of the quadratic variation of $\psi$ is given by

(29) $[d\psi\,d\psi]=d\tilde{\lambda}^{2}.$

Relation (29) does not contain the explicit information on the full quadratic
variation, since both copies of $\psi$ are (implicitly) evaluated at the same
point $x$. Information on the covariation at different points is for instance
contained in (57) below. In fact, Lemma 4.4 and Lemma 4.5 can be seen as
characterizing $[d\psi\,Ad\psi]$ for specific pseudo-differential operators
$A$.

###  3.2. Definition of the proxy corrector $\tilde{\phi}$ via an Itô SDE in
$\tilde{\lambda}^{2}$

Since our main result deals with the behavior of nonlinear coordinates, we
follow [19] in adopting a rather strict differential geometric language: It is
natural to interpret the coordinate directions $\xi$ as a linear form, that
is, a cotangent vector. Likewise, it is natural to interpret the differential
$\nabla$ as converting a scalar field into a cotangent vector field. It
follows from (20) that the cotangent vector field $\nabla\phi$ is linear in
$\xi$, so that we interpret $\nabla\phi$ as field with values in (linear)
endomorphisms of cotangent space. Hence (20) may be rewritten as

(30) $\nabla\cdot a_{L}({\rm id}+\nabla\phi_{L})=0,$

where the product in the flux $a_{L}({\rm id}+\nabla\phi_{L})$ is the
composition of endomorphisms. In line with this, we will interpret $\phi$ as a
tangent vector field with $\phi=\phi^{i}e_{i}$, where $\\{e_{i}\\}_{i=1,2}$
denotes the Cartesian basis of the tangent space151515Upper indices enumerate
the components of a tangent vector, whereas lower indices enumerate those of a
cotangent vector, and we employ Einstein’s convention of summation over
repeated indices provided one is up and the other down. of the plane. On the
other hand, the matrix representation of $\nabla\phi$ w. r. t. the (dual)
Cartesian basis $\\{e^{j}\\}_{j=1,2}$ of cotangent space, defined through
$(F\xi)_{j}=F_{j}^{i}\xi_{i}$ where $\xi=\xi_{j}e^{j}$, is given by
$(\nabla\phi)_{j}^{i}=\partial_{j}\phi^{i}$.

We now recall the definitions from [19, (2), (3) & (4)]. We shall often
suppress the explicit dependence on $L$ and rather consider the objects as
functions or Schwartz distributions of the variable $\tilde{\lambda}^{2}$. The
random Schwartz distributions (in $\tilde{\lambda}^{2}$ and in space) $d\phi$
and $d\sigma$ with values in tangent vectors are defined161616$d\phi$ is not
to be confused with the distributional derivative of $\phi$ w. r. t.
$\tilde{\lambda}^{2}$; it is an approximation thereof, as we explain below. In
the continuum approach from [19] that we use here, it corresponds to the
incremental quantity $\phi^{\prime}$ in [6]. via the spatial Helmholtz
decomposition

(31) $\displaystyle\tilde{\lambda}\nabla d\phi+d\psi J=J\nabla
d\sigma\quad\mbox{and}\quad\mathbb{E}d\phi=\mathbb{E}d\sigma=0.$

Since by linearity, the random $d\phi$ and $d\sigma$ inherit stationarity in
space from $d\psi$, the second part of (31) ensures
uniqueness171717Alternatively, the definition could be formulated in Fourier
space, as done in Appendix A.1.. In Appendix A.1 we derive the two local
relations

(32) $\displaystyle\operatorname{\rm tr}\nabla
d\phi=0\quad\mbox{and}\quad\tilde{\lambda}\operatorname{\rm tr}J\nabla
d\phi=-d\psi.$

Since $\nabla d\phi$ and $\nabla d\sigma$ are Schwartz distributions with
values in the endomorphisms of cotangent space, $\tilde{\lambda}$ is a smooth
scalar, and $J$ a constant endomorphisms, (31) makes pathwise sense. Since
$\tilde{\lambda}$ is deterministic, $d\phi$ and $d\sigma$ inherit the property
of having independent increments (in $\tilde{\lambda}^{2}$) from $d\psi$,
which allows us to consider them as drivers (in the jargon of SDEs).

Equipped with these drivers, we define the proxy corrector $\tilde{\phi}$ and
the proxy flux corrector181818in this paper, $\tilde{\sigma}$ does not play a
role $\tilde{\sigma}$ as a process in $\tilde{\lambda}^{2}$ with values in
stationary (smooth) tangent vector fields in $x$, via the Itô191919As opposed
to (31), Itô calculus is required in (33) in order to give sense to the
product $\tilde{\phi}^{i}\partial_{i}d\phi$ because of the lack of regularity
in $\tilde{\lambda}^{2}$. differential equation

(33) $\displaystyle d\tilde{\phi}$
$\displaystyle=(1+\tilde{\phi}^{i}\partial_{i})d\phi,\quad\tilde{\phi}|_{\tilde{\lambda}^{2}=1}=0,$
(34) $\displaystyle d\tilde{\sigma}$
$\displaystyle=d\sigma+\tilde{\sigma}^{i}\partial_{i}d\phi+d\psi\,\tilde{\phi}+\frac{1}{2\tilde{\lambda}}J\tilde{\phi}\,d\tilde{\lambda}^{2},\quad\tilde{\sigma}|_{\tilde{\lambda}^{2}=1}=0.$

Note that (31) implies $\tilde{\lambda}\triangle d\phi$ $=\nabla\cdot d\psi
J$, an equation that amounts to the homogenization, and the linearization in
$\varepsilon$, of the equation for the (distributional) derivative of $\phi$
w. r. t. $\tilde{\lambda}^{2}$. Note also that the operation
$1+\tilde{\phi}^{i}\partial_{i}$ amounts to taking the two-scale expansion.
Hence (33) implements the concept of a scale-by-scale homogenization, on the
level of the corrector itself.

Recall from homogenization theory that the pair $(\phi,\sigma)$ of corrector
and flux corrector is characterized by the Helmholtz decomposition $a({\rm
id}+\nabla\phi)=\lambda{\rm id}+J\nabla\sigma$. It was shown – on a few pages
and in a self-contained way – in [19, (5)] that the residuum $f$ arising from
the approximation $(\tilde{\phi},\tilde{\sigma})$, defined through

(35) $\displaystyle a({\rm id}+\nabla\tilde{\phi})=\tilde{\lambda}\,{\rm
id}+J\nabla\tilde{\sigma}+f,$

which is a process in $\tilde{\lambda}^{2}$ with values in stationary
endomorphism fields in $x$, satisfies

(36) $\displaystyle\mathbb{E}f=0\quad{\rm
and}\quad\mathbb{E}|f|^{2}\lesssim\varepsilon^{2}\tilde{\lambda},$

see [19, Proposition 1]. We remark that (36) states that the residuum $f$ is
$O(\varepsilon\tilde{\lambda}^{-\frac{1}{2}})$ times smaller than the
effective diffusivity $\lambda\approx\tilde{\lambda}$, to which it should be
compared in view of (35). Since in view of Lemma 3.4, the proxy corrector
$\tilde{\phi}$ is $O(\varepsilon\tilde{\lambda}^{-1})$ times smaller than
linear growth $L$, the approach of [6] amounts to a quantitative stochastic
homogenization result of order $\varepsilon\tilde{\lambda}^{-\frac{1}{2}}$
$\sim\varepsilon^{\frac{1}{2}}\ln^{-\frac{1}{4}}L$, cf. (28). Apart from (36),
the present paper is self-contained (and reasonably short).

A continuum variance decomposition, i. e. a continuum decomposition of an
underlying Gaussian field, often the Gaussian free field $\psi$ like here,
according to spatial scales $L$ is also used in quantum field theory (where
the underlying noise is thermal instead of environmental). For instance, in
case of the $\Phi^{4}_{d}$ model for $d=2,3,$[3] use it to control
(exponential) moments of the field $\phi$ with help of suitable martingales in
$\ln L$, analogous to $\tilde{\phi}$. The variance decomposition also induces
a Hamilton-Jacobi-type evolution equation for the effective Hamiltonian in
$\ln L$, known as Polchinski equation. In [13, Section 2.3], to cite another
recent work, proxys for the solution of the Polchinski equation are
constructed by truncation of an expansion (analogous to an expansion in
$\varepsilon$ here), and the evolution of the residuum is monitored (like $f$
in [19, (7)]).

###  3.3. Intermittency of the proxy Jacobian matrix $\tilde{F}={\rm
id}+\nabla\tilde{\phi}$

In line with (22), we set

(37) $\displaystyle\tilde{F}:={\rm id}+\nabla\tilde{\phi};$

following [6, (52) & (79)], we shall argue in the proof of Theorem 2.1 that

(38)
$\displaystyle\mathbb{E}|F-\tilde{F}|^{2}\leq\mathbb{E}|f|^{2}\stackrel{{\scriptstyle(\ref{wr03})}}{{\lesssim}}\varepsilon^{2}\tilde{\lambda}.$

Intermittency is more drastically seen on the level of the proxies:

######  Theorem 3.1 (Intermittency for the proxies).

We have

(39) $\displaystyle\mathbb{E}(\det\tilde{F}-1)^{2}$
$\displaystyle\lesssim\varepsilon^{2},\phantom{\frac{1}{2}}$ (40)
$\displaystyle\mathbb{E}|\tilde{F}|^{2}$ $\displaystyle\approx
2\tilde{\lambda},\phantom{\frac{1}{2}}$ (41)
$\displaystyle\mathbb{E}|\tilde{F}|^{4}$
$\displaystyle\approx\frac{8}{3}\tilde{\lambda}^{3}+\frac{4}{3}$

as $\varepsilon\ll 1$ uniformly in $L\geq 1$, and thus in particular in our
regime $\tilde{\lambda}\gg 1$ and $\varepsilon\ll 1$.

Clearly, the fourth-moment asymptotics (41) is an intermittency statement for
the proxy $\tilde{F}$ of much stronger form than the statement (25) on second-
moment tails of the harmonic coordinates $F$. In view of the error estimate
(38), which is on the level of second moment, only the estimate in the
following proposition transfers from $\tilde{F}$ to $F$.

######  Proposition 3.2.

We have

(42)
$\displaystyle\mathbb{E}I(|\tilde{F}|^{2}\leq\hat{r}\,\mathbb{E}|\tilde{F}|^{2})|\tilde{F}|^{2}\ll\mathbb{E}|\tilde{F}|^{2}$

provided $\varepsilon^{2}\ll 1$, $\tilde{\lambda}\gg 1$ and

(43) $\displaystyle\hat{r}\ll
e^{-\sqrt{2\ln\tilde{\lambda}}}\sqrt{\tilde{\lambda}}.$

The statement (42) is not an immediate consequence of Theorem 3.1 but follows
from an analysis of the backward Kolmogorov equation for $|\tilde{F}|^{2}$. We
will derive the SDE for $|\tilde{F}|^{2}$ in the proof of (52).

The strategy that leads to Theorem 3.1 is in the spirit of [6] (in its
continuum version [19]): It is based on the Itô SDE

(44) $\displaystyle d\tilde{F}=\tilde{F}\nabla
d\phi+\tilde{\phi}^{i}\nabla\partial_{i}d\phi,\quad\tilde{F}|_{\tilde{\lambda}^{2}=1}={\rm
id},$

which follows by applying $\nabla$ to (33), and noting that
$\partial_{i}d\phi\otimes\nabla\tilde{\phi}^{i}$ $=(\nabla\tilde{\phi})\nabla
d\phi$, where the product on the r. h. s. denotes the composition of
endomorphisms (of cotangent space). In particular, $\tilde{F}$ is a
martingale. First, we use Itô calculus on (44) to derive SDEs on
$\det\tilde{F}$ and $|\tilde{F}|^{2}$ as functions of $\tilde{\lambda}^{2}$,
and then use these to derive ODEs for $\mathbb{E}\det\tilde{F}$,
$\mathbb{E}(\det\tilde{F})^{2}$, $\mathbb{E}|\tilde{F}|^{2}$, and
$\mathbb{E}|\tilde{F}|^{4}$. Finally, we use these ODEs to derive the precise
asymptotics for $\tilde{\lambda}\gg 1$.

The outcome of our rigorous analysis allows for the following informal
interpretation: As $\varepsilon\ll 1$, the first r. h. s. term in (44) is
dominant, that is, $\tilde{F}$ is close to $\tilde{F}_{exp}$, where
$\tilde{F}_{exp}$ is the stochastic exponential defined through

(45) $\displaystyle d\tilde{F}_{exp}=\tilde{F}_{exp}\nabla
d\phi\;\;\mbox{and}\;\;\tilde{F}_{exp}|_{\tilde{\lambda}^{2}=1}={\rm id}.$

More precisely, $\tilde{F}_{exp}$ is the stochastic exponential of the
martingale $\int_{1}^{\cdot}\nabla d\phi$. Since by (31)
$\tilde{\lambda}^{-1}\nabla d\phi$ is the Helmholtz projection of $d\psi J$,
and $\psi=\int_{1}^{\cdot}d\psi$ is the Gaussian free field, (see the second
item in (32) for a more direct relation between $\tilde{\lambda}^{-1}\nabla
d\phi$ and $d\psi$), we may assimilate $\tilde{F}_{exp}$ to a Gaussian
multiplicative chaos.

However, the tensor $\nabla d\phi$ has a special structure, as reflected by
its quadratic variation202020which is a four-tensor $\diamond$, see definition
(60). Since $\operatorname{\rm tr}\nabla d\phi=0$, which is the first item in
(32), and in view of the property (77) of $\diamond$, we have

(46) $\displaystyle d\det\tilde{F}_{exp}=0\quad\mbox{so
that}\quad\det\tilde{F}_{exp}=1,$

see Section 4 for all details. Likewise, the property (66) of $\diamond$
yields212121see below for the standard definition of $F^{*}$

(47) $\displaystyle
d|\tilde{F}_{exp}|^{2}=\frac{1}{2}|\tilde{F}_{exp}|^{2}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+2\operatorname{\rm
tr}\tilde{F}_{exp}^{*}\tilde{F}_{exp}\nabla d\phi.$

In a second stage this implies, appealing to the finer structure (65) of
$\diamond$,

(48) $\displaystyle
d|\tilde{F}_{exp}|^{4}=\big{(}\frac{3}{2}|\tilde{F}_{exp}|^{4}-2(\det\tilde{F}_{exp})^{2}\big{)}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+\mbox{martingale},$

where $\frac{3}{2}$ arises as $2\times\frac{1}{2}+2^{2}\times\frac{1}{8}$ with
$\frac{1}{2}$ coming from (47) and $\frac{1}{8}$ coming from (65). It is the
integrating factor $\frac{1}{2\tilde{\lambda}^{2}}$ in (47) that is
responsible for the growth of order
$(\tilde{\lambda}^{2})^{\frac{1}{2}}=\tilde{\lambda}$ in (40); this was
already clear from [6]. Hence the factor $\frac{1}{2}$ in (47) matters.
Intermittency arises from the fact that the factor $\frac{3}{2}$ in (48) is
(strictly) larger than twice the factor $\frac{1}{2}$ of (47). Hence the
factor $\frac{3}{2}$ in (48) matters (relative to the one in (47)). In this
sense, the tensorial stochastic exponential defined through (48) behaves as if
$\tilde{F}_{exp}$ and $\nabla d\phi$ were scalar. In fact, $\tilde{F}_{exp}$
is close to being rank-one in view of the (near-)degeneracy of
$\tilde{F}_{exp}$ in form of
$|\det\tilde{F}_{exp}|=1\ll\tilde{\lambda}^{2}=\mathbb{E}|\tilde{F}_{exp}|^{2}$,
which in turn is a consequence of (46) & (47).

In the upcoming Lemma 3.3, the endomorphism $F^{t}$ of tangent space denotes
the transpose of the endomorphism $F$ of cotangent space (transposition is a
purely linear algebra operation), whereas the endomorphism $F^{*}$ on
cotangent space denotes its adjoint w. r. t. the inner product $\cdot$. It
also features the adjugate222222In coordinates $F=F_{j}^{i}e_{i}\otimes e^{j}$
we have $\operatorname{\rm adj}F=F_{2}^{2}e_{1}\otimes
e^{1}+F_{1}^{1}e_{2}\otimes e^{2}-F_{1}^{2}e_{1}\otimes
e^{2}-F_{2}^{1}e_{2}\otimes e^{1}$. $\operatorname{\rm adj}F:=(\det F)F^{-1}$,
which in $d=2$ is linear in $F$.

######  Lemma 3.3 (ODEs for moments of $\tilde{F}$ and $det\tilde{F}$).

We have

(49) $\displaystyle d\mathbb{E}\det\tilde{F}$ $\displaystyle=0,$ (50)
$\displaystyle d\mathbb{E}(\det\tilde{F})^{2}$
$\displaystyle=\mathbb{E}((\operatorname{\rm
adj}\tilde{F})^{t}\otimes\tilde{\phi})\bullet((\operatorname{\rm
adj}\tilde{F})^{t}\otimes\tilde{\phi})\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}},$
(51) $\displaystyle d\mathbb{E}|\tilde{F}|^{2}$
$\displaystyle=\big{(}\frac{1}{2}\mathbb{E}|\tilde{F}|^{2}+\frac{\mathbb{E}|\tilde{\phi}|^{2}}{2L^{2}}\big{)}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}},$
(52) $\displaystyle d\mathbb{E}|\tilde{F}|^{4}$
$\displaystyle=\big{(}\frac{3}{2}\mathbb{E}|\tilde{F}|^{4}-2\mathbb{E}(\det\tilde{F})^{2}\big{)}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}$
$\displaystyle+\big{(}\mathbb{E}|\tilde{\phi}|^{2}|\tilde{F}|^{2}+4\mathbb{E}(\tilde{F}^{*t}\otimes\tilde{\phi})\bullet(\tilde{F}^{*t}\otimes\tilde{\phi})\big{)}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}},$

where $\bullet$ is a universal symmetric positive semi-definite bilinear form
on the tensor space $(\mbox{endomorphisms of tangent
space})\otimes(\mbox{tangent space})$.

Here, universal means that $\bullet$ is independent of any parameter like
$\varepsilon$ and $L$; $\bullet$ is defined in (61), and characterized as
being universal in Appendix A.2. The identities of Lemma 3.3 are much more
explicit than needed for Theorem 3.1. As we know from [6], see Lemma 3.4,
$\tilde{\phi}$ is of order $\frac{\varepsilon L}{\tilde{\lambda}}$; hence in
view of (50) and (52), $\det\tilde{F}$ is of lower order w. r. t.
$|\tilde{F}|^{2}$. For the same reason, the constants in front of the terms
involving $\tilde{\phi}$ are irrelevant; so is the specific form of $\bullet$.
However, for this we need to capture $\tilde{\phi}=O(\frac{\varepsilon
L}{\tilde{\lambda}})$ up to fourth moments, as already done in [6, (72)] and
[19, (30) & (38)] based on (33) and once more Itô calculus. Again, we are more
explicit here, although the factors don’t matter for the asymptotic scaling.

######  Lemma 3.4 (Evolution of moments of $\tilde{\phi}$).

We have

(53) $\displaystyle d\mathbb{E}|\tilde{\phi}|^{2}$
$\displaystyle=(\frac{1}{2}\mathbb{E}|\tilde{\phi}|^{2}+L^{2})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}},$
(54) $\displaystyle d\mathbb{E}|\tilde{\phi}|^{4}$
$\displaystyle=(\frac{3}{2}\mathbb{E}|\tilde{\phi}|^{4}+4L^{2}\mathbb{E}|\tilde{\phi}|^{2})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}.$

Equipped with these evolution equations for $\mathbb{E}|\tilde{\phi}|^{2}$ and
$\mathbb{E}|\tilde{\phi}|^{4}$, we derive the precise asymptotics for these
quantities, which already appeared in [6, (69) & (70)] and [19, (31) & (39)].

######  Lemma 3.5 (Integration for $\tilde{\phi}$).

We have

(55) $\displaystyle\mathbb{E}|\tilde{\phi}|^{2}$
$\displaystyle\approx\frac{\varepsilon^{2}}{2}\frac{L^{2}}{\tilde{\lambda}^{2}},$
(56) $\displaystyle\mathbb{E}|\tilde{\phi}|^{4}$
$\displaystyle\approx\frac{\varepsilon^{4}}{2}\frac{L^{4}}{\tilde{\lambda}^{4}}$

as $L\gg 1$ and uniformly in $\varepsilon\leq 1$.

Note that while (53) & (54) determine the evolution of
$\mathbb{E}|\tilde{\phi}|^{2}$ and $\mathbb{E}|\tilde{\phi}|^{4}$, the ODEs of
Lemma 3.3 do not close on the level of $\mathbb{E}\det\tilde{F}$,
$\mathbb{E}(\det\tilde{F})^{2}$, $\mathbb{E}|\tilde{F}|^{2}$,
$\mathbb{E}|\tilde{F}|^{4}$. However, both lemmas combined provide precise
asymptotics for these quantities. The rest of the paper is organized as
follows. In order to prove Lemma 3.3 and Lemma 3.4, we identify a number of
quadratic variations for the drivers $d\phi$ and its derivatives. We introduce
the necessary notation in Section 4.1. Therein we also collect the necessary
identities, while the main computational work is deferred to Appendix A.
Proofs of the above results are given in Section 4.2 (where we derive the
differential equations for moments), in Section 4.3 (where identify the
asymptotic), and Section 4.4 (where we analyze a backward Kolmogorov equation
for $|\tilde{F}|^{2}$ as needed for (42)).

##  4\. Proofs

###  4.1. Identification of quadratic variations

In order to establish Lemmas 3.3 and 3.4, in view of (33) and (44) and Itô
calculus, we need to identify the quadratic variations of the drivers $\nabla
d\psi$, $d\phi$, $\nabla d\phi$, and $\nabla^{2}d\phi$ by relating them to the
one of $d\psi$, see (29). The underlying principle that a derivative gains a
factor of $L^{-1}$ is best seen for $\psi$, i. e.

(57) $[\nabla d\psi\cdot\nabla d\psi]=\frac{[d\psi\,d\psi]}{L^{2}}.$

Indeed, on the level of covariance functions and their Fourier-transforms we
have

$c_{\nabla\psi_{L_{+}}-\nabla\psi_{L}}=-\nabla^{2}c_{\psi_{L_{+}}-\psi_{L}}\quad{\rm
so~{}that}\quad\operatorname{\rm
tr}\mathcal{F}c_{\nabla\psi_{L_{+}}-\nabla\psi_{L}}=|k|^{2}\mathcal{F}c_{\psi_{L_{+}}-\psi_{L}}~{}{\rm
distributionally}.$

From the latter identity, it follows that

$\frac{1}{L_{+}^{2}}\mathbb{E}(\psi_{L_{+}}-\psi_{L})^{2}\leq\mathbb{E}|\nabla\psi_{L_{+}}-\nabla\psi_{L}|^{2}\leq\frac{1}{L^{2}}\mathbb{E}(\psi_{L_{+}}-\psi_{L})^{2}$

for $L+\geq L$. Using the characterization of $[d\psi\,d\psi]$ and $[\nabla
d\psi\cdot\nabla d\psi]$ in Footnote 14, we have

$\displaystyle\int_{\ell_{1}}^{\ell_{2}}[\nabla d\psi\cdot\nabla d\psi]$
$\displaystyle=\lim_{{\rm fineness}(\\{L_{k}\\})\rightarrow
0}\sum_{k}\mathbb{E}|\nabla\psi_{L_{k}}-\nabla\psi_{L_{k-1}}|^{2}$
$\displaystyle=\lim_{{\rm fineness}(\\{L_{k}\\})\rightarrow
0}\sum_{k}\mathbb{E}\frac{1}{L_{k}^{2}}(\psi_{L_{k}}-\psi_{L_{k-1}})^{2}$
$\displaystyle=\int_{\ell_{1}}^{\ell_{2}}\frac{[d\psi\,d\psi]}{L^{2}},$

which establishes (57).

We will now collect the necessary formulas for $d\phi$, $\nabla d\phi$ and
$\nabla^{2}d\phi$. In view of (31), the most natural object out of these three
is $\nabla d\phi$ as it is directly related to $d\psi$. Since we move most
computations to the appendix, we however start the presentation with $d\phi$.

######  Lemma 4.1 (Quadratic variation of $d\phi$).

Given a cotangent vector $\xi$, it holds232323We use the symbol . to denote a
canonical pairing between primal and dual space, here
$\xi.\dot{x}=\xi_{i}\dot{x}^{i}$, to be distinguished from the inner product
$\dot{x}\cdot\dot{y}=\sum_{i=1}^{2}\dot{x}^{i}\dot{y}^{i}$.

(58)
$\displaystyle[\xi.d\phi\,\xi.d\phi]=|\xi|^{2}\frac{L^{2}d\tilde{\lambda}^{2}}{2\tilde{\lambda}^{2}}.$

As a special case of this identity, we have

(59) $\displaystyle[d\phi\cdot
d\phi]=L^{2}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}.$

A proof of Lemma 4.1 is given after the next lemma. Identifying the other
variations is more involved because the processes are tensor valued (fields).
We thus consider general linear forms $G$ on these tensor spaces, which act as
observables. All quadratic variations come as positive semi-definite bilinear
forms on the linear space of $G$’s; like all quadratic variations of processes
with independent increments, these forms are deterministic. Since the
processes arise from evaluating a spatially stationary field in $x$, these
forms are independent of $x$, see Footnote 14. As a consequence, since $\nabla
d\phi$ is the increment of such a process with values in the space of
endomorphisms of cotangent space, the corresponding quadratic variation is
such a deterministic form on the dual space, which is canonically
isomorphic242424 via the pairing $(\xi\otimes\dot{x}).F$ $:=(F\xi).\dot{x}$
for a tangent vector $\dot{x}$, a cotangent vector $\xi$, and an endomorphism
$F$ of cotangent space, and where the . on the r. h. s. denotes the natural
pairing between cotangent and tangent vectors to

$\displaystyle\mbox{endomorphisms of tangent space}\cong(\mbox{cotangent
space})\otimes(\mbox{tangent space}).$

Likewise, the quadratic variation of $\nabla^{2}d\phi$ is a deterministic and
$x$-independent positive semi-definite bilinear form on $(\mbox{cotangent
space})\otimes(\mbox{tangent space})\otimes(\mbox{tangent space})$. Let us
summarize this in the following definition.

######  Definition 4.2.

  1. i)

For $G$ $\in(\mbox{cotangent space})\otimes(\mbox{tangent space})$, we write

(60) $\displaystyle G\diamond
G\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}\coloneqq[G.\nabla
d\phi\,G.\nabla d\phi],$

where $G.\nabla d\phi$ denotes the natural pairing which componentwise is
given by $G_{j}^{i}\partial_{i}d\phi^{j}$.

  2. ii)

For $\textit{{G}}\in(\mbox{cotangent space})\otimes(\mbox{tangent
space})\otimes(\mbox{tangent space})$, we let

(61)
$\displaystyle\textit{{G}}\bullet\textit{{G}}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}\coloneqq[\textit{{G}}.\nabla^{2}d\phi\,\textit{{G}}.\nabla^{2}d\phi],$

where $\textit{{G}}.\nabla^{2}d\phi$ denotes the natural pairing which
componentwise is given by
$\textit{{G}}^{ij}_{k}\partial_{i}\partial_{j}d\phi^{k}$.

Note that via polarization, (60) and (61) indeed define symmetric bilinear
forms. There is no need to track the mixed covariation of zeroth, first, and
second derivative:

######  Remark 4.3.

For every $\xi\in\mbox{cotangent space}$, $G$ $\in(\mbox{cotangent
space})\otimes(\mbox{tangent space})$ and $\textit{{G}}\in(\mbox{cotangent
space})\otimes(\mbox{tangent space})\otimes(\mbox{tangent space})$ the mixed
covariations $[\xi.d\phi\,G.\nabla d\phi]$ and $[G.\nabla
d\phi\,\textit{{G}}.\nabla^{2}d\phi]$ vanish because of parity. By this we
mean that via (31), the process $\nabla d\phi$ inherits the evenness (in law)
of $\psi$ under point reflections in space; which implies that $d\phi$ and
$\nabla\partial_{i}d\phi$ are odd. Since their co-variation is deterministic
by independence of increments, it must vanish.

In Subsections A.1 and A.2 we identify natural bases that diagonalize these
forms, based on their invariance under the action of rotations: In (125), we
introduce such a basis $(E^{n})_{n=1,2,3,4}$ of $(\mbox{cotangent
space})\otimes(\mbox{tangent space})$. In (151) we define a suitable basis
$(\textit{{E}}^{n})_{n=1,\ldots,6}$ of $(\mbox{cotangent
space})\otimes(\mbox{tangent space})\otimes_{sym}(\mbox{tangent space})$,
where the subscript $sym$ indicates the symmetry in the last two factors, to
which we may restrict in view of $\nabla^{2}d\phi$ in view of the symmetry of
second derivatives. In the rest of this section we record some contractions of
these forms for later use.

We start with the contractions of $\diamond$ that are relevant for computing
$d|\tilde{F}|^{2}$ and $d|\tilde{F}|^{4}$. For $d|\tilde{F}|^{2}$, we
represent the (squared) Frobenius norm $|F|^{2}=\operatorname{\rm
tr}F^{*}F=\operatorname{\rm tr}FF^{*}$ of an endomorphism as the following sum
of quadratic form

(62) $\displaystyle|F|^{2}=\sum_{i,j=1}^{2}(e^{i}\otimes e_{j}.F)(e^{i}\otimes
e_{j}.F).$

In view of the first r. h. s. term of (44), we apply (62) to
$F=\tilde{F}\nabla d\phi$, so that we actually need to monitor the more
general quadratic form on the space of endomorphisms of cotangent space

(63) $\displaystyle F\mapsto\sum_{i,j=1}^{2}(Ge^{i}\otimes
e_{j}.F)(Ge^{i}\otimes e_{j}.F),$

where the endomorphism $G$ on tangent space is the placeholder for
$\tilde{F}^{t}$; the outcome is given in (66). We now turn to
$d|\tilde{F}|^{4}$; in order to identify the contribution of the martingale
term in (47) to (48), we need to characterize the quadratic variation of
$G.\nabla d\phi$ for a symmetric endomorphism $G$ on tangent space, which
plays the role of $(F^{*}F)^{t}$; see (65) for the outcome. Two contractions
of $\diamond$ are relevant for $d|\tilde{\phi}|^{2}$ and $d|\phi|^{4}$: In
order to identify the contribution of the second r. h. s. term of (33) to
$|\tilde{\phi}|^{2}$, we need to characterize the quadratic variation of
$\dot{x}\cdot\nabla d\phi^{i}$ $=(e^{i}\otimes\dot{x}).\nabla d\phi$, for a
tangent vector $\dot{x}$ playing the role of $\tilde{\phi}$, see (67) for the
result. For $|\tilde{\phi}|^{4}$ we need (68). Finally, the contraction (69)
is needed to pass from the quadratic variation of $\nabla d\phi$ to the one of
$d\phi$. The factor $\frac{1}{2}$ in (66), (67), and (69) has the same origin.
The factors $\frac{1}{2}$ in (66) and $\frac{1}{8}$ in (65) are crucial.

######  Lemma 4.4 (Quadratic variation of $\nabla d\phi$ for
$|\tilde{F}|^{2}$, $|\tilde{F}|^{4}$, $|\tilde{\phi}|^{2}$,
$|\tilde{\phi}|^{4}$).

For an endomorphism $G$ of tangent space we have in terms of the trace-free
part $G_{sym}$ $-{\textstyle\frac{1}{2}}(\operatorname{\rm tr}G_{sym}){\rm
id}$ of its symmetric part $G_{sym}=\frac{1}{2}(G+G^{*})$ and its skew-
symmetric part $G_{skew}:=\frac{1}{2}(G-G^{*})$

(64) $\displaystyle G\diamond
G=\frac{1}{4}|G_{sym}-{\textstyle\frac{1}{2}}(\operatorname{\rm
tr}G_{sym}){\rm id}|^{2}+\frac{1}{2}|G_{skew}|^{2},$

which we use in form of

(65) $G\diamond G=\frac{1}{8}(\operatorname{\rm tr}G)^{2}-\frac{1}{2}\det
G\quad\mbox{provided}\;G\;\mbox{is symmetric}.$

As a consequence of (64) we have252525 The product $Ge^{i}\otimes e_{j}$
denotes the composition of endomorphisms (on tangent space), that is,
$Ge^{i}\otimes e_{j}=e^{i}\otimes Ge_{j}$. In general, we often suppress
brackets based on the convention that $\otimes$ binds more tightly than
composition, pairing, and inner products like $\cdot$, $\diamond$, or
$\bullet$.

(66) $\sum_{i,j=1}^{2}Ge^{i}\otimes e_{j}\diamond Ge^{i}\otimes
e_{j}=\frac{1}{2}|G|^{2}$

and for any cotangent vector $\xi$ and any tangent vector $\dot{x}$

(67) $\displaystyle\sum_{i=1}^{2}e^{i}\otimes\dot{x}\diamond
e^{i}\otimes\dot{x}$ $\displaystyle=\frac{1}{2}|\dot{x}|^{2},$ (68)
$\displaystyle\dot{x}^{*}\otimes\dot{x}\diamond\dot{x}^{*}\otimes\dot{x}$
$\displaystyle=\frac{1}{8}|\dot{x}|^{4},$ (69)
$\displaystyle\sum_{i=1}^{2}\xi\otimes e_{i}\diamond\xi\otimes e_{i}$
$\displaystyle=\frac{1}{2}|\xi|^{2},$

where $\dot{x}^{*}$ denotes the cotangent vector associated to $\dot{x}$ by
the inner product, in coordinates $(\dot{x}^{*})_{i}=\dot{x}^{i}$.

###### Proof of Lemma 4.4..

Argument for (64). According to Appendix A.1, both $\diamond$ and the standard
inner product given by

(70) $\displaystyle G:G^{\prime}=\operatorname{\rm tr}G^{*}G^{\prime}$

are diagonal w. r. t. the decomposition of endomorphism space

(71) $\displaystyle\mbox{trace-free
symmetric}\oplus\mbox{skew}\oplus\mbox{isotropic}.$

More precisely, it follows from (126) and (127) & (128) that $\diamond$ is
$\frac{1}{4}$ times $:$ on the first component, whereas the factor is
$\frac{1}{2}$ on the second, and $0$ on the last. Since
$G_{sym}-\frac{1}{2}(\operatorname{\rm tr}G_{sym}){\rm id}$ and $G_{skew}$ are
the $:$-orthogonal projections on the first and second component,
respectively, (64) follows.

Argument for (65). Passing from (64) to (65) reduces to the identity for
symmetric matrices $G$

$\displaystyle 2|G-\frac{1}{2}(\operatorname{\rm tr}G){\rm
id}|^{2}=(\operatorname{\rm tr}G)^{2}-4\det G,$

which by definition of $|\cdot|^{2}$ is equivalent to
$|G|^{2}=(\operatorname{\rm tr}G)^{2}-2\det G$, which in terms of the
eigenvalues amounts to
$\lambda_{1}^{2}+\lambda_{2}^{2}=(\lambda_{1}+\lambda_{2})^{2}-2\lambda_{1}\lambda_{2}$
and thus holds true.

Argument for (67). In view of the invariance of $\diamond$ under the action
(122) of rotations we have in particular for the rotation $J$ by
$\frac{\pi}{2}$

$\displaystyle e^{i}\otimes\dot{x}\diamond
e^{i}\otimes\dot{x}=J.e^{i}\otimes\dot{x}\diamond
J.e^{i}\otimes\dot{x}=J^{-t}e^{i}\otimes J\dot{x}\diamond J^{-t}e^{i}\otimes
J\dot{x}.$

Since $\\{\pm J^{-t}e^{i}\\}_{i=1,2}$ is a permutation of $\\{\pm
e^{i}\\}_{i=1,2}$, the quadratic form in $\dot{x}$ on the l. h. s. of (67) is
invariant under $J$, and thus isotropic. Hence it is enough to establish (67)
for $\dot{x}=e_{j}$, that is

(72) $\displaystyle\sum_{i=1}^{2}e^{i}\otimes e_{j}\diamond e^{i}\otimes
e_{j}=\frac{1}{2}.$

We now derive (72) from (64). For this purpose, we rewrite (64) as

(73) $\displaystyle G\diamond
G=\frac{1}{4}|G|^{2}-\frac{1}{8}\big{(}(\operatorname{\rm
tr}G)^{2}-(\operatorname{\rm tr}J^{*}G)^{2}\big{)},$

which follows from the $:$-related identities

$\displaystyle|G_{sym}-\frac{1}{2}(\operatorname{\rm tr}G_{sym}){\rm
id}|^{2}+|G_{skew}|^{2}$ $\displaystyle=|G-\frac{1}{2}(\operatorname{\rm
tr}G){\rm id}|^{2}=|G|^{2}-\frac{1}{2}(\operatorname{\rm tr}G)^{2},$
$\displaystyle|G_{skew}|^{2}$ $\displaystyle=|\frac{1}{2}(\operatorname{\rm
tr}J^{*}G)J|^{2}=\frac{1}{2}(\operatorname{\rm tr}J^{*}G)^{2}.$

It remains to note that $G^{i}:=e^{i}\otimes e_{j}$ satisfies $|G^{i}|^{2}=1$
and thus $\sum_{i=1}^{2}|G^{i}|^{2}=2$. It also satisfies $(\operatorname{\rm
tr}G^{i})^{2}=\delta^{i}_{j}$ while $(\operatorname{\rm
tr}J^{*}G^{i})^{2}=1-\delta^{i}_{j}$ and thus
$\sum_{i=1}^{2}((\operatorname{\rm tr}G^{i})^{2}-(\operatorname{\rm
tr}J^{*}G^{i})^{2})=0$.

Argument for (68). This follows from (73) for $G=\dot{x}^{*}\otimes\dot{x}$
which satisfies $|G|^{2}=|\dot{x}|^{4}$, $\operatorname{\rm
tr}G=|\dot{x}|^{2}$ and $\operatorname{\rm tr}J^{*}G=0$.

Argument for (69). We read off (64) that $\diamond$ is invariant under
$G\mapsto G^{*}$. Since $(e^{i}\otimes\dot{x})^{*}=\dot{x}^{*}\otimes e_{i}$,
(69) thus follows from (67) for $\dot{x}^{*}=\xi$.

Argument for (66). Noting that $Ge^{i}\otimes e_{j}=e^{i}\otimes Ge_{j}$ we
see that the l. h. s. of (66) is given by

$\displaystyle\sum_{j=1}^{2}\sum_{i=1}^{2}e^{i}\otimes Ge_{j}\diamond
e^{i}\otimes
Ge_{j}\stackrel{{\scriptstyle(\ref{wr65bis})}}{{=}}\frac{1}{2}\sum_{j=1}^{2}|Ge_{j}|^{2}=\frac{1}{2}|G|^{2},$

as desired. ∎

We are now ready to characterize also the quadratic variation of $d\phi$.

###### Proof of Lemma 4.1..

For the purpose of this argument, we introduce the tangent-vector valued
process262626which is not $L\mapsto\phi_{L}$, see Footnote 16
$L\mapsto\phi_{L}^{\prime}$ that is obtained from integrating $d\phi$. We
start as for (57) and observe that on the level of covariance functions we
have

$-\nabla^{2}c_{\xi.(\phi_{L_{+}}^{\prime}-\phi_{L}^{\prime})}=c_{\nabla(\xi.(\phi_{L_{+}}^{\prime}-\phi_{L}^{\prime}))}\quad\mbox{and
thus}\quad-\triangle
c_{\xi.(\phi_{L_{+}}^{\prime}-\phi_{L}^{\prime})}=\sum_{i=1}^{2}c_{\xi.\partial_{i}(\phi_{L_{+}}^{\prime}-\phi_{L}^{\prime})}.$

As for $\nabla\tilde{\phi}_{L}$, we interpret $\nabla\phi_{L}^{\prime}$ as
having values in the space of endomorphisms of cotangent space
$\cong(\mbox{tangent space})\otimes(\mbox{cotangent space})$, which implies
$\xi.\partial_{i}\phi_{L}^{\prime}=(\xi\otimes
e_{i}).\nabla\phi_{L}^{\prime}$. Hence on the Fourier level, the above
translates to

$|k|^{2}\mathcal{F}c_{\xi.(\phi_{L_{+}}^{\prime}-\phi_{L}^{\prime})}=\sum_{i=1}^{2}\mathcal{F}c_{(\xi\otimes
e_{i}).(\nabla\phi_{L_{+}}^{\prime}-\nabla\phi_{L}^{\prime})}.$

Therefore, the same argument that lead to (57) yields

$\displaystyle L^{2}[\xi.d\phi\,\xi.d\phi]$
$\displaystyle=\sum_{i=1}^{2}[(\xi\otimes e_{i}).\nabla d\phi\,(\xi\otimes
e_{i}).\nabla d\phi]$
$\displaystyle\stackrel{{\scriptstyle(\ref{wr51})}}{{=}}\sum_{i=1}^{2}\xi\otimes
e_{i}\diamond\xi\otimes
e_{i}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}\stackrel{{\scriptstyle(\ref{wr65})}}{{=}}|\xi|^{2}\frac{d\tilde{\lambda}^{2}}{2\tilde{\lambda}^{2}},$

as desired. ∎

The quadratic variation $\bullet$ arises from the second r. h. s. term of the
Itô SDE (44). The following contraction comes up when computing
$d|\tilde{F}|^{2}$, and is easily made explicit.

######  Lemma 4.5 (Quadratic variation of $\nabla^{2}d\phi$ for
$d|\tilde{F}|^{2}$).

For any tangent vector $\dot{x}$ we have

(74) $\sum_{i,j=1}^{2}e^{i}\otimes e_{j}\otimes\dot{x}\bullet e^{i}\otimes
e_{j}\otimes\dot{x}=\frac{1}{2}|\dot{x}|^{2}.$

###### Proof.

We note that the l. h. s. of (74) defines a (non-negative) quadratic form on
tangent space. Since $\bullet$ is invariant under the action (141) of
rotations, and in particular under $J$, we have

$\displaystyle J.e^{i}\otimes e_{j}\otimes\dot{x}\bullet J.e^{i}\otimes
e_{j}\otimes\dot{x}=J^{-t}e^{i}\otimes Je_{j}\otimes J\dot{x}\bullet
J^{-t}e^{i}\otimes Je_{j}\otimes J\dot{x}.$

Since $\\{(J^{-t}e^{i},Je_{j})\\}_{i=1,2}$ is a permutation of
$\\{(e^{i},e_{j})\\}_{i=1,2}$ modulo the sign, we learn that this quadratic
form is invariant under $J$. Hence it must be isotropic, so that it is enough
to show

(75) $\displaystyle\sum_{i,j=1}^{2}e^{i}\otimes e_{j}\otimes e_{1}\bullet
e^{i}\otimes e_{j}\otimes e_{1}=\frac{1}{2}.$

To this purpose, we use (154) and (155) to express $e^{i}\otimes e_{j}\otimes
e_{1}$ in terms of the $\textit{{E}}^{1},\ldots,\textit{{E}}^{6}$:

$\displaystyle 4\,e^{1}\otimes e_{1}\otimes e_{1}$
$\displaystyle=-\textit{{E}}^{1}+\textit{{E}}^{3}+\textit{{E}}^{5},$
$\displaystyle 4\,e^{1}\otimes e_{2}\otimes e_{1}$
$\displaystyle=_{sym}\textit{{E}}^{2}-\textit{{E}}^{4}+\textit{{E}}^{6},$
$\displaystyle 4\,e^{2}\otimes e_{1}\otimes e_{1}$
$\displaystyle=\textit{{E}}^{2}+3\,\textit{{E}}^{4}-\textit{{E}}^{6},$
$\displaystyle 4\,e^{2}\otimes e_{2}\otimes e_{1}$
$\displaystyle=_{sym}\textit{{E}}^{1}-\textit{{E}}^{3}+\textit{{E}}^{5},$

where the subscript sym indicates that the identity holds only modulo
symmetrization of the last two factors in the tensor product $(\mbox{cotangent
space})\otimes(\mbox{tangent space})\otimes(\mbox{tangent space})$. In view of
(156) we have

$\displaystyle
2(-\textit{{E}}^{1}+\textit{{E}}^{3}+\textit{{E}}^{5})\bullet(-\textit{{E}}^{1}+\textit{{E}}^{3}+\textit{{E}}^{5})$
$\displaystyle=1+1=2,$ $\displaystyle
2(\textit{{E}}^{2}-\textit{{E}}^{4}+\textit{{E}}^{6})\bullet(\textit{{E}}^{2}-\textit{{E}}^{4}+\textit{{E}}^{6})$
$\displaystyle=1+1=2,$ $\displaystyle
2(\textit{{E}}^{2}+3\textit{{E}}^{4}-\textit{{E}}^{6})\bullet(\textit{{E}}^{2}+3\textit{{E}}^{4}-\textit{{E}}^{6})$
$\displaystyle=1+3^{2}=10,$ $\displaystyle
2(\textit{{E}}^{1}-\textit{{E}}^{3}+\textit{{E}}^{5})\bullet(\textit{{E}}^{1}-\textit{{E}}^{3}+\textit{{E}}^{5})$
$\displaystyle=1+1=2,$

so that the sum is $16$, which yields (75) ∎

Finally, we turn to the contributions of both $\diamond$ and $\bullet$ to
$d\det\tilde{F}$. Note that in $d=2$, the determinate $\det$ is a bilinear
form on the space of endomorphisms $F$ of cotangent space, as it can be
written as

(76) $\displaystyle\det F=F^{1}_{1}F^{2}_{2}-F_{1}^{2}F_{2}^{1}=(e^{1}\otimes
e_{1}.F)(e^{2}\otimes e_{2}.F)-(e^{1}\otimes e_{2}.F)(e^{2}\otimes e_{1}.F).$

The next lemma implies in particular that the contraction of $\det$ with
$\diamond$, which is a canonical pairing since they are bilinear forms on dual
spaces, and important since it arises as Itô correction, vanishes. In other
words, $\det$ is a harmonic bilinear form w. r. t. to the Euclidean structure
defined by $\diamond$. Since via Itô calculus, the representation (76) applied
to $F=\tilde{F}$ will give rise to (76) with $F$ replaced by the composition
$\tilde{F}\nabla d\phi$, we need (77) with $G=\tilde{F}^{t}$.

######  Lemma 4.6 (Quadratic variation of $\nabla d\phi$ and $\nabla^{2}d\phi$
for $d\det\tilde{F}$).

For any endomorphism $G$ of tangent space and any tangent vector $\dot{x}$ we
have

(77) $\displaystyle Ge^{1}\otimes e_{1}\diamond Ge^{2}\otimes
e_{2}-Ge^{1}\otimes e_{2}\diamond Ge^{2}\otimes e_{1}=0,$ (78) $\displaystyle
e^{1}\otimes e_{1}\otimes\dot{x}\bullet e^{2}\otimes
e_{2}\otimes\dot{x}-e^{1}\otimes e_{2}\otimes\dot{x}\bullet e^{2}\otimes
e_{1}\otimes\dot{x}=0.$

###### Proof of Lemma 4.6..

Argument for (77). Note that the l. h. s. of (77), which we rewrite in terms
of $G_{i}:=Ge_{i}$ as

$\displaystyle e^{1}\otimes G_{1}\diamond e^{2}\otimes G_{2}-e^{1}\otimes
G_{2}\diamond e^{2}\otimes G_{1},$

is a skew-symmetric bilinear form in the pair of tangent vectors
$(G_{1},G_{2})$, and thus a scalar multiple of $\det G$. Hence it is enough to
check (77) for $G={\rm id}$:

(79) $\displaystyle e^{1}\otimes e_{1}\diamond e^{2}\otimes e_{2}=e^{1}\otimes
e_{2}\diamond e^{2}\otimes e_{1}.$

In order to establish (79), we appeal to the polarized version of (73), that
is

(80) $\displaystyle G\diamond
G^{\prime}=\frac{1}{4}G:G^{\prime}-\frac{1}{8}\big{(}(\operatorname{\rm
tr}G)(\operatorname{\rm tr}G^{\prime})-(\operatorname{\rm
tr}J^{*}G)(\operatorname{\rm tr}J^{*}G^{\prime})\big{)},$

which we use for $(G,G^{\prime})=(e^{1}\otimes e_{1},e^{2}\otimes e_{2})$ and
$(G,G^{\prime})=(e^{1}\otimes e_{2},e^{2}\otimes e_{1})$. For both couples,
the first r. h. s. term of (80) vanishes. For the first couple,
$(\operatorname{\rm tr}G)(\operatorname{\rm tr}G^{\prime})=1\times 1$ while
$(\operatorname{\rm tr}J^{*}G)(\operatorname{\rm tr}J^{*}G^{\prime})=0\times
0$. Since for the second couple $(J^{*}G,J^{*}G^{\prime})=(e^{1}\otimes
J^{*}e_{2},e^{2}\otimes J^{*}e_{1})=(e^{1}\otimes e_{1},-e^{2}\otimes e_{2})$,
we have $(\operatorname{\rm tr}G)(\operatorname{\rm tr}G^{\prime})=0\times 0$
and $(\operatorname{\rm tr}J^{*}G)(\operatorname{\rm
tr}J^{*}G^{\prime})=1\times(-1)$. Hence both sides of (79) are identical.

Argument for (78). By linearity, we need to show

(81) $\displaystyle e^{1}\otimes e_{1}\otimes e_{i}\bullet e^{2}\otimes
e_{2}\otimes e_{j}=e^{1}\otimes e_{2}\otimes e_{i}\bullet e^{2}\otimes
e_{1}\otimes e_{j}.$

For $i\not=j$, both sides of (81) vanish: Indeed, consider the
$\bullet$-product on either side, and note that in one factor, the index $2$
appears an even number of times, while in the other, it appears odd times.
Hence by (154) and modulo symmetrizing in the last two factors one factor is
in the span of $\\{\textit{{E}}^{1},\textit{{E}}^{3},\textit{{E}}^{5}\\}$,
while by (155) the other is in the one of
$\\{\textit{{E}}^{2},\textit{{E}}^{4},\textit{{E}}^{6}\\}$. Hence by (156),
the $\bullet$-product vanishes.

For $i=j$ in (81), we first note that by the invariance of $\bullet$ under the
action (141) of rotations, and by the symmetry of $\bullet$, (81) reduces to

$\displaystyle e^{1}\otimes e_{1}\otimes e_{1}\bullet e^{2}\otimes
e_{2}\otimes e_{1}=e^{2}\otimes e_{1}\otimes e_{2}\bullet e^{1}\otimes
e_{2}\otimes e_{2}.$

By (154) and dividing by $16$, this assumes the form

$\displaystyle(-\textit{{E}}^{1}+\textit{{E}}^{3}+\textit{{E}}^{5})\bullet(\textit{{E}}^{1}-\textit{{E}}^{3}+\textit{{E}}^{5})=(\textit{{E}}^{1}-\textit{{E}}^{3}+\textit{{E}}^{5})\bullet(\textit{{E}}^{1}+3\textit{{E}}^{3}-\textit{{E}}^{5}).$

It follows from (156) that both sides take the value $-1$. ∎

###  4.2. Identification of Itô SDEs

We now give the proofs of Lemma 3.4 and Lemma 3.3, which state ODEs for (first
and second) moments of $|\tilde{\phi}|^{2}$, $\det\tilde{F}$, and
$|\tilde{F}|^{2}$. In fact, we shall derive Itô SDEs for these quantities
themselves and their squares (in case of the latter without keeping track of
the martingale term), so that the lemmas follow after taking the expectation.

###### Proof of Lemma 3.4..

Identification of $d|\tilde{\phi}|^{2}$. We claim that

(82) $\displaystyle d|\tilde{\phi}|^{2}=2\tilde{G}.\nabla
d\phi+2\tilde{\phi}\cdot
d\phi+(\frac{1}{2}|\tilde{\phi}|^{2}+L^{2})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}},$

where $\tilde{G}$ is the endomorphism of tangent space (of Gram type and thus
symmetric)

(83)
$\displaystyle\tilde{G}:=\tilde{\phi}^{*}\otimes\tilde{\phi},\quad\mbox{in
coordinates}\;\;\tilde{G}^{j}_{i}=\tilde{\phi}^{i}\tilde{\phi}^{j}.$

Note that since the two first r. h. s. terms of (82) are of vanishing
expectation, (82) implies (53).

Applying Itô’s formula to (33) and appealing to Remark 4.3 we obtain

$\displaystyle d|\tilde{\phi}|^{2}$ $\displaystyle=$ $\displaystyle
2\tilde{\phi}\cdot d\tilde{\phi}+[d\tilde{\phi}\cdot d\tilde{\phi}]$
$\displaystyle\stackrel{{\scriptstyle(\ref{wr04})}}{{=}}$ $\displaystyle
2\tilde{\phi}\cdot
d\phi+2\tilde{\phi}^{i}\tilde{\phi}^{j}\partial_{i}d\phi^{j}+[d\phi\cdot
d\phi]+\tilde{\phi}^{i}\tilde{\phi}^{j}[\partial_{i}d\phi\cdot\partial_{j}d\phi].$

Since $\partial_{i}d\phi^{k}=(e^{k}\otimes e_{i}).\nabla d\phi$, we obtain by
definitions (60) and (83)

$\displaystyle d|\tilde{\phi}|^{2}$ $\displaystyle=$ $\displaystyle
2\tilde{\phi}\cdot d\phi+2\tilde{G}.\nabla d\phi+[d\phi\cdot
d\phi]+\sum_{k=1}^{2}e^{k}\otimes\tilde{\phi}\diamond
e^{k}\otimes\tilde{\phi}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}.$

It remains to insert (59) and (67) to obtain (82).

Identification of $d\mathbb{E}|\tilde{\phi}|^{4}$. Applying Itô’s formula to
(82), appealing to Remark 4.3, and inserting definition (60) yields

$\displaystyle
d|\tilde{\phi}|^{4}=(2\tilde{G})\diamond(2\tilde{G})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+[2\tilde{\phi}\cdot
d\phi\,2\tilde{\phi}\cdot
d\phi]+2|\tilde{\phi}|^{2}(\frac{1}{2}|\tilde{\phi}|^{2}+L^{2})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+\mbox{martingale}.$

Inserting (58) and (68) (recall the definition $G$ in (83)) yields

$\displaystyle d|\tilde{\phi}|^{4}$
$\displaystyle=\frac{1}{2}|\tilde{\phi}|^{4}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+2|\tilde{\phi}|^{2}L^{2}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+2|\tilde{\phi}|^{2}(\frac{1}{2}|\tilde{\phi}|^{2}+L^{2})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+\mbox{martingale}$
$\displaystyle=(\frac{3}{2}|\tilde{\phi}|^{4}+4L^{2}|\tilde{\phi}|^{2})\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+\mbox{martingale},$

which indeed implies the claim. ∎

###### Proof of Lemma 3.3..

Identification of $d\det\tilde{F}$. We claim that

(84) $\displaystyle d\det\tilde{F}=\tilde{\phi}^{i}(\operatorname{\rm
adj}\tilde{F})^{t}.\nabla\partial_{i}d\phi.$

In particular, it follows from (84) that $\det F$ is a martingale, which
implies (49). The adjugate $\operatorname{\rm adj}F=(\det F)F^{-1}$, an
endomorphism of tangent space, naturally arises in (84) as the differential of
the determinate: If $F$ depends on some parameter $s$, we have

(85) $\displaystyle\frac{d}{ds}\det F=(\det F)\operatorname{\rm
tr}F^{-1}\frac{d}{ds}F=(\operatorname{\rm adj}F)^{t}.\frac{d}{ds}F.$

Using (85), and appealing to the definitions (60) and (61), we obtain from
(44) by Itô’s formula272727Recall that we omit brackets based on the
convention that $\otimes$ binds more tightly than any other product.

$\displaystyle d\det\tilde{F}$ $\displaystyle=(\operatorname{\rm
adj}\tilde{F})^{t}.\tilde{F}\nabla d\phi+\tilde{\phi}^{i}(\operatorname{\rm
adj}\tilde{F})^{t}.\nabla\partial_{i}d\phi\phantom{\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}}$
$\displaystyle+\big{(}\tilde{F}^{t}e^{1}\otimes
e_{1}\diamond\tilde{F}^{t}e^{2}\otimes e_{2}-\tilde{F}^{t}e^{1}\otimes
e_{2}\diamond\tilde{F}^{t}e^{2}\otimes
e_{1}\big{)}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}$
$\displaystyle+\big{(}e^{1}\otimes e_{1}\otimes\tilde{\phi}\bullet
e^{2}\otimes e_{2}\otimes\tilde{\phi}-e^{1}\otimes
e_{2}\otimes\tilde{\phi}\bullet e^{2}\otimes
e_{1}\otimes\tilde{\phi}\big{)}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}},$

where we already used Remark 4.3. By (77) and (78) also the remaining
quadratic variation terms in the two last lines vanish. Since by definition of
the adjunct, $(\operatorname{\rm adj}\tilde{F})^{t}.\tilde{F}\nabla d\phi$
$=(\det\tilde{F}){\rm tr}\nabla d\phi$, which vanishes by (32), this yields
(84).

Identification of $d(\det\tilde{F})^{2}$. We claim that

(86) $d(\det\tilde{F})^{2}=(\operatorname{\rm
adj}\tilde{F})^{t}\otimes\tilde{\phi}\bullet(\operatorname{\rm
adj}\tilde{F})^{t}\otimes\tilde{\phi}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}+\text{martingale},$

which also implies (50). Indeed, we apply Itô’s formula to (84), in which we
write

$\displaystyle\tilde{\phi}^{i}(\operatorname{\rm
adj}\tilde{F})^{t}.\nabla\partial_{i}d\phi=(\operatorname{\rm
adj}\tilde{F})^{t}\otimes\tilde{\phi}.\nabla^{2}d\phi,$

and insert the definition (61). This yields (86).

Identification of $d|\tilde{F}|^{2}$. We claim

(87) $\displaystyle d|\tilde{F}|^{2}=2(\tilde{F}^{*}\tilde{F})^{t}.\nabla
d\phi+2\tilde{F}^{*t}\otimes\tilde{\phi}.\nabla^{2}d\phi+\big{(}\frac{1}{2}|\tilde{F}|^{2}+\frac{|\tilde{\phi}|^{2}}{2L^{2}}\big{)}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}},$

which implies (51).

We appeal to the representation (62) and apply Itô’s formula in conjunction
with Definition 4.2 and Remark 4.3 to (44), which yields

$\displaystyle d|\tilde{F}|^{2}$ $\displaystyle=2\operatorname{\rm
tr}\tilde{F}^{*}\tilde{F}\nabla d\phi+2\tilde{\phi}^{i}\operatorname{\rm
tr}\tilde{F}^{*}\nabla\partial_{i}d\phi$
$\displaystyle+\sum_{i,j=1}^{2}\big{(}\tilde{F}^{t}e^{i}\otimes
e_{j}\diamond\tilde{F}^{t}e^{i}\otimes
e_{j}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+e^{i}\otimes
e_{j}\otimes\tilde{\phi}\bullet e^{i}\otimes
e_{j}\otimes\tilde{\phi}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}\big{)}.$

Appealing to the formulas (66), in conjunction with
$|\tilde{F}^{t}|^{2}=|\tilde{F}|^{2}$, and (74) we obtain (87).

Identification of $d|\tilde{F}|^{4}$. We claim

(88) $\displaystyle d|\tilde{F}|^{4}$
$\displaystyle=\frac{3}{2}|\tilde{F}|^{4}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}-2\det{\tilde{F}}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+|\tilde{F}|^{2}|\tilde{\phi}|^{2}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}+4\,\tilde{F}^{*t}\otimes\tilde{\phi}\bullet\tilde{F}^{*t}\otimes\tilde{\phi}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}$
$\displaystyle+\text{martingale},$

which implies (52). Indeed, from (87) and Itô’s formula we obtain

$\displaystyle d|\tilde{F}|^{4}$
$\displaystyle=|\tilde{F}|^{4}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+|\tilde{F}|^{2}|\tilde{\phi}|^{2}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}$
$\displaystyle+4(\tilde{F}^{*}\tilde{F})^{t}\diamond(\tilde{F}^{*}\tilde{F})^{t}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}}+4\,\tilde{F}^{*t}\otimes\tilde{\phi}\bullet\tilde{F}^{*t}\otimes\tilde{\phi}\frac{d\tilde{\lambda}^{2}}{L^{2}\tilde{\lambda}^{2}}+\text{martingale}.$

We now appeal to formula (65) to the symmetric
$G=(\tilde{F}^{*}\tilde{F})^{t}$ to the effect of

(89)
$(\tilde{F}^{*}\tilde{F})^{t}\diamond(\tilde{F}^{*}\tilde{F})^{t}=\frac{1}{8}(\operatorname{\rm
tr}(\tilde{F}^{*}\tilde{F})^{t})^{2}-\frac{1}{2}\det(\tilde{F}^{*}\tilde{F})^{t}=\frac{1}{8}|\tilde{F}|^{4}-\frac{1}{2}(\det\tilde{F})^{2},$

so that (88) follows. ∎

###  4.3. Derivation of asymptotics

We now integrate the odes that we derived in the last section to prove Lemma
3.5 and the asymptotics in Theorem 3.1. We start with the former.

###### Proof of Lemma 3.5..

Growth of $\mathbb{E}|\tilde{\phi}|^{2}$. Our goal is to prove (55). To this
end, we let $x\coloneqq\tilde{\lambda}^{2}$, which by (28) means
$x=1+\varepsilon^{2}\ln L$, and $a\coloneqq\mathbb{E}|\tilde{\phi}|^{2}$.
Equation (53) may be rewritten as

$x^{\frac{1}{2}}\frac{d}{dx}\frac{a}{x^{\frac{1}{2}}}=\frac{da}{dx}-\frac{1}{2}\frac{a}{x}=\frac{\exp(\frac{2}{\varepsilon^{2}}(x-1))}{x}$

and integrated explicitly, i. e.

(90) $\displaystyle
a=x^{\frac{1}{2}}\int_{1}^{x}dy\frac{\exp(\frac{2}{\varepsilon^{2}}(y-1))}{y^{\frac{3}{2}}}=\frac{\exp(\frac{2}{\varepsilon^{2}}(x-1))}{x}\int_{1}^{x}dy(\frac{x}{y})^{\frac{3}{2}}\exp(\frac{2}{\varepsilon^{2}}(y-x)).$

To read off the asymptotics, it is convenient to change variables to
$-\widehat{y}=\frac{1}{\varepsilon^{2}}(y-x)$ and thus
$-\widehat{x}=\frac{1}{\varepsilon^{2}}(1-x)$, which in view of (28) is
equivalent to $\widehat{x}=\ln L\gg 1$, so that

(91)
$\int_{1}^{x}dy(\frac{x}{y})^{\frac{3}{2}}\exp(\frac{2}{\varepsilon^{2}}(y-x))=\varepsilon^{2}\int_{0}^{\widehat{x}}d\widehat{y}(\frac{\varepsilon^{2}\widehat{x}+1}{\varepsilon^{2}(\widehat{x}-\widehat{y})+1})^{\frac{3}{2}}\exp(-2\widehat{y}).$

Due to the presence of $\exp(-2\widehat{y})$, uniformly in $\varepsilon\leq
1$, the main contribution to the integral comes from $\widehat{y}\lesssim 1$.
For these $\widehat{y}$ we have
$\frac{\varepsilon^{2}\widehat{x}+1}{\varepsilon^{2}(\widehat{x}-\widehat{y})+1}\approx
1$ in our regime $\widehat{x}\gg 1$. Hence the integral is approximated as
follows

(92)
$\int_{0}^{\widehat{x}}d\widehat{y}(\frac{\varepsilon^{2}\widehat{x}+1}{\varepsilon^{2}(\widehat{x}-\widehat{y})+1})^{\frac{3}{2}}\exp(-2\widehat{y})\approx\int_{0}^{\infty}d\widehat{y}\exp(-2\widehat{y})=\frac{1}{2},$

so that we learn

$a\approx\frac{\varepsilon^{2}}{2}\frac{\exp(\frac{2}{\varepsilon^{2}}(x-1))}{x}.$

By definition of $x$, $\widehat{x}$ and $a$ this translates into (55).

Growth of $\mathbb{E}|\tilde{\phi}|^{4}$. We will now prove (56). We continue
to use the abbreviations $x=\tilde{\lambda}^{2}$ and
$a=\mathbb{E}|\tilde{\phi}|^{2}$ and introduce
$b\coloneqq\mathbb{E}|\tilde{\phi}|^{4}$ so that (54) assumes the compact form

$x^{\frac{3}{2}}\frac{d}{dx}\frac{b}{x^{\frac{3}{2}}}=\frac{db}{dx}-\frac{3}{2}\frac{b}{x}=4\frac{\exp(\frac{2}{\varepsilon^{2}}(x-1))}{x}a.$

Integrating and inserting (91) yields

(93) $\displaystyle b$
$\displaystyle=4x^{\frac{3}{2}}\int_{1}^{x}dy\frac{\exp(\frac{2}{\varepsilon^{2}}(y-1))}{y^{2}}\int_{1}^{y}dz\frac{\exp(\frac{2}{\varepsilon^{2}}(z-1))}{z^{\frac{3}{2}}}$
$\displaystyle=4\frac{\exp(\frac{4}{\varepsilon^{2}}(x-1))}{x^{2}}\int_{1}^{x}dy(\frac{x}{y})^{\frac{7}{2}}\exp(\frac{2}{\varepsilon^{2}}(y-x))\int_{1}^{y}dz(\frac{y}{z})^{\frac{3}{2}}\exp(\frac{2}{\varepsilon^{2}}(z-x)).$

We continue to transform variables according to
$-\widehat{x}=\frac{1}{\varepsilon^{2}}(1-x)$,
$-\widehat{y}=\frac{1}{\varepsilon^{2}}(y-x)$ and now also
$-\widehat{z}=\frac{1}{\varepsilon^{2}}(z-x)$ to the effect of

$\displaystyle\int_{1}^{x}dy(\frac{x}{y})^{\frac{7}{2}}\exp(\frac{2}{\varepsilon^{2}}(y-x))\int_{1}^{y}dz(\frac{y}{z})^{\frac{3}{2}}\exp(\frac{2}{\varepsilon^{2}}(z-x))$
(94)
$\displaystyle=\varepsilon^{4}\int_{0}^{\widehat{x}}d\widehat{y}(\frac{\varepsilon^{2}\widehat{x}+1}{\varepsilon^{2}(\widehat{x}-\widehat{y})+1})^{\frac{7}{2}}\exp(-2\widehat{y})\int_{\widehat{y}}^{\widehat{x}}d\widehat{z}(\frac{\varepsilon^{2}(\widehat{x}-\widehat{y})+1}{\varepsilon^{2}(\widehat{x}-\widehat{z})+1})^{\frac{3}{2}}\exp(-2\widehat{z}).$

Again, in our regime $\hat{x}\gg 1$, the main contribution comes from
$\hat{y},\hat{z}\lesssim 1$, so that the double integral is

$\displaystyle\approx\int_{0}^{\infty}d\widehat{y}\exp(-2\widehat{y})\int_{\widehat{y}}^{\infty}d\widehat{z}\exp(-2\widehat{z})=\frac{1}{2}\int_{0}^{\infty}d\widehat{y}\exp(-4\widehat{y})=\frac{1}{8},$

so that

$b\approx\frac{\varepsilon^{4}}{2}\frac{\exp(\frac{4}{\varepsilon^{2}}(x-1))}{x^{2}}\quad\text{as}~{}\widehat{x}\gg
1.$

By definition of $x$, $\widehat{x}$ and $b$ this translates into (56). ∎

We note that the l. h. s. integral in (92) is monotone increasing in
$\varepsilon$ and thus estimated as

$\displaystyle\int_{0}^{\widehat{x}}d\widehat{y}(\frac{\varepsilon^{2}\widehat{x}+1}{\varepsilon^{2}(\widehat{x}-\widehat{y})+1})^{\frac{3}{2}}\exp(-2\widehat{y})$
(95)
$\displaystyle\leq\int_{0}^{\widehat{x}}d\widehat{y}(\frac{\widehat{x}+1}{\widehat{x}-\widehat{y}+1})^{\frac{3}{2}}\exp(-2\widehat{y})\leq\int_{0}^{\infty}d\widehat{y}(1+\widehat{y})^{\frac{3}{2}}\exp(-2\widehat{y})<\infty.$

Hence an inspection of the proof of (55) establishes the pre-asymptotic
estimate (96) below. Since the inner integral in (4.3) is largest for
$\widehat{y}=0$, it is also estimated by the constant on the r. h. s. of
(4.3); the outer integral is then treated like (4.3) with the exponent
$\frac{3}{2}$ replaced by $\frac{7}{2}$. This yields the pre-asymptotic
estimate (97).

######  Remark 4.7.

[see [6, Lemma 4]]. We have for $\varepsilon\leq 1$

(96) $\displaystyle\mathbb{E}|\tilde{\phi}|^{2}$
$\displaystyle\lesssim\varepsilon^{2}\frac{L^{2}}{\tilde{\lambda}^{2}},$ (97)
$\displaystyle\mathbb{E}|\tilde{\phi}|^{4}$
$\displaystyle\lesssim\varepsilon^{4}\frac{L^{4}}{\tilde{\lambda}^{4}}.$

We collected all ingredients to prove the asymptotics stated in Theorem 3.1.

###### Proof of asymptotics in Theorem 3.1..

Asymptotics of $\mathbb{E}|\tilde{F}|^{2}$. As in the proof of Lemma 3.5, we
use the convention $x=\tilde{\lambda}^{2}$ and now consider
$A\coloneqq\mathbb{E}|\tilde{F}|^{2}$ to rewrite (51) as

$x^{\frac{1}{2}}\frac{d}{dx}\frac{A}{x^{\frac{1}{2}}}=\frac{dA}{dx}-\frac{1}{2}\frac{A}{x}=\frac{1}{2}\frac{a}{x}\exp(-\frac{2}{\varepsilon^{2}}(x-1)).$

By virtue of (90) and the initial condition $F=\rm id$ and thus $A=2$ at $x=1$
we have

$\displaystyle\frac{A}{x^{\frac{1}{2}}}-2$
$\displaystyle=\frac{1}{2}\int_{1}^{x}dy\frac{a}{y^{\frac{3}{2}}}\exp(-\frac{2}{\varepsilon^{2}}(y-1))$
$\displaystyle\stackrel{{\scriptstyle(\ref{ho50})}}{{=}}\frac{1}{2}\int_{1}^{x}dy\frac{1}{y^{\frac{5}{2}}}\int_{1}^{y}dz(\frac{y}{z})^{\frac{3}{2}}\exp(\frac{2}{\varepsilon^{2}}(z-y)).$

We continue to change the inner variable according to
$-\widehat{z}=\frac{1}{\varepsilon^{2}}(z-y)$, but change the outer one
according to $-\widehat{y}=\frac{1}{\varepsilon^{2}}(1-y)$, rather like
$-\widehat{x}=\frac{1}{\varepsilon^{2}}(1-x)$. This yields

(98)
$\frac{A}{x^{\frac{1}{2}}}-2=\frac{\varepsilon^{2}}{2}\int_{0}^{\widehat{x}}d\widehat{y}\frac{\varepsilon^{2}}{(\varepsilon^{2}\widehat{y}+1)^{\frac{5}{2}}}\int_{0}^{\widehat{y}}d\widehat{z}(\frac{\varepsilon^{2}\widehat{y}+1}{\varepsilon^{2}(\widehat{y}-\widehat{z})+1})^{\frac{3}{2}}\exp(-2\widehat{z}).$

By (4.3), the inner integral is $\lesssim 1$. Hence the double integral is

$\displaystyle\lesssim\int_{0}^{\infty}d\widehat{y}\frac{\varepsilon^{2}}{(\varepsilon^{2}\widehat{y}+1)^{\frac{5}{2}}}=\int_{1}^{\infty}dy\frac{1}{y^{\frac{5}{2}}}<\infty.$

Therefore in the regime $\varepsilon\ll 1$, (98) yields

$A\approx 2{x^{\frac{1}{2}}},$

which translates back to (40).

Rough upper bound on $\mathbb{E}|\tilde{F}|^{4}$. By the universality of
$\bullet$ established in Appendix A.2, the last term in (52) is estimated by
the penultimate one:

$\displaystyle\mathbb{E}(\tilde{F}^{*t}\otimes\tilde{\phi}\bullet\tilde{F}^{*t}\otimes\tilde{\phi})\lesssim\mathbb{E}|\tilde{\phi}|^{2}|\tilde{F}|^{2};$

by Cauchy-Schwarz and (97), the latter is estimated as follows

$\displaystyle\frac{1}{L^{2}}\mathbb{E}|\tilde{\phi}|^{2}|\tilde{F}|^{2}\lesssim\frac{\varepsilon^{2}}{\tilde{\lambda}^{2}}\mathbb{E}^{\frac{1}{2}}|\tilde{F}|^{4}.$

Hence with the abbreviations $B\coloneqq\mathbb{E}|\tilde{F}|^{4}$ and
$C\coloneqq\mathbb{E}(\det\tilde{F})^{2}$, (52) yields

(99)
$\displaystyle|\frac{dB}{dx}-(\frac{3}{2}B-2C)\frac{1}{x}|\lesssim\varepsilon^{2}\frac{B^{\frac{1}{2}}}{x^{2}}.$

Since $C\geq 0$, (99) in particular entails the differential inequality

$x^{\frac{3}{2}}\frac{d}{dx}\frac{B}{x^{\frac{3}{2}}}=\frac{dB}{dx}-\frac{3}{2}\frac{B}{x}\lesssim\varepsilon^{2}\frac{B^{\frac{1}{2}}}{x^{2}}=\varepsilon^{2}\frac{1}{x^{\frac{5}{4}}}(\frac{B}{x^{\frac{3}{2}}})^{\frac{1}{2}},$

which we rewrite as

$\frac{d}{dx}(\frac{B}{x^{\frac{3}{2}}})^{\frac{1}{2}}\lesssim\varepsilon^{2}\frac{1}{x^{\frac{11}{4}}}.$

Integration with the initial condition $B=4$ at $x=1$ gives

(100)
$\displaystyle(\frac{B}{x^{\frac{3}{2}}})^{\frac{1}{2}}-2\lesssim\varepsilon^{2}\quad\mbox{and
thus}\quad B\lesssim x^{\frac{3}{2}}.$

Asymptotics of $\mathbb{E}(\det\tilde{F}-1)^{2}$. We recall the definitions
$B=\mathbb{E}|\tilde{F}|^{4}$ and $C=\mathbb{E}(\det\tilde{F})^{2}$; by the
same argument as above, that is, universality of $\bullet$, Cauchy-Schwarz,
and (97), the ODE (50) yields

$\displaystyle|\frac{dC}{dx}|\lesssim\varepsilon^{2}\frac{B^{\frac{1}{2}}}{x^{2}}.$

By the upper bound (100), this implies

(101)
$\displaystyle|\frac{dC}{dx}|\lesssim\varepsilon^{2}\frac{1}{x^{\frac{5}{4}}}.$

Integration with the initial condition $C=1$ at $x=1$ gives

(102) $\displaystyle|C-1|\lesssim\varepsilon^{2},$

which turns into (39) in view of $\mathbb{E}\det\tilde{F}=1$ as a consequence
of (49).

Asymptotics of $\mathbb{E}|\tilde{F}|^{4}$. We return to (99) and combine it
with (101) to obtain a differential inequality rewrite on $\frac{3}{2}B-2C$,
and use (100) to estimate the r. h. s.:

$\displaystyle|\frac{d}{dx}(\frac{3}{2}B-2C)-\frac{3}{2}(\frac{3}{2}B-2C)\frac{1}{x}|\lesssim\varepsilon^{2}\frac{1}{x^{\frac{5}{4}}}.$

We rewrite this once more as

$\displaystyle|\frac{d}{dx}\frac{1}{x^{\frac{3}{2}}}(\frac{3}{2}B-2C)|\lesssim\varepsilon^{2}\frac{1}{x^{\frac{11}{4}}}.$

Integration with the initial condition $B=4$ and $C=1$ at $x=1$ gives

$\displaystyle|\frac{1}{x^{\frac{3}{2}}}(\frac{3}{2}B-2C)-4|\lesssim\varepsilon^{2},$

which together with (102) yields

$\displaystyle|\frac{1}{x^{\frac{3}{2}}}(\frac{3}{2}B-2)-4|\lesssim\varepsilon^{2}.$

It is the weaker version

$\displaystyle B\approx\frac{8}{3}x^{\frac{3}{2}}+\frac{4}{3}$

that corresponds to (41). ∎

###  4.4. Proof of main result

We now complete the proof of Theorem 3.1 by establishing (42). The asymptotics
were already established in Section 4.3.

###### Proof of Proposition 3.2 — Non equi-integrability of
$(\mathbb{E}|\tilde{F}|^{2})^{-\frac{1}{2}}\tilde{F}$..

The same Itô argument that lead from (87) to (52) also implies that that for
any test function/observable $\zeta$ of $r=|\tilde{F}|^{2}$

$\displaystyle d\mathbb{E}\zeta(|\tilde{F}|^{2})$
$\displaystyle=\mathbb{E}\Big{(}\frac{d\zeta}{dr}(|\tilde{F}|^{2})\big{(}\frac{1}{2}|\tilde{F}|^{2}+\frac{|\tilde{\phi}|^{2}}{2L^{2}})$
(103)
$\displaystyle+\frac{1}{2}\frac{d^{2}\zeta}{dr^{2}}(|\tilde{F}|^{2})\big{(}\frac{1}{2}|\tilde{F}|^{4}-2(\det\tilde{F})^{2}+\frac{(2\tilde{F}^{*t}\otimes\tilde{\phi})\bullet(2\tilde{F}^{*t}\otimes\tilde{\phi})}{L^{2}}\big{)}\Big{)}\frac{d\tilde{\lambda}^{2}}{\tilde{\lambda}^{2}};$

in fact, (52) amounts to (4.4) for $\zeta(r)=r^{2}$. Since according to
Appendix A.2, the form $\bullet$ is universal, we have

$\displaystyle|\tilde{F}^{*t}\otimes\tilde{\phi}\bullet\tilde{F}^{*t}\otimes\tilde{\phi}|\lesssim|\tilde{\phi}|^{2}|\tilde{F}|^{2}\quad\mbox{next
to}\quad|{\rm det}\tilde{F}|\lesssim|\tilde{F}|^{2}.$

Hence we obtain the differential inequality (where $r=|\tilde{F}|^{2}$ is the
argument of $\zeta$ and its derivatives)

(104)
$\displaystyle\Big{|}\frac{d}{d\tau}\mathbb{E}\zeta-\mathbb{E}\big{(}\frac{1}{2}|\tilde{F}|^{2}\frac{d\zeta}{dr}+\frac{1}{4}|\tilde{F}|^{4}\frac{d^{2}\zeta}{dr^{2}}\big{)}\Big{|}\lesssim\mathbb{E}\Big{(}|\det\tilde{F}||\tilde{F}|^{2}|\frac{d^{2}\zeta}{dr^{2}}|+\frac{|\tilde{\phi}|^{2}}{L^{2}}\big{(}|\frac{d\zeta}{dr}|+|\tilde{F}|^{2}|\frac{d^{2}\zeta}{dr^{2}}|\big{)}\Big{)}$

in terms of the time and space-like variable

(105) $\displaystyle\tau=\ln\tilde{\lambda}^{2}\quad\mbox{and}\quad
r=|\tilde{F}|^{2},$

respectively.

This motivates to consider a $\tau$-dependent $\zeta=\zeta(\tau,r)$ that
satisfies the backward equation

(106)
$\displaystyle\frac{\partial\zeta}{\partial\tau}+\frac{r}{2}\frac{\partial\zeta}{\partial
r}+\frac{r^{2}}{4}\frac{\partial^{2}\zeta}{\partial r^{2}}=0$

so that by the (pre-asymptotic w. r. t. $L$) bounds (39) & (40) and (96), and
using the notation (105), (104) implies

$\displaystyle|\frac{d}{d\tau}\mathbb{E}\zeta(\tau,|\tilde{F}|^{2})|\lesssim\sup_{r}r|\frac{\partial^{2}\zeta}{\partial
r^{2}}|+\varepsilon^{2}e^{-\tau}\sup_{r}|\frac{\partial\zeta}{\partial r}|.$

We will use this after integration in $\tau^{\prime}\in(0,\tau)$ and appealing
to $|\tilde{F}_{\tau=0}|^{2}=2$, see (33):

(107)
$\displaystyle\mathbb{E}\zeta(\tau,|\tilde{F}|^{2})\lesssim\zeta(0,2)+\int_{0}^{\tau}d\tau^{\prime}\sup_{r}r|\frac{\partial^{2}\zeta}{\partial
r^{2}}(\tau^{\prime},r)|+\varepsilon^{2}\int_{0}^{\tau}d\tau^{\prime}e^{-\tau^{\prime}}\sup_{r}|\frac{\partial\zeta}{\partial
r}(\tau^{\prime},r)|.$

We now construct a suitable solution $\zeta$ of the backwards parabolic
equation (106), which amounts to a duality argument in the jargon of PDE
theory. To this purpose, we pass to the natural282828Note that in view of
(105), this amounts to the natural scale $r=\tilde{\lambda}\hat{r}$ for
$r=|\tilde{F}|^{2}$; $\zeta=\hat{r}\hat{\zeta}$ means that $\hat{\zeta}$
modulates the properly rescaled second moment of $\tilde{F}$. variables

(108) $\displaystyle
r=e^{\frac{\tau}{2}}\hat{r}\stackrel{{\scriptstyle(\ref{ao04})}}{{=}}\tilde{\lambda}\hat{r}\quad\mbox{and}\quad\zeta=\hat{r}\hat{\zeta},$

in the sense of $\zeta(\tau,e^{\frac{\tau}{2}}\hat{r})$
$=\hat{r}\hat{\zeta}(\tau,\hat{r})$, which implies
$\frac{\partial\zeta}{\partial\tau}+\frac{r}{2}\frac{\partial\zeta}{\partial
r}$ $=\hat{r}\frac{\partial\hat{\zeta}}{\partial\tau}$. Hence by
$r^{2}\frac{\partial}{\partial r^{2}}$
$=\hat{r}^{2}\frac{\partial^{2}}{\partial\hat{r}^{2}}$ and
$\frac{\partial^{2}\zeta}{\partial\hat{r}^{2}}$
$=\hat{r}\frac{\partial^{2}\hat{\zeta}}{\partial\hat{r}^{2}}+2\frac{\partial\hat{\zeta}}{\partial\hat{r}}$,
the form of (106) is preserved:

(109)
$\displaystyle\frac{\partial\hat{\zeta}}{\partial\tau}+\frac{\hat{r}}{2}\frac{\partial\hat{\zeta}}{\partial\hat{r}}+\frac{\hat{r}^{2}}{4}\frac{\partial^{2}\hat{\zeta}}{\partial\hat{r}^{2}}=0.$

The scale-invariant (in $\hat{r}$) nature of (109) suggests to pass to
logarithmic independent variable

(110) $\displaystyle\hat{\sigma}=\ln\hat{r}.$

Indeed, because of
$\hat{r}\frac{\partial}{\partial\hat{r}}=\frac{\partial}{\partial\hat{\sigma}}$
and $\hat{r}^{2}\frac{\partial}{\partial\hat{r}^{2}}$
$=\frac{\partial^{2}}{\partial\hat{\sigma}^{2}}-\frac{\partial}{\partial\hat{\sigma}}$,
(109) turns into the constant-coefficient equation

(111)
$\displaystyle\frac{\partial\hat{\zeta}}{\partial\tau}+\frac{1}{4}\frac{\partial\hat{\zeta}}{\partial\hat{\sigma}}+\frac{1}{4}\frac{\partial^{2}\hat{\zeta}}{\partial\hat{\sigma}^{2}}=0.$

For momentarily fixed $\tau\gg 1$ and $\hat{\sigma}\gg 1$ (as above we use
$\tau^{\prime}\in(0,\tau)$ and $\hat{\sigma}^{\prime}$ as active variables
instead) we consider the $C^{2}$ terminal292929as opposed to initial, in view
of the backwards nature of (111) data given by

(115)
$\displaystyle\hat{\zeta}(\tau,\hat{\sigma}^{\prime})\left\\{\begin{array}[]{ll}=1&\mbox{for}\;\hat{\sigma}^{\prime}\leq\hat{\sigma}\\\
\in[0,1]&\mbox{for all}\;\hat{\sigma}^{\prime}\\\
=0&\mbox{for}\;\hat{\sigma}^{\prime}\geq\hat{\sigma}+1\end{array}\right\\}\quad\mbox{and}\quad|\frac{\partial^{2}\hat{\zeta}}{d\hat{\sigma}^{\prime
2}}(\tau,\hat{\sigma}^{\prime})|\lesssim 1.$

These terminal data are chosen such that the l. h. s. of (107) controls the
second moments of $\tilde{F}$ up to tails; indeed, by the definitions (106),
(108), and (110), we have

(116)
$\displaystyle\mathbb{E}\zeta(\tau,|\tilde{F}|^{2})\geq\frac{1}{\tilde{\lambda}}\mathbb{E}I(|\tilde{F}|^{2}\leq\tilde{\lambda}\hat{r})|\tilde{F}|^{2}.$

We now choose the parameter $\hat{\sigma}=\ln\hat{r}$ so large that the first
r. h. s. term of (107) satisfies

(117)
$\displaystyle\zeta(\tau^{\prime}=0,r^{\prime}=2)\stackrel{{\scriptstyle(\ref{ao06}),(\ref{ao10})}}{{=}}2\hat{\zeta}(\tau^{\prime}=0,\hat{\sigma}^{\prime}=\ln
2)\ll 1.$

To this purpose we note that the (translation invariant) semi-group kernel for
(111) with terminal data at $\tau^{\prime}=\tau$ is a Gaussian of mean
$-\frac{1}{4}(\tau-\tau^{\prime})$ and variance
$\frac{1}{2}(\tau-\tau^{\prime})$.303030This is conveniently seen by making a
Gaussian Ansatz with general mean and variance that lead to the specific form
claimed above. Hence in view of (115) in form of
$\hat{\zeta}(\tau,\hat{\sigma}^{\prime})$ $\leq
I(\hat{\sigma}^{\prime}<\hat{\sigma}+1)$, we have in terms of the error
function

$\displaystyle\hat{\zeta}(\tau^{\prime},\hat{\sigma}^{\prime})\leq
F_{\mathcal{N}(0,1)}\Big{(}\frac{-\hat{\sigma}^{\prime}+\hat{\sigma}+1-\frac{1}{4}(\tau-\tau^{\prime})}{\sqrt{\frac{1}{2}(\tau-\tau^{\prime})}}\Big{)}$

and thus $\hat{\zeta}(\tau^{\prime}=0,\hat{\sigma}^{\prime}=\ln 2)$ $\leq
F_{\mathcal{N}(0,1)}(\frac{1-\ln
2+\hat{\sigma}-\frac{\tau}{4}}{\sqrt{\frac{\tau}{2}}})$. Hence (117) is
satisfied provided $\frac{\tau}{4}-\hat{\sigma}$ $\gg\sqrt{\tau}$, which in
view (105) and (110) amounts to (43).

We now turn to the two last r. h. s. terms of (107) and shall establish

(118)
$\displaystyle\int_{0}^{\tau}d\tau^{\prime}\sup_{r}r|\frac{\partial^{2}\zeta}{\partial
r^{2}}(\tau^{\prime},r)|$ $\displaystyle\lesssim\frac{1}{\sqrt{\tau}},$ (119)
$\displaystyle\int_{0}^{\tau}d\tau^{\prime}e^{-\tau^{\prime}}\sup_{r}|\frac{\partial\zeta}{\partial
r}(\tau^{\prime},r)|$ $\displaystyle\lesssim 1.$

For this we note that by (108) and (110)

$\displaystyle\frac{\partial\zeta}{\partial r}$
$\displaystyle=e^{-\frac{\tau^{\prime}}{2}}\frac{\partial}{\partial\hat{r}}\hat{r}\hat{\zeta}=e^{-\frac{\tau^{\prime}}{2}}(\frac{\partial}{\partial\hat{\sigma}^{\prime}}+1)\hat{\zeta},$
$\displaystyle r\frac{\partial^{2}\zeta}{\partial r^{2}}$
$\displaystyle=e^{-\frac{\tau^{\prime}}{2}}\hat{r}\frac{\partial^{2}}{\partial\hat{r}^{2}}\hat{r}\hat{\zeta}=e^{-\frac{\tau^{\prime}}{2}}(\frac{\partial}{\partial\hat{\sigma}^{\prime}}+1)\frac{\partial\hat{\zeta}^{\prime}}{\partial\hat{\sigma}^{\prime}}.$

In order to control suprema in $\sigma^{\prime}$ of
$(\frac{\partial}{\partial\hat{\sigma}}+1)\hat{\zeta}$ and
$(\frac{\partial}{\partial\hat{\sigma}}+1)\frac{\partial\hat{\zeta}}{\partial\tilde{\sigma}}$
as a function of $\tau^{\prime}$, the drift term in (111) is irrelevant, so
that we may appeal to standard estimates on the (backward) heat equation. From
these we learn that the estimates on the terminal data, cf. (115), both
uniform
$\sup_{\hat{\sigma}^{\prime}}(|\hat{\zeta}|+|\frac{\partial\hat{\zeta}}{\partial\hat{\sigma}^{\prime}}|+|\frac{\partial^{2}\hat{\zeta}}{\partial\hat{\sigma}^{\prime
2}}|)$ $\lesssim 1$ and integral $\int
d\hat{\sigma}^{\prime}(|\frac{\partial\hat{\zeta}}{\partial\hat{\sigma}^{\prime}}|+|\frac{\partial^{2}\hat{\zeta}}{\partial\hat{\sigma}^{\prime
2}}|)$ $\lesssim 1$, lead to

$\displaystyle|\hat{\zeta}|\lesssim
1,\quad|\frac{\partial\hat{\zeta}}{\partial\hat{\sigma}^{\prime}}|\lesssim\frac{1}{\sqrt{1+\tau-\tau^{\prime}}},\quad|\frac{\partial^{2}\hat{\zeta}}{\partial\hat{\sigma}^{\prime
2}}|\lesssim\frac{1}{1+\tau-\tau^{\prime}}.$

These estimates immediately imply (119). For (118), we appeal to
$\int_{0}^{\tau}d\tau^{\prime}e^{-\frac{\tau^{\prime}}{2}}$
$\frac{1}{\sqrt{1+\tau-\tau^{\prime}}}$ $\lesssim\frac{1}{\sqrt{\tau}}$, which
is elementary.

In view of (40) and $\varepsilon^{2}\ll 1$, (42) follows from inserting (116),
(117), (118), and (119) into (107). ∎

Finally, we collected all ingredients to establish Theorem 2.1.

###### Proof of Theorem 2.1..

Asymptotics of $\mathbb{E}|\det F|$. Applying Cauchy-Schwarz to the elementary
inequality $|\det F-\det\tilde{F}|$ $\lesssim(|F|+|\tilde{F}|)|F-\tilde{F}|$
$\lesssim|\tilde{F}||F-\tilde{F}|+|F-\tilde{F}|^{2}$ we obtain

(120) $\displaystyle\mathbb{E}|\det
F-\det\tilde{F}|\lesssim(\mathbb{E}|\tilde{F}|^{2})^{\frac{1}{2}}(\mathbb{E}|F-\tilde{F}|^{2})^{\frac{1}{2}}+\mathbb{E}|F-\tilde{F}|^{2}.$

In order to establish the estimate (38) on $\mathbb{E}|F-\tilde{F}|^{2}$, we
apply the divergence to the decomposition (35) to obtain

$\nabla\cdot a({\rm id}+\nabla\tilde{\phi})=\nabla\cdot f$

so that by (20) and the stationarity of $\tilde{\phi}$

$\nabla\cdot a(\nabla\phi-\nabla\tilde{\phi})=-\nabla\cdot
f\quad\mbox{and}\quad\mathbb{E}\nabla\phi=\mathbb{E}\nabla\tilde{\phi}=0.$

Hence because of the form (19) of $a$ (which implies $\xi\cdot
a\xi=|\xi|^{2}$), the energy estimate on the level of stationary gradient
fields of vanishing expectation takes the form

$\displaystyle\mathbb{E}|\nabla\phi-\nabla\tilde{\phi}|^{2}\leq\mathbb{E}|f|^{2},$

which by the definitions (22) & (37) and the estimate (36) yields (38).

Inserting (38) and (40) into (120), we obtain in the regime of $\varepsilon\ll
1$

(121) $\displaystyle\mathbb{E}|\det
F-\det\tilde{F}|\lesssim(\varepsilon+\varepsilon^{2})\tilde{\lambda}\sim\varepsilon\tilde{\lambda}.$

Note that this type of argument also implies (23), see [6, (103)]. Using
$\mathbb{E}|\det F|$ $\leq\mathbb{E}|\det F-\det\tilde{F}|$
$+(\mathbb{E}(\det\tilde{F}-1)^{2})^{\frac{1}{2}}$ $+1$, into which we insert
(121) and (39), we obtain in the regime of $\varepsilon\ll 1$

$\displaystyle\mathbb{E}|\det F|\lesssim\varepsilon\tilde{\lambda}+1,$

which in view of (23), also in form of $\tilde{\lambda}\approx\lambda$,
assumes the desired form of (24).

The failure of equi-integrability for $(\mathbb{E}|F|^{2})^{-\frac{1}{2}}F$.
We start from the elementary inequality

$\frac{1}{2}|F|^{2}I(|F|^{2}\leq\frac{1}{4}r)\leq|\tilde{F}|^{2}I(|\tilde{F}|^{2}\leq
r)+|F-\tilde{F}|^{2}.$

Choosing $r=\hat{r}\mathbb{E}|\tilde{F}|^{2}$ with $\hat{r}$ as in (43) and
taking the expectation, we obtain from (38) and (42) in the regime
$\varepsilon\ll 1$

$\mathbb{E}|F|^{2}I(|F|^{2}\leq\frac{1}{4}\hat{r}\mathbb{E}|\tilde{F}|^{2})\ll\mathbb{E}|\tilde{F}|^{2}+\tilde{\lambda}\stackrel{{\scriptstyle(\ref{mr05})}}{{\sim}}\tilde{\lambda}.$

In view of (23), this implies the desired (25). ∎

##  Appendix A Identification of quadratic variation of the drivers

In this appendix, we identify the quadratic variations of $\nabla d\phi$ and
$\nabla^{2}d\phi$. More specifically, we diagonalize the positive semi-
definite quadratic forms $\diamond$ and $\bullet$ introduced in Definition
4.2. In particular, we will learn that both are universal313131i. e.
independent of any parameter like $L$ or $\varepsilon$. This is an exercise in
representation theory for the group of rotations, which we reproduce for the
convenience of the reader. It is natural to start with $\nabla d\phi$ in view
of its definition via $d\psi$, see (31), which is fully characterized by (29).

###  A.1. Quadratic variation of $\nabla d\phi$

We will now identify $\diamond$ arising from (60). A less representation-
theoretic argument for the contraction (66) was given in [19, (26)]. In view
of the canonical isomorphism

$\displaystyle(\mbox{cotangent space})\otimes(\mbox{tangent
space})\cong\mbox{endomorphisms of tangent space},$

we (also) view $\diamond$ as a symmetric bilinear form on the latter. By
isotropy, $\diamond$ is invariant under the action of the rotation group via
conjugation of the endomorphisms:

(122) $\displaystyle Q.G=QGQ^{-1}=QGQ^{*}\quad\mbox{or equivalently}\quad
Q.(\xi\otimes\dot{x})=Q^{-t}\xi\otimes Q\dot{x}.$

This group action leaves the following decomposition (71) of endomorphism
space invariant. This motivates to introduce the following basis on this four-
dimensional space

(125) $\displaystyle\begin{array}[]{ll}E^{1}:=e^{1}\otimes e_{1}-e^{2}\otimes
e_{2},&E^{2}:=e^{1}\otimes e_{2}+e^{2}\otimes e_{1},\\\\[4.30554pt]
E^{3}:=e^{1}\otimes e_{2}-e^{2}\otimes e_{1},&E^{4}:=e^{1}\otimes
e_{1}+e^{2}\otimes e_{2}.\end{array}$

Indeed, $\\{E^{1},E^{2}\\}$ is a basis of the space of trace-free symmetric
endomorphisms, $E^{3}=J$ is a skew endomorphism323232We denote the rotation by
$\frac{\pi}{2}$ (in the positive direction) on tangent and cotangent space
both with $J$., and $E^{4}={\rm id}$ is an isotropic endomorphism. This choice
is natural in many ways: $E^{1},\ldots,E^{4}$ are orthogonal w. r. t. the
inner product $:$ defined in (70), more precisely

(126) $\displaystyle E^{m}:E^{n}=0\quad\mbox{unless}~{}m=n\quad\mbox{and}\quad
E^{m}:E^{m}=2,$

as used in the argument for (66).

We claim that this basis is orthogonal also w. r. t. $\diamond$, i. e.

(127) $\displaystyle E^{m}\diamond E^{n}=0\quad\mbox{unless}\quad
m=n\quad\mbox{and}\quad$ (128) $\displaystyle 2E^{1}\diamond
E^{1}=2E^{2}\diamond E^{2}=E^{3}\diamond E^{3}=1\quad\mbox{and}\quad
E^{4}\diamond E^{4}=0.$

We start with the first component in (71) by establishing conformality:

(129) $\displaystyle E^{1}\diamond E^{1}=E^{2}\diamond
E^{2}\quad\mbox{and}\quad E^{1}\diamond E^{2}=0.$

We give an abstract argument for (129) which seems a bit pretentious here but
is useful when appealing to it in Subsection A.2. Let us denote by
$Q_{\theta}$ the rotation by $\frac{1}{2}\theta$; the factor of $\frac{1}{2}$
in front of the angle is natural in view of $(\mbox{trace free symmetric})$
$=(\mbox{constant})\times(\mbox{reflection})$, since angles between reflection
lines should be measured mod $\pi$. In view of

(130)
$\displaystyle\frac{dQ_{\theta}}{d\theta}\Big{|}_{\theta=0}=\frac{1}{2}J\quad\mbox{and}\quad\frac{dQ_{\theta}^{-1}}{d\theta}\Big{|}_{\theta=0}=\frac{dQ_{-\theta}}{d\theta}\Big{|}_{\theta=0}=-\frac{1}{2}J,$

the generator of the action (122) is given by

$\displaystyle\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.G=\frac{1}{2}(JG-
GJ).$

The definitions (125) of $E^{1},E^{2}$ are made such that they are related by
the generator:

(131) $\displaystyle\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.E^{1}=E^{2}$

By the action properties $Q_{\theta}^{-1}.G=Q_{-\theta}.G$ and
$Q_{\theta+\theta^{\prime}}.G$ $=Q_{\theta}.Q_{\theta^{\prime}}.G$, (131)
entails

(132)
$\displaystyle\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.E^{2}=-E^{1}\quad\mbox{and}\quad\frac{d^{2}}{d\theta^{2}}\Big{|}_{\theta=0}Q_{\theta}.E^{1}=-E^{1}.$

We obtain the second part of (129) from applying
$\frac{d}{d\theta}\big{|}_{\theta=0}$ to $Q_{\theta}E^{1}\diamond
Q_{\theta}E^{1}=E^{1}\diamond E^{1}$, which follows from the invariance of
$\diamond$, and then appealing to (131) and the symmetry of $\diamond$.
Likewise, we obtain the first part of (129) from applying
$\frac{d^{2}}{d\theta^{2}}\big{|}_{\theta=0}$ and appealing to (131) and
(132).

We now consider the mixed terms w. r. t. (71). Since $E^{3}$ is a fixed point
under conjugation (with rotations), $G\mapsto E^{3}\diamond G$ is an invariant
linear form on the (dual of) endomorphism space, and thus is a multiple of the
trace. Hence the first and the second components of (71) are orthogonal under
$\diamond$:

(133) $\displaystyle E^{1}\diamond E^{3}=E^{2}\diamond E^{3}=0.$

Next, we argue that $E^{4}$ is in the null space of $\diamond$. To this
purpose, we consider the Fourier transform333333note that its argument $k$
naturally is a cotangent vector of (31), that is,

(134) $\displaystyle\tilde{\lambda}{\mathcal{F}}d\phi\otimes
k=i{\mathcal{F}}\psi J+{\mathcal{F}}d\sigma\otimes Jk.$

Contracting (134) with $k\otimes k^{*}$ by the skew symmetry of $J$ yields
$\tilde{\lambda}|k|^{2}k.{\mathcal{F}}d\phi=0$ and thus343434since
$\operatorname{\rm tr}\dot{x}\otimes\xi=\xi.\dot{x}$ $\operatorname{\rm
tr}{\mathcal{F}}d\phi\otimes k=0$, which undoing the Fourier transform implies
the first item in (32). By definition (125) this means

(135) $\displaystyle E^{4}.\nabla d\phi=0.$

In view of definition (60), this implies

(136) $\displaystyle E^{4}\diamond G=0\quad\mbox{for all}\;G.$

Because of stationarity, the covariation of the pair
$(\partial_{i}d\phi^{k},\partial_{j}d\phi^{l})$ is invariant under permuting
$i$ and $j$. In particular, the covariation of the pairs
$(\partial_{1}d\phi^{1},\partial_{2}d\phi^{2})$ and
$(\partial_{2}d\phi^{1},\partial_{1}d\phi^{2})$ coincide. By definition (60)
this translates into

$\displaystyle e^{1}\otimes e_{1}\diamond e^{2}\otimes e_{2}=e^{1}\otimes
e_{2}\diamond e^{2}\otimes e_{1}.$

Using (125) to express the $e^{i}\otimes e_{j}$’s in terms of the $E^{n}$’s
yields $(E^{4}+E^{1})\diamond(E^{4}-E^{1})$ $=(E^{2}$
$+E^{3})\diamond(E^{2}-E^{3})$, which by (133) and (136) reduces to

(137) $\displaystyle E^{3}\diamond E^{3}=E^{1}\diamond E^{1}+E^{2}\diamond
E^{2}.$

It remains to identify $E^{3}\diamond E^{3}$. To this purpose, we contract
(134) with $J(-ik)\otimes k^{*}$ to the effect of
$\tilde{\lambda}|k|^{2}J(-ik).{\mathcal{F}}d\phi$ $=i{\mathcal{F}}d\psi
JJ(-ik).k^{*}$; since $J^{2}=-{\rm id}$, the latter coincides with
$-|k|^{2}{\mathcal{F}}d\psi$, so that we obtain
$\tilde{\lambda}\operatorname{\rm tr}{\mathcal{F}}d\phi\otimes J(-ik)$
$=-{\mathcal{F}}d\psi$, which in real space amounts to the second item in
(32). By definition (125) we have353535 Since here $J=e_{1}\otimes
e^{2}-e_{2}\otimes e^{1}$, we have $JF=F^{t}e_{1}\otimes
e^{2}-F^{t}e_{2}\otimes Fe^{1}$ and thus $\operatorname{\rm
tr}JF=e_{1}.Fe^{2}-e_{2}.Fe^{1}$ $=(e^{2}\otimes e_{1}-e^{1}\otimes
e_{2}).F=-E^{3}.F$ $\operatorname{\rm tr}JF=-E^{3}.F$; hence the second item
in (32) assumes the form of

(138) $\displaystyle\tilde{\lambda}E^{3}.\nabla d\phi=d\psi.$

In view of definition (60), this means $E^{3}\diamond
E^{3}\,d\tilde{\lambda}^{2}=[d\psi\,d\psi]$ so that by the normalization of
the latter, cf. (29), we thus obtain

(139) $\displaystyle E^{3}\diamond E^{3}=1.$

Now (127) and (128) follow from (129), (133), (136), (137), and (139).

###  A.2. Quadratic variation of $\nabla^{2}d\phi$

In this subsection we identify the bilinear form $\bullet$ introduced in
definition (61). The contraction (75) of $\bullet$ was already identified in
[19, (27)].

Recall that by definition (61), $\bullet$ is a positive semi-definite
symmetric bilinear form on the 8-dimensional tensor space

(140) $\displaystyle\mbox{cotangent space}\otimes\mbox{tangent
space}\otimes\mbox{tangent space}.$

By isotropy, $\bullet$ is invariant w. r. t. the natural action of rotations
$Q$ characterized by

(141) $\displaystyle
Q.(\xi\otimes\dot{x}\otimes\dot{y}):=(Q^{-t}\xi)\otimes(Q\dot{x})\otimes(Q\dot{y})$

and extended by linearity to (140). Analogous to Subsection A.1, there are
natural invariant subspace of (140) under (141), namely

(142) trace-free symmetric
$\displaystyle:=\\{\textit{{E}}|\textit{{E}}^{ij}_{k}\;\mbox{invariant under
permutations of}\;i,j,k\;\mbox{and}\;\textit{{E}}^{ij}_{i}=0\\},$ (143)
isotropic $\displaystyle:=\\{\xi\otimes(e_{1}\otimes e_{1}+e_{2}\otimes
e_{2})|\xi\;\mbox{cotangent vector}\\},$

which obviously have trivial intersection. Clearly, (143) is two-dimensional,
and a basis is given by $\\{\textit{{E}}^{3},\textit{{E}}^{4}\\}$, see (151)
since these tensors can be rewritten as

(144) $\displaystyle\textit{{E}}^{3}=e^{1}\otimes(e_{1}\otimes
e_{1}+e_{2}\otimes
e_{2})\quad\mbox{and}\quad\textit{{E}}^{4}=e^{2}\otimes(e_{2}\otimes
e_{2}+e_{1}\otimes e_{1}).$

As for the dimension of (142), we note that the space of (totally) symmetric
three-tensors is isomorphic to polynomials (in two variables) that are
homogeneous of degree 3, and thus is of dimension 4; the trace condition means
that these polynomials are harmonic, which reduces the dimension to 2. Hence
$\\{\textit{{E}}^{1},\textit{{E}}^{2}\\}$, see (151), is a basis of this
space363636Note that by the evident total symmetry of $\textit{{E}}^{1}$,
$\textit{{E}}^{2}$, only one of the trace conditions has to be checked; it is
convenient to contract the first two indices..

We shall argue that $\bullet$ is diagonal w. r. t.

(151)
<EMAIL_ADDRESS>e_{1}\otimes e_{1}+e^{2}\otimes e_{1}\otimes e_{2}+e^{2}\otimes e_{2}\otimes
e_{1}+e^{1}\otimes e_{2}\otimes e_{2},\\\
\textit{{E}}^{2}&\coloneqq\phantom{1}-&e^{2}\otimes e_{2}\otimes
e_{2}+e^{1}\otimes e_{2}\otimes e_{1}+e^{1}\otimes e_{1}\otimes
e_{2}+e^{2}\otimes e_{1}\otimes e_{1},\\\
\textit{{E}}^{3}&\coloneqq&e^{1}\otimes e_{1}\otimes e_{1}+e^{1}\otimes
e_{2}\otimes e_{2},\\\ \textit{{E}}^{4}&\coloneqq\phantom{1}&e^{2}\otimes
e_{2}\otimes e_{2}+e^{2}\otimes e_{1}\otimes e_{1},\\\
\textit{{E}}^{5}&\coloneqq\phantom{+}2&e^{1}\otimes e_{1}\otimes
e_{1}+e^{2}\otimes e_{1}\otimes e_{2}+e^{2}\otimes e_{2}\otimes e_{1},\\\
\textit{{E}}^{6}&\coloneqq\phantom{+}2&e^{2}\otimes e_{2}\otimes
e_{2}+e^{1}\otimes e_{2}\otimes e_{1}+e^{1}\otimes e_{1}\otimes
e_{2}.\end{array}$

Let us continue to explain why this choice is natural. Clearly, (151) comes in
pairs related by rotation373737 We recall that $J$ is characterized by
$Je_{1}=e_{2}$ and $Je_{2}=-e_{1}$, so that its transpose is given by
$J^{t}e^{2}=e^{1}$ and $J^{t}e^{1}=-e^{2}$, and therefore the inverse of its
transpose behaves like $J$ itself: $J^{-t}e_{1}=e_{2}$ and
${}^{-t}Je_{2}=-e_{1}$. As a consequence, the action $J.$ on $e^{i}\otimes
e_{j}\otimes e_{k}$ replaces an index $1$ by $2$, and an index $2$ by $1$ up
to a minus sign. by the angle $\frac{\pi}{2}$

(152)
$\displaystyle\textit{{E}}^{2}=J.\textit{{E}}^{1},\quad\textit{{E}}^{4}=J.\textit{{E}}^{3},\quad\textit{{E}}^{6}=J.\textit{{E}}^{5}.$

Also the pair $\\{\textit{{E}}^{5},\textit{{E}}^{6}\\}$ is natural: Comparing
(151) to (125), we realize383838We recall that the subscript sym indicates
that the identity holds modulo symmetrization of the last two factors in the
triple tensor product.

$\displaystyle\textit{{E}}^{5}=_{sym}2\,E^{4}\otimes
e_{1}\quad\mbox{and}\quad\textit{{E}}^{6}=_{sym}2\,E^{4}\otimes e_{2},$

which implies

$\displaystyle\textit{{E}}^{5}.\nabla^{2}d\phi=2\,\partial_{1}E^{4}.\nabla
d\phi\quad\mbox{and}\quad\textit{{E}}^{6}.\nabla^{2}d\phi=2\,\partial_{2}E^{4}.\nabla
d\phi.$

Hence we learn from (135) that

(153)
$\displaystyle\textit{{E}}^{5}\bullet\textit{{G}}=\textit{{E}}^{6}\bullet\textit{{G}}=0\quad\mbox{for
all}\;\textit{{G}}.$

These 6 elements indeed form a basis of the 6-dimensional sub-space relevant
for $\bullet$

$\displaystyle(\mbox{cotangent space})\otimes(\mbox{tangent
space})\otimes_{sym}(\mbox{tangent space}),$

which follows from expressing the standard basis as

(154) $\displaystyle 4\,e^{1}\otimes e_{1}\otimes e_{1}$
$\displaystyle=-\textit{{E}}^{1}+\phantom{3\,}\textit{{E}}^{3}+\textit{{E}}^{5},$
$\displaystyle 4\,e^{1}\otimes e_{2}\otimes e_{2}$
$\displaystyle=\phantom{-}\textit{{E}}^{1}+3\,\textit{{E}}^{3}-\textit{{E}}^{5},$
$\displaystyle 4\,e^{2}\otimes e_{1}\otimes e_{2}=_{sym}2\,(e^{2}\otimes
e_{1}\otimes e_{2}+e^{2}\otimes e_{2}\otimes e_{1})$
$\displaystyle=\phantom{-}\textit{{E}}^{1}-\phantom{3\,}\textit{{E}}^{3}+\textit{{E}}^{5}.$

Applying $J.$ to these identities we obtain by definition (141) and the
relation (152)

(155) $\displaystyle 4\,e^{2}\otimes e_{2}\otimes e_{2}$
$\displaystyle=-\textit{{E}}^{2}+\phantom{3\,}\textit{{E}}^{4}+\textit{{E}}^{6},$
$\displaystyle 4\,e^{2}\otimes e_{1}\otimes e_{1}$
$\displaystyle=\phantom{-}\textit{{E}}^{2}+3\,\textit{{E}}^{4}-\textit{{E}}^{6},$
$\displaystyle 4\,e^{1}\otimes e_{2}\otimes e_{1}$
$\displaystyle=_{sym}\phantom{-}\textit{{E}}^{2}-\phantom{3\,}\textit{{E}}^{4}+\textit{{E}}^{6}.$

We now give the argument that indeed

(156)
$\displaystyle\textit{{E}}^{m}\bullet\textit{{E}}^{n}=0\quad\mbox{unless}\;m=n\quad\mbox{and}\quad\textit{{E}}^{m}\bullet\textit{{E}}^{m}=\frac{1}{2}\quad\mbox{for}\;m\leq
4,$

where $\textit{{E}}^{m}\bullet\textit{{E}}^{m}=0$ for $m\in\\{5,6\\}$ by
(153). We first leverage the invariance of $\bullet$ under the natural action
of rotations, cf. (141). By the argument of Subsection A.1 based on the action
of rotations on a two-dimensional subspace, which now is (142), and the
parameterization of the rotation group via $Q_{\theta}$, which now is the
rotation by the angle $-\frac{\theta}{3}$ (as opposed to $\frac{1}{2}\theta$
before), so that the first item in (152) assumes the form of (131), and the
invariance of a symmetric bilinear form, which now is $\bullet$, we obtain
conformality

$\displaystyle\textit{{E}}^{1}\bullet\textit{{E}}^{1}=\textit{{E}}^{2}\bullet\textit{{E}}^{2}\quad\mbox{and}\quad\textit{{E}}^{1}\bullet\textit{{E}}^{2}=0,$

and the same statement for $\\{1,2\\}$ replaced by $\\{3,4\\}$ and a rotation
by $\theta$ instead of $-\frac{\theta}{3}$.

We now give a similar, slightly more involved, argument for

(157)
$\displaystyle\textit{{E}}^{m}\bullet\textit{{E}}^{n}=0\quad\mbox{for}\;m\in\\{1,2\\}\;\mbox{and}\;n\in\\{3,4\\},$

which amounts to showing that the rotation-invariant sub-spaces (142) and
(143) are orthogonal w. r. t. $\bullet$. Using (141) for $Q=Q_{\theta}$,
applying $\frac{d}{d\theta}\big{|}_{\theta=0}$, and appealing to Leibniz’ rule
and to (130), we obtain393939it is crucial that the derivative is applied to
the action $Q_{\theta}.$, not just the rotation $Q_{\theta}$

$\displaystyle\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.=-J^{t}\otimes{\rm
id}\otimes{\rm id}+{\rm id}\otimes J\otimes{\rm id}+{\rm id}\otimes{\rm
id}\otimes J$

to which we apply $J.$, see (141), which by $J^{2}=-{\rm id}$ and thus by
$J^{-t}=-J^{t}$ yields

$\displaystyle J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.=-{\rm
id}\otimes J\otimes J+J^{t}\otimes{\rm id}\otimes J+J^{t}\otimes J\otimes{\rm
id}.$

We use this to formulate two identities on ${\rm
id}+J.\frac{d}{d\theta}\big{|}_{\theta=0}Q_{\theta}.$, namely

$\displaystyle({\rm id}+J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.)$
$\displaystyle={\rm id}\otimes{\rm id}\otimes{\rm id}-{\rm id}\otimes J\otimes
J+J^{t}\otimes{\rm id}\otimes J+J^{t}\otimes J\otimes{\rm id},$

which entails

$\displaystyle({\rm id}+J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.)$
$\displaystyle\big{(}{\rm id}\otimes{\rm id}\otimes{\rm id}+{\rm id}\otimes
J\otimes J\big{)}=0.$

Applying these identities to $e^{1}\otimes e_{1}\otimes e_{1}$ we learn by
definition (151) and $Je_{1}=e_{2}$, $J^{t}e^{1}=-e^{2}$

$\displaystyle\textit{{E}}^{1}\in{\rm im}\big{(}{\rm
id}+J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.\big{)}\quad\mbox{and}\quad\textit{{E}}^{3}\in{\rm
ker}\big{(}{\rm id}+J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.\big{)}.$

Since $J.$ commutes with ${\rm
id}+J.\frac{d}{d\theta}\big{|}_{\theta=0}Q_{\theta}.$ this entails by (152)

$\displaystyle\textit{{E}}^{2}\in{\rm im}\big{(}{\rm
id}+J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.\big{)}\quad\mbox{and}\quad\textit{{E}}^{4}\in{\rm
ker}\big{(}{\rm id}+J.\frac{d}{d\theta}\Big{|}_{\theta=0}Q_{\theta}.\big{)}.$

This yields (157) since ${\rm
id}+J.\frac{d}{d\theta}\big{|}_{\theta=0}Q_{\theta}.$ is symmetric w. r. t.
$\bullet$, which follows from the fact that $J.$ is skew (since $J.$ preserves
$\bullet$ and by $J^{2}=-{\rm id}$) and that
$\frac{d}{d\theta}\big{|}_{\theta=0}Q_{\theta}.$ is skew (by Leibniz’ rule for
$\bullet$). Hence (157), and therefore the first item in (156), are
established.

Towards the second item in (156), we now claim that by an additional symmetry
of $\bullet$ we have

(158)
$\displaystyle\textit{{E}}^{1}\bullet\textit{{E}}^{1}-\textit{{E}}^{3}\bullet\textit{{E}}^{3}=\textit{{E}}^{2}\bullet\textit{{E}}^{2}-\textit{{E}}^{4}\bullet\textit{{E}}^{4}=0$

so that
$\textit{{E}}^{1}\bullet\textit{{E}}^{1}=\textit{{E}}^{2}\bullet\textit{{E}}^{2}=\textit{{E}}^{3}\bullet\textit{{E}}^{3}=\textit{{E}}^{4}\bullet\textit{{E}}^{4}$.
Indeed, by stationarity of $d\phi$, which implies that
$[\partial_{i}\partial_{j}d\phi\,\partial_{k}\partial_{l}d\phi]$ is invariant
under permuting $i,j,k,l$, and by definition (61) of $\bullet$, the latter
enjoys the additional symmetry

(159) $(\xi\otimes e_{i}\otimes e_{j})\bullet(\xi^{\prime}\otimes e_{k}\otimes
e_{l})\quad\mbox{is invariant under permuting}\;i,j,k,l.$

Rewriting the first item in (151) as

$\displaystyle\textit{{E}}^{1}=-e^{1}\otimes(e_{1}\otimes e_{1}-e_{2}\otimes
e_{2})+e^{2}\otimes(e_{1}\otimes e_{2}+e_{2}\otimes e_{1}),$

so that together with the first item in (144) we have

$\displaystyle\textit{{E}}^{1}+\textit{{E}}^{3}$
$\displaystyle=_{sym}\phantom{+}2e^{1}\otimes e_{2}\otimes e_{2}+2e^{2}\otimes
e_{1}\otimes e_{2},$ $\displaystyle\textit{{E}}^{1}-\textit{{E}}^{3}$
$\displaystyle=_{sym}-2e^{1}\otimes e_{1}\otimes e_{1}+2e^{2}\otimes
e_{1}\otimes e_{2},$

we obtain by the symmetry of $\bullet$

(160)
$\displaystyle\textit{{E}}^{1}\bullet\textit{{E}}^{1}-\textit{{E}}^{3}\bullet\textit{{E}}^{3}$
$\displaystyle=$ $\displaystyle-4(e^{1}\otimes e_{1}\otimes
e_{1})\bullet(e^{1}\otimes e_{2}\otimes e_{2})-4(e^{2}\otimes e_{1}\otimes
e_{2})\bullet(e^{1}\otimes e_{1}\otimes e_{1})$ $\displaystyle+4(e^{1}\otimes
e_{2}\otimes e_{2})\bullet(e^{2}\otimes e_{1}\otimes e_{2})+4(e^{2}\otimes
e_{1}\otimes e_{2})\bullet(e^{2}\otimes e_{1}\otimes e_{2}).$

We now appeal to the invariance of $\bullet$ under (141) with $Q=J$, in order
to rewrite the two last terms in (160) as

$\displaystyle(e^{1}\otimes e_{2}\otimes e_{2})\bullet(e^{2}\otimes
e_{1}\otimes e_{2})$ $\displaystyle=(e^{2}\otimes e_{1}\otimes
e_{1})\bullet(e^{1}\otimes e_{2}\otimes e_{1}),$ $\displaystyle(e^{2}\otimes
e_{1}\otimes e_{2})\bullet(e^{2}\otimes e_{1}\otimes e_{2})$
$\displaystyle=(e^{1}\otimes e_{2}\otimes e_{1})\bullet(e^{1}\otimes
e_{2}\otimes e_{1}).$

Hence we learn from (159) that the first and last, and the second and third
term in (160) cancel. This establishes the first equality in (158). In view of
(152), the second equality follows by invariance of $\bullet$ under the action
of $J$.

In order to conclude the proof of (156), we argue

(161)
$\displaystyle\textit{{E}}^{3}\bullet\textit{{E}}^{3}+\textit{{E}}^{4}\bullet\textit{{E}}^{4}=1,$

which by (152) implies
$\textit{{E}}^{3}\bullet\textit{{E}}^{3}=\textit{{E}}^{4}\bullet\textit{{E}}^{4}=\frac{1}{2}$.
To this purpose we introduce

(164)
<EMAIL_ADDRESS>e_{2}\otimes e_{2}+e^{2}\otimes e_{1}\otimes e_{2}+e^{2}\otimes e_{2}\otimes
e_{1},\\\ \textit{{E}}^{8}&\coloneqq-&2e^{2}\otimes e_{1}\otimes
e_{1}+e^{1}\otimes e_{2}\otimes e_{1}+e^{1}\otimes e_{1}\otimes
e_{2},\end{array}$

which are defined such that, see (125),

$\displaystyle\textit{{E}}^{7}=_{sym}-2E^{3}\otimes
e_{2}\quad\mbox{and}\quad\textit{{E}}^{8}=_{sym}-2E^{3}\otimes e_{1}.$

This results in

$\displaystyle\textit{{E}}^{7}.\nabla^{2}d\phi=2\partial_{2}E^{3}.\nabla
d\phi\quad\mbox{and}\quad\textit{{E}}^{8}.\nabla^{2}d\phi=-2\partial_{1}E^{3}.\nabla
d\phi,$

which by (138) implies

$\displaystyle\tilde{\lambda}\textit{{E}}^{7}.\nabla^{2}d\phi=-2\partial_{2}d\psi\quad\mbox{and}\quad\tilde{\lambda}\textit{{E}}^{8}.\nabla^{2}d\phi=-2\partial_{1}d\psi.$

Hence the characterization (57) of $[\nabla d\psi\cdot\nabla d\psi]$
translates by definition (61) of $\bullet$ into

(165)
$\displaystyle\textit{{E}}^{7}\bullet\textit{{E}}^{7}+\textit{{E}}^{8}\bullet\textit{{E}}^{8}=4.$

Because of the representation, cf. (151) and (164),

$\displaystyle\textit{{E}}^{7}=\textit{{E}}^{5}-2\textit{{E}}^{3}\quad\mbox{and}\quad\textit{{E}}^{8}=\textit{{E}}^{6}-2\textit{{E}}^{4}$

and appealing to (153), (165) reduces to (161).

## Acknowledgements

The authors thank Princeton University for its warm hospitality in Fall 2023,
where this work was started.

## References

  * [1] S. Armstrong, A. Bou-Rabee, and T. Kuusi.  Superdiffusive central limit theorem for a Brownian particle in a critically-correlated incompressible random drift.  _arXiv preprint_ arXiv:2404.01115 (2024).
  * [2] S. Armstrong, and V. Vicol.  Anomalous diffusion by fractal homogenization  _arXiv preprint_ arXiv:2305.05048 (2023).
  * [3] N. Barashkov and M. Gubinelli.  A variational method for $\Phi^{4}_{3}$.  _Duke Math. J._ 169 (17) 3339 - 3415, 15 November 2020.
  * [4] J. Bricmont, and A. Kupiainen.  Random walks in asymmetric random environments.  _Communications in mathematical physics_ 142 (1991): 345-420.
  * [5] J. Burczak, L. Székelyhidi Jr., and B. Wu.  Anomalous dissipation and Euler flows.  _arXiv preprent_ arXiv:2310.02934 (2023).
  * [6] G. Chatzigeorgiou, P. Morfe, F. Otto, and L. Wang.  The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off.  _arXiv preprint_ arXiv:2212.14244 (2022).
  * [7] G. Cannizzaro, D. Erhard, and F. Toninelli, Fabio.  The stationary AKPZ equation: logarithmic superdiffusivity.  _Comm. Pure Appl. Math._ 76 (2023), no.11, 3044–3103.
  * [8] G. Cannizzaro, L. Haunschmid-Sibitz, and F Toninelli.  $\log t$-Superdiffusivity for a Brownian particle in the curl of the 2D GFF.  _The Annals of Probability_ 50.6 (2022): 2475-2498.
  * [9] G. de Lima Feltes, Guilherme, and H. Weber.  Brownian particle in the curl of 2-d stochastic heat equations.  _Journal of Statistical Physics_ 191.2 (2024): 16.
  * [10] A. Fannjiang.  Anomalous diffusion in random flows. In Mathematics of multiscale materials.  (Minneapolis, MN, 1995–1996), volume 99 of _IMA Vol. Math. Appl._ , pages 81–99. Springer, New York, 1998.
  * [11] B. Fehrman.  Large-scale regularity in stochastic homogenization with divergence-free drift.  _The Annals of Applied Probability_ 33.4 (2023): 2559-2599.
  * [12] D. S. Fisher, D. Friedan, Z. Qiu, S. J. Shenker, and S. H. Shenker.  Random walks in two-dimensional random environments with constrained drift forces.  _Phys. Rev. A, 31:3841–3845_ , Jun 1985.
  * [13] M. Gubinelli, and S.-J. Meyer.  The FBSDE approach to sine-Gordon up to $6\pi$.  _arXiv preprint_ arXiv:2401.13648 (2024).
  * [14] M. Gubinelli, and N. Perkowski.  The infinitesimal generator of the stochastic Burgers equation.  _Probab. Theory Related Fields_ 178 (2020), no.3-4, 1067–1124.
  * [15] G. Kozma, and B. Tóth.  Central limit theorem for random walks in doubly stochastic random environment: $H_{-1}$ suffices.  _The Annals of Probability_ 2017, Vol. 45, No. 6B, 4307–4347.
  * [16] S.M. Kozlov.  The averaging of random operators.(Russian)  _Mat. Sb. (N.S.)_ 109(151)(1979), no.2, 188–202, 327.
  * [17] T. Komorowski, C. Landim, and S Olla.  _Fluctuations in Markov processes: time symmetry and martingale approximation._ Springer-Verlag Berlin Heidelberg, 2012.
  * [18] J.-F. Le Gall,  _Brownian motion, martingales, and stochastic calculus._ Springer-Verlag Berlin Heidelberg, 2013.
  * [19] P. Morfe, F. Otto, and C. Wagner.  The Gaussian free-field as a stream function: continuum version of the scale-by-scale homogenization result.  _arXiv preprint_ arXiv:2404.00709 (2024).
  * [20] K. Oelschlager.  Homogenization of a Diffusion Process in a Divergence-Free Random Field.  _Ann. Probab._ 16 (3) 1084 - 1126, July, 1988.
  * [21] G. C. Papanicolaou and S. R. S. Varadhan.  Boundary value problems with rapidly oscillating random coefficients.Random fields, Vol. I, II (Esztergom, 1979), 835–873.  Colloq. Math. Soc. János Bolyai, 27 _North-Holland Publishing Co., Amsterdam-New York_ , 1981.
  * [22] A.-S. Sznitman, and O. Zeitouni.  An invariance principle for isotropic diffusions in random environment.  _Inventiones mathematicae_ 164.3 (2006): 455-567.
  * [23] B. Tóth.  Quenched Central Limit Theorem for Random Walks in doubly stochastic random environment  _The Annals of Probability_ 2018, Vol. 46, No. 6, 3558–3577.
  * [24] B. Tóth and B. Valkó.  Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d= 2.  _Journal of Statistical Physics_ 147 (2012): 113-131.