# A Harris Theorem for Enhanced Dissipation, and an Example of Pierrehumbert

William Cooperman  Courant Institute of Mathematical Sciences, New York
University, NY 10003<EMAIL_ADDRESS>,  Gautam Iyer  Department of
Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213.
<EMAIL_ADDRESS>and  Seungjae Son  Department of Mathematical Sciences,
Carnegie Mellon University, Pittsburgh, PA 15213<EMAIL_ADDRESS>
###### Abstract.

In many situations, the combined effect of advection and diffusion greatly
increases the rate of convergence to equilibrium – a phenomenon known as
_enhanced dissipation_. Here we study the situation where the advecting
velocity field generates a random dynamical system satisfying certain _Harris
conditions_. If $\kappa$ denotes the strength of the diffusion, then we show
that with probability at least $1-o(\kappa^{N})$ enhanced dissipation occurs
on time scales of order $\lvert\ln\kappa\rvert$, a bound which is known to be
optimal. Moreover, on long time scales, we show that the rate of convergence
to equilibrium is almost surely  _independent_ of diffusivity. As a
consequence we obtain enhanced dissipation for the randomly shifted
alternating shears introduced by Pierrehumbert ’94.

###### Key words and phrases:

enhanced dissipation, mixing

###### 1991 Mathematics Subject Classification:

Primary: 37A25. Secondary: 60J05, 76R99.

This work has been partially supported by the National Science Foundation
under grants DMS-2108080 to GI, DMS-2303355 to WC, and the Center for
Nonlinear Analysis.

##  1\. Introduction

###  1.1. Main Results

We begin by stating our results. Following this, we will survey the literature
and place our work in the context of existing results. Let $u$ be a (possibly
time dependent) divergence free vector on the torus, $\kappa>0$, and $\rho$
solve the advection diffusion equation

(1.1) $\partial_{t}\rho+u\cdot\nabla\rho-\kappa\Delta\rho=0\,,$

on the $d$-dimensional torus $\mathbb{T}^{d}$. Multiplying (1.1) by $\rho$,
using the fact that $\nabla\cdot u=0$, integrating and using the Poincaré
inequality implies

(1.2) $\lVert\rho(\cdot,t)-\bar{\rho}\rVert_{L^{2}}\leqslant
e^{-\lambda_{1}\kappa t}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{2}}\,,$

where

(1.3) $\bar{\rho}=\int_{\mathbb{T}^{d}}\rho_{0}(x)\,dx\,,$

is the (constant) equilibrium solution to (1.1) and $\lambda_{1}>0$ is the
smallest non-zero eigenvalue of the negative Laplacian. _Enhanced dissipation_
is the phenomenon where solutions to (1.1) converge to equilibrium faster than
the upper bound (1.2). Our main result shows that if the flow of $u$ generates
a random dynamical system (RDS) that satisfies the Harris conditions (stated
below), then enhanced dissipation occurs. The enhanced dissipation rate is
optimal at short times with large probability, and is almost surely
_independent_ of the diffusivity for long times.

######  Theorem 1.1.

Suppose the flow of $u$ generates a random dynamical system that satisfies the
Harris conditions stated in Assumptions 2.1–2.4, below. For any $\alpha>0$,
$q<\infty$ there exists $\gamma>0$ and a $\kappa$-dependent random variable
$D_{\kappa}$ such that for every initial data $\rho_{0}\in
L^{1}(\mathbb{T}^{d})$, the solution to (1.1) satisfies

(1.4)
$\lVert\rho(\cdot,t)-\bar{\rho}\rVert_{L^{\infty}}\leqslant\frac{D_{\kappa}}{\kappa^{\frac{d}{2}+\alpha}}e^{-\gamma
t}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{1}}\,.$

Moreover, there exist a $\kappa$-independent, deterministic, constant
$\bar{D}_{q}$ such that

(1.5) $\bm{E}D_{\kappa}^{q}\leqslant\bar{D}_{q}\,.$

######  Remark 1.2.

The bound (1.5) implies that for any $\beta>0$,
$\bm{P}(D_{\kappa}\geqslant\kappa^{-\beta})\leqslant\kappa^{\beta
q}\bar{D}_{q}$. Using this in (1.4) will show that with probability at least
$1-\bar{D}_{q}\kappa^{\beta q}$, we have

(1.6)
$\lVert\rho(\cdot,t)-\bar{\rho}\rVert_{L^{\infty}}\leqslant\frac{1}{\kappa^{\frac{d}{2}+\alpha+\beta}}e^{-\gamma
t}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{1}}\,,$

for all $t\geqslant 0$, and all $\rho_{0}\in L^{1}$.

An equivalent probabilistic formulation of this result is as follows. Consider
the Markov process defined by the SDE

(1.7)
$dX^{\kappa}_{t}(x)=-u(X^{\kappa}_{t}(x),t)\,dt+\sqrt{2\kappa}\,dW_{t}\,,\quad
X^{\kappa}_{0}(x)=x\,,$

on the torus $\mathbb{T}^{d}$. Since $\nabla\cdot u=0$, the (unique)
stationary distribution of $X^{\kappa}$ is the Lebesgue measure. Let
$p^{\kappa}_{t}(x,y)$ denote the transition density of $X^{\kappa}_{t}$, and
recall the  _uniform mixing time_ [LP17, MT06] is defined by

(1.8)
$t_{\mathrm{mix}}^{\infty}(X^{\kappa},\varepsilon)=\inf\Big{\\{}t\geqslant
0\nonscript\>\Big{|}\nonscript\>\mathopen{}\allowbreak\sup_{x\in\mathbb{T}^{d}}\lVert
p^{\kappa}_{t}(x,\cdot)-1\rVert_{L^{\infty}}<\varepsilon\Big{\\}}\,.$

It is easy to see that the  _uniform mixing time_ of $X^{\kappa}$ satisfies

(1.9)
$t_{\mathrm{mix}}^{\infty}(X^{\kappa},\varepsilon)\leqslant\frac{C\lvert\ln\varepsilon\rvert}{\kappa}\,,$

for some constant $C>0$. Theorem 1.1 is equivalent to the following uniform
mixing time estimate.

######  Theorem 1.3.

Suppose the flow of $u$ generates a random dynamical system satisfying the
Harris conditions (Assumptions 2.1–2.4, below). For any $\alpha>0$, $q<\infty$
there exists $\gamma>0$ and a $\kappa$-dependent random variable $D_{\kappa}$
such that

(1.10)
$t_{\mathrm{mix}}^{\infty}(X^{\kappa},\varepsilon)\leqslant\frac{1}{\gamma}\ln\Big{(}\frac{D_{\kappa}}{\varepsilon\kappa^{\frac{d}{2}+\alpha}}\Big{)}$

almost surely. Moreover, there exists a $\kappa$-independent (deterministic)
constant $\bar{D}_{q}$ such that (1.5) holds.

######  Remark 1.4.

We clarify that $t_{\mathrm{mix}}^{\infty}(X^{\kappa},\varepsilon)$ is random
as it depends on $u$ (it is, of course, independent of $W$). For any fixed
$\varepsilon>0$ (independent of $\kappa$), Chebychev’s inequality (1.5) and
(1.10) show that for any $\beta>0$ we have

(1.11)
$\bm{P}\Big{(}t_{\mathrm{mix}}^{\infty}(X^{\kappa},\varepsilon)\leqslant\frac{1}{\gamma}\ln\Big{(}\frac{1}{\varepsilon\kappa^{\frac{d}{2}+\alpha+\beta}}\Big{)}\Big{)}\geqslant
1-\bar{D}_{q}\kappa^{\beta q}\,.$

Moreover, as $\varepsilon\to 0$, we obtain the $\kappa$-independent uniform
mixing time bound

(1.12) $\lim_{\varepsilon\to
0}\frac{t_{\mathrm{mix}}(X^{\kappa},\varepsilon)}{\lvert\ln\varepsilon\rvert}\leqslant\frac{1}{\gamma}\,,\quad\text{almost
surely}\,.$

######  Remark 1.5.

While Assumptions 2.1–2.4 are easy to state (see Section 2.3, below), they
aren’t easy to verify in practice. Recent papers [BBPS22, BCZG22] instead
assume certain conditions which are stronger than Harris conditions, but are
easier to verify. For convenience of the reader, we state these conditions in
Section 2.4, below.

######  Remark 1.6 (Pulsed diffusions).

We can also obtain similar results for pulsed diffusions. Namely, define the
Markov process $Y^{\kappa}$ by

(1.13)
$Y^{\kappa}_{n+1}=\varphi_{n+1}(Y^{\kappa}_{n})+\zeta^{\kappa}_{n+1}\,,$

where $\zeta^{\kappa}_{n}$ are i.i.d. periodized Gaussians with variance
$\kappa$, and $\varphi_{n}$ is a RDS on $\mathbb{T}^{d}$, independent of
$\zeta^{\kappa}$. If the RDS $\varphi_{n}$ satisfies the Harris conditions
(Assumption 2.1–2.3), then we obtain the same mixing time bound (1.10) for the
process $Y^{\kappa}$. The proof is similar to the proof of Theorem 1.3, and in
this setting several technical steps become much simpler.

As an immediate consequence, we can show enhanced dissipation if $u$ is
obtained by randomly shifting and alternating sinusoidal shears.

######  Corollary 1.7.

Let $A>0$, $d=2$ and $\zeta_{n}$ a sequence of i.i.d. random variables that
are uniformly distributed on $[0,1]$. For $n\in\mathbb{N}$ and $t\in[2n,2n+2)$
define

(1.14)
$u(x,t)=\begin{cases}A\sin(2\pi(x_{2}-\zeta_{2n}))\,e_{1}&t\in[2n,2n+1)\,,\\\
A\sin(2\pi(x_{1}-\zeta_{2n+1}))\,e_{2}&t\in[2n+1,2n+2)\,,\end{cases}$

where $x=(x_{1},x_{2})\in\mathbb{T}^{2}$, and $e_{i}$ is the $i$-th standard
basis vector. Then, almost surely, we have the enhanced dissipation bound
(1.4) for a random variable $D_{\kappa}$ that satisfies the
$\kappa$-independent bounds (1.5). Consequently, with probability at least
$1-\bar{D}_{q}\kappa^{\beta q}$, the enhanced dissipation bound (1.6) holds.

This example was introduced by Pierrehumbert [Pie94] and [BCZG22] recently
verified that it satisfies conditions that are stronger than our Harris
assumptions. As a result the Corollary 1.7 follows immediately from Theorem
1.1 and [BCZG22] (see Proposition 2.9, below). We mention, however, that even
though [BCZG22] abstractly check Assumption 2.2 (the existence of a Lyapunov
function) for (1.14), the system is simple enough that a Lyapunov function can
be constructed explicitly. We do this in Proposition 9.1 in Section 9, below.

######  Remark 1.8.

Instead of using a sine shear profile in (1.14), we can use shears with a
piecewise linear profile. In this case the results in [CFIN23] will show that
Assumption 2.1–2.4 are satisfied, and so Theorems 1.1, 1.3 apply and will give
the enhanced dissipation bound (1.4), and the equivalent uniform mixing time
bound (1.10).

###  1.2. Motivation and Literature Review

We now survey the literature and place our results in the context of existing
results. _Enhanced dissipation_ is a phenomenon that can be observed in every
day life: Pour some cream in your coffee. If left alone, it will take hours to
mix. Stir it a little and it mixes right away. This effect arises due to the
interaction between the advection (stirring) and diffusion, and plays an
important role in many applications concerning hydrodynamic stability and
turbulence and occurs on scales ranging from micro fluids to meteorological /
cosmological [LTD11, Thi12, Are84, SSA04].

To describe this mathematically, let $u$ be the velocity field of the ambient
incompressible fluid, and $\rho$ denote the concentration of a passively
advected solute with molecular diffusivity $\kappa>0$. The evolution of $\rho$
is governed by the advection diffusion equation (1.1). For simplicity, in this
paper we only consider (1.1) with periodic boundary conditions on the
$d$-dimensional torus $\mathbb{T}^{d}$.

If the ambient fluid is incompressible, the velocity field $u$ satisfies
divergence free condition

(1.15) $\nabla\cdot u=0\,.$

In this case, an elementary energy estimate shows (1.2) and hence the $L^{2}$
distance of the concentration from the equilibrium distribution decreases at
most exponentially with a rate proportional to $\kappa$.

Of course, (1.2) is only a crude upper bound. In many practical situations one
expects the convergence to happen much faster than (1.2). Indeed, the
advection term $u\cdot\nabla\rho$ typically causes filamentation and moves
energy towards small scales. The diffusion term $\kappa\Delta\rho$ damps small
scales faster, and the combination of these two effects leads to enhanced
dissipation – faster convergence of $\rho(\cdot,t)$ to $\bar{\rho}$.

Several authors have proved enhanced dissipation by showing all solutions to
(1.1) satisfy the decay estimate

(1.16a)
$\lVert\rho(\cdot,t)-\bar{\rho}\rVert_{L^{2}}\leqslant\exp\Big{(}-\Big{(}\frac{t}{T(\kappa)}-1\Big{)}^{+}\Big{)}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{2}}\,,$
for every $t\geqslant 0$, and some time scale $T(\kappa)$ for which (1.16b)
$\lim_{\kappa\to 0}\kappa T(\kappa)=0\,.$

Seminal work of Constantin et al. [CKRZ08] (see also [Zla10, KSZ08]) shows
that if $u$ is time independent, then such a $T(\kappa)$ exists if and only if
$u\cdot\nabla$ has no eigenfunctions in $H^{1}$. For shear flows classical
work of Kelvin [Kel87] shows one can choose $T(\kappa)=\kappa^{-\alpha}$ for
some $\alpha<1$. There are now several results studying enhanced dissipation
in more generality and for nonlinear equations (see [FNW04, Wei19, BCZ17,
CZD21, FMN23, CZG23, ABN22, CH23, Sei23]).

The purpose of this paper is to further investigate the link between enhanced
dissipation and mixing properties of $u$. Recall, a velocity field $u$ is said
to be _exponentially mixing_ if, in the absence of diffusion, a dye that is
initially localized to ball of size $\varepsilon$ will get spread throughout
the torus in time $O(\lvert\ln\varepsilon\rvert)$ (see for instance [SOW06]).
In the presence of diffusion, a dye localized to a point gets spread to a ball
of size $O(\sqrt{\kappa})$ in time $O(1)$. If $u$ is exponentially mixing,
then this dye is spread throughout the torus by the flow in time
$O(\lvert\ln\kappa\rvert)$. As a result, in this case we expect (1.16a) should
hold with $T(\kappa)=O(\lvert\ln\kappa\rvert)$.

Surprisingly, this is not easy to prove, and is an open question in this
generality. Currently available results [FI19, Fen19, CZDE20] show that if $u$
is exponentially mixing, then one can choose
$T(\kappa)=O(\lvert\ln\kappa\rvert^{2})$ in (1.16a). For a few specific
exponentially mixing systems, available results [BBPS21, ELM23, ILN23] show
that one can choose $T(\kappa)=O(\lvert\ln\kappa\rvert)$ in (1.16a). However,
to the best of our knowledge, there is no general theorem (in discrete or
continuous time) that shows that for any exponentially mixing flow one can
choose $T(\kappa)=O(\lvert\ln\kappa\rvert)$ in (1.16a).

One elementary observation is that if almost every realization of the
stochastic flows $X^{\kappa}$ is exponentially mixing, then one has enhanced
dissipation as in (1.16a) with $T(\kappa)=C\lvert\ln\kappa\rvert$ for some $C$
that can be explicitly computed in terms of the mixing rate. Thus, a natural
question to ask is is whether or not the notion of exponentially mixing is
stable with respect to $\kappa$.

######  Question 1.9.

If $u$ is exponentially mixing, then for sufficiently small $\kappa>0$ must
almost every realization of $X^{\kappa}$ be exponentially mixing (with a
controlled rate)?

Since the notion of exponentially mixing involves the long time behavior, it
is not easy to determine the answer to Question 1.9. We instead look for
stronger conditions on $u$ which will will guarantee that for all sufficiently
small $\kappa>0$, almost every realization $X^{\kappa}$ is exponentially
mixing (with a controlled rate).

A general principle that is well known to the Sinai school is that for _random
dynamical systems (RDS)_ , geometric ergodicity of the two point process
implies almost sure exponential mixing (see [DKK04, BBPS22], or the proof of
Lemma 3.3, below). One could then ask whether or not this property is stable
in $\kappa$.

######  Question 1.10.

If the two point process associated to the flow of a random dynamical system
$u$ is geometrically ergodic, then is the two point processes associated to
the SDE (1.7) also geometrically ergodic for all small $\kappa>0$?

If Question 1.10 is answered affermatively, then for all sufficiently small
$\kappa>0$, must almost every realization $X^{\kappa}$ be exponentially
mixing. Not surprisingly, this question is also hard to answer. Geometric
ergodicity involves questions about long time limits which are not stable as
$\kappa$ varies. There is, however, a classical result of Harris [Har55, MT09]
that proves geometric ergodicity of a Markov process provided there is a
Lyapunov function, and a small set. A version of this condition turns out to
be stable in $\kappa$ (see Lemma 3.1, below) which in turn leads to almost
sure exponential mixing of the flows $X^{\kappa}$ (see Lemma 3.3, below),
which in turn yields Theorem 1.1.

Finally, we mention that if $u$ is regular, and uniformly bounded in time,
then we must have $T(\kappa)\geqslant O(\kappa)$ in (1.16a) (see for instance
[MD18, Poo96, BBPS21, Sei22]). When $u$ is irregular, one can even choose
$T(\kappa)=O(1)$. This is known as anomalous dissipation, and examples of this
were recently proved in [DEIJ22, CCS22, AV23]. In this paper we only consider
regular velocity fields, and so are in a situation where anomalous dissipation
can not ocur.

### Plan of this paper.

In Section 2.3 we define our notation and state Assumption 2.1–2.4 used in
Theorems 1.1 and 1.3. In Section 3 we state the three lemmas (Lemmas 3.1–3.3)
that will quickly yield Theorems 1.1 and 1.3, and use these lemmas to prove
Theorems 1.1 and 1.3. The first of these lemmas (Lemma 3.1) is the main new
contribution of this paper and which guarantees that our assumptions imply the
Harris conditions hold for all sufficiently small $\kappa>0$. The proof of
Lemma 3.1 is split up into two steps – the existence of a Lyapunov function in
Section 4, and the existence of a $\kappa$-independent small set in Section 5.
Following this, we prove Lemma 3.1 in Section 6. The proof of Lemma 3.2 uses a
quantitative version of Harris’s theorem [HM11] and is presented in Section 7.
The proof of Lemma 3.3 is based on the idea in [DKK04] and is presented in
Section 8. Finally, we conclude this paper by explicitly finding a Lyapunov
function for the randomly shifted alternating shear example in 1.7.

##  2\. Notation and Preliminaries.

In this section we set up our notational convention and state the assumptions
required for Theorem 1.1 and 1.3.

###  2.1. Notation and setup.

We will now define a setup that chooses the velocity field $u$ randomly from a
finite dimensional family of $C^{2}$, incompressible vector fields, and
repeats this choice on after time intervals of length $1$. Let $\mathscr{M}$
be a complete smooth Riemannian manifold, and
$\mathscr{U}\colon\mathscr{M}\times\mathbb{T}^{d}\times[0,1]\to\mathbb{R}^{d}$
be a $C^{2}$-function such that

(2.1) $\nabla_{x}\cdot\mathscr{U}(\xi,x,t)=0\,.$

Now let $(\Omega_{0},\mathcal{F}_{0},\bm{P}_{0})$ be a probability space and
let $\omega=(\omega_{1},\omega_{2},\dots)$ be a sequence of
$\mathscr{M}$-valued random variables whose distribution is absolutely
continuous with respect to the volume measure on $\mathscr{M}$. Define the
(random) velocity field $u$ by

(2.2)
$u(x,t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\mathscr{U}(\omega_{n},x,t-n)\quad\text{when
}t\in[n,n+1)\,,$

where, following standard convention, we suppress the dependence of $u$ on
$\omega$. The flow of $u$ is defined by the ODE

(2.3) $\partial_{t}X^{0}_{t}=u(X^{0}_{t},t)\,,\qquad X^{0}_{0}=\mathrm{Id}\,.$

Restricting $X^{0}$ to integer times gives a Markov process, which we refer to
as the RDS generated by $u$.

To define the processes $X^{\kappa}$, let
$(\Omega_{W},\mathcal{F}_{W},\bm{P}_{W})$ be a probability space, and $W$ be a
$\mathbb{T}^{d}$-valued Brownian motion on this space. We will now consider
the product space $\Omega=\Omega_{W}\times\Omega_{0}$ with the product
$\sigma$-algebra $\mathcal{F}=\mathcal{F}_{W}\otimes\mathcal{F}_{0}$ and
product measure
$\bm{P}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\bm{P}_{W}\otimes\bm{P}_{0}$.
By a slight abuse of notation, we will sometimes treat random variables on
each of the coordinate spaces $\Omega_{0}$, $\Omega_{W}$ as random variables
on the product space $\Omega$ by composing with the corresponding coordinate
projection. In this sense, we may treat $u,W$ as independent processes on
$\Omega$, and let $X^{\kappa}_{t}$ be the solution of (1.7) on
$\mathbb{T}^{d}$.

###  2.2. Two point and projective chains.

In order to state the Harris conditions, we need to define the  _two point_
and  _projective chains_ , and formalize the dependence on the noise history.
Given $\xi=(\xi_{1},\dots,\xi_{n})\in\mathscr{M}^{n}$, define
$\mathscr{U}_{n}\colon\mathscr{M}^{n}\times\mathbb{T}^{d}\times[0,n)\to\mathbb{R}^{d}$
by

(2.4) $\mathscr{U}_{n}(\xi,x,t)=\mathscr{U}(\xi_{m},x,t-m)\quad\text{if
}t\in[m,m+1)\,.$

Now define
$\mathscr{X}_{n}\colon\mathscr{M}^{n}\times\mathbb{T}^{d}\to\mathbb{T}^{d}$ to
be the flow of $\mathscr{U}_{n}$ after time $n$. That is, define

(2.5) $\mathscr{X}_{n}(\xi,x)=\varPhi_{n}(x)\,,$

where $\varPhi$ is defined by

(2.6)
$\partial_{t}\varPhi_{t}=\mathscr{U}_{n}(\xi,\varPhi_{t},t)\,,\quad\varPhi_{0}=\mathrm{Id}\,.$

Given $\xi\in\mathscr{M}^{n}$, $(x,v)\in T\mathbb{T}^{d}$ (the tangent bundle
of the torus), and $y\in\mathbb{T}^{d}$ with $\lVert v\rVert=1$, $y\neq x$, we
define the derivative, projective and two point maps by

(2.7)
$\mathscr{A}_{n}(\xi,x)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}D_{x}\mathscr{X}_{n}(\xi,x)\,,\quad\hat{\mathscr{X}}_{n}(\xi,x,v)=\Big{(}\mathscr{X}_{n}(\xi,x),\frac{\mathscr{A}_{n}(\xi,x)v}{\lVert\mathscr{A}_{n}(\xi,x)v\rVert}\Big{)}$

and

(2.8)
$\mathscr{X}_{n}^{(2)}(\xi,x,y)=(\mathscr{X}_{n}(\xi,x),\mathscr{X}_{n}(\xi,y))\,,$

respectively. Note that $\hat{\mathscr{X}_{n}}$ takes values on
$\mathscr{M}^{n}\times S\mathbb{T}^{d}$ where $S\mathbb{T}^{d}$ is the unit
sphere bundle (see for instance [dC92]) defined by

(2.9)
$S\mathbb{T}^{d}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\\{(x,v)\in
T\mathbb{T}^{d}\nonscript\>|\nonscript\>\mathopen{}\allowbreak\lVert
v\rVert=1\\}\,,$

The map $\mathscr{X}^{(2)}_{n}$ takes values on
$\mathscr{M}^{n}\times\mathbb{T}^{d,(2)}$ where

(2.10)
$\displaystyle\mathbb{T}^{d,(2)}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\mathbb{T}^{d}\times\mathbb{T}^{d}-\Delta\,,$

and
$\Delta\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\\{(x,y)\in\mathbb{T}^{d}\times\mathbb{T}^{d}\nonscript\>|\nonscript\>\mathopen{}\allowbreak
x=y\\}$.

Notice that the RDS generated by $u$ is precisely the Markov process

(2.11)
$X^{0}_{n}(x)=\mathscr{X}_{n}(\underline{\omega_{n}},x)\,,\quad\text{where}\quad\underline{\omega_{n}}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}(\omega_{1},\dots,\omega_{n})\,.$

We also define the derivative, projective and two point processes by

(2.12)
$A^{0}_{n}(x)=\mathscr{A}_{n}(\underline{\omega_{n}},x)\,,\quad\hat{X}_{n}(x,v)=\hat{\mathscr{X}}_{n}(\underline{\omega_{n}},x,v)\,,\quad
X^{0,(2)}_{n}(x,y)=\mathscr{X}^{(2)}_{n}(\underline{\omega_{n}},x,y)\,,$

where as before $(x,v)\in S\mathbb{T}^{d}$ and $(x,y)\in\mathbb{T}^{d,(2)}$.
We denote the $n$-step transition kernel of these processes by

(2.13) $\displaystyle P_{0}^{n}(x,A)=\bm{P}_{0}[X_{n}^{0}(x)\in A]\,,$ (2.14)
$\displaystyle\quad\hat{P}_{0}^{n}((x,v),A)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\bm{P}_{0}\Big{[}\hat{X}_{n}^{0}(x,v)\in
A\Big{]}\,,$ (2.15) $\displaystyle
P_{0}^{(2),n}((x,y),A)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\bm{P}_{0}\big{[}X_{n}^{0,(2)}\in
A\big{]}\,.$

For brevity, when $n=1$ we will drop the superscript.

For $\kappa>0$, we define the Markov process $X^{\kappa}$ obtained by
restricting the solution to (1.7) to integer times. Given $n\in\mathbb{N}$,
let $P^{n}_{\kappa}(x,\cdot)$ denote the transition probability of
$X^{\kappa}_{n}$. That is, for $n\in\mathbb{N}$ we define

(2.16)
$P^{n}_{\kappa}(x,A)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\bm{P}(X^{\kappa}_{n}(x)\in
A)\,.$

As before, when $n=1$, we drop the superscript and simply write $P_{\kappa}$
for $P^{1}_{\kappa}$. The two point process $X^{\kappa,(2)}$ is defined
analogously by

(2.17) $X^{\kappa,(2)}_{n}(x,y)=(X^{\kappa}_{n}(x),X^{\kappa}_{n}(y))\,,$

and its $n$-step transition kernel will be denoted by $P_{\kappa}^{(2),n}$.

###  2.3. Harris Conditions.

Harris theorems typically assume the existence of a Lyapunov function and a
small set, and show that the associated system is uniformly geometrically
ergodic [MT09, Har55, HM11]. We will now state these conditions for the two
point chain $P^{(2)}_{0}$ defined in (2.15). We reiterate that our assumptions
only concern the RDS generated by the flow of $u$ with $\kappa=0$.

######  Assumption 2.1.

The two point chain with kernel $P_{0}^{(2)}$ is Feller, topologically
irreducible, strongly aperiodic.

Our next assumption concerns the existence of a Lyapunov function for the two
point chain near a small neighborhood of the diagonal. To state this, let
$s>0$ and define a punctured neighborhood of the diagonal $\Delta(s)$ by

(2.18)
$\Delta(s)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\\{(x,y)\in\mathbb{T}^{d}\times\mathbb{T}^{d}\nonscript\>|\nonscript\>\mathopen{}\allowbreak
0<d(x,y)<s\\}\,,$

where $d(x,y)$ is the torus distance between $x$ and $y$. On the flat torus it
is easy to see that for $x,y\in\Delta(1/2)$, we can find a unique $v\in
T_{x}\mathbb{T}^{d}$ such that $\exp_{x}(v)=y$, where $\exp_{x}$ denotes the
exponential map (see for instance [dC92]).

######  Assumption 2.2.

There exists a Lyapunov function $V\colon\mathbb{T}^{d,(2)}\to[1,\infty)$ and
constants $\tilde{\gamma}\in(0,1)$, $s_{*}\in(0,1/2)$ such that

(2.19) $P^{(2)}_{0}V<\tilde{\gamma}V\quad\text{on }\Delta(s_{*})\,.$

Moreover, there exists $p\in(0,1)$ such that the function $V$ is of the form

(2.20)
$V(x,y)=d(x,y)^{-p}\psi\Big{(}x,\frac{\exp_{x}^{-1}(y)}{d(x,y)}\Big{)}\quad\text{on
}\Delta(s_{*})\,.$

where $\psi:S\mathbb{T}^{d}\to\mathbb{R}^{+}$ is a continuous, strictly
positive function.

The standard Harris Theorem [MT09] requires the assumptions 2.1, 2.2, and the
existence of a small set for $P_{0}^{(2)}$, and shows ergodicity of the chain.
However, for Lemma 3.3, we will need to show that for sufficiently small
$\kappa>0$ there exists a small set with $\kappa$-independent bound on the
minorizing measure. We are presently unable to do this assuming only the
existence of a small set for $P_{0}^{(2)}$. We can, however, prove this under
a slightly stronger condition which is not hard to verify in practice.

######  Assumption 2.3.

There exist $n\geqslant 1$ and
$(\xi_{*},x_{*})\in\mathscr{M}^{n}\times\mathscr{\mathbb{T}}^{d,(2)}$ such
that the following hold.

  1. (1)

There exists $c,\varepsilon>0$ such that for every $\xi\in\mathscr{M}^{n}$
with $\lvert\xi-\xi_{*}\rvert<\varepsilon$, we have $\rho_{n}(\xi)\geqslant
c>0$. Here $\rho_{n}$ is the density of $\underline{\omega_{n}}$ on
$\mathscr{M}^{n}$.

  2. (2)

The map
$\mathscr{X}^{(2)}_{n}(\cdot,x_{*})\colon\mathscr{M}^{n}\to\mathscr{\mathbb{T}}^{d,(2)}$
is a submersion at $\xi=\xi_{*}\,.$

Finally, we need a uniform bound on the velocity field so that the gradients
of the diffeomorphisms $\mathscr{X}_{n}$ are controlled uniformly in the
noise.

######  Assumption 2.4.

The function
$\mathscr{U}\colon\mathscr{M}\times\mathbb{T}^{d}\times[0,1]\to\mathbb{R}^{d}$
is such that

(2.21)
$\sup_{(\xi,t)\in\mathscr{M}\times[0,1]}\lVert\mathscr{U}(\xi,\cdot,t)\rVert_{C^{2}(\mathbb{T}^{d})}<\infty\\\
\quad\text{ and
}\quad\sup_{(x,t)\in\mathbb{T}^{d}\times[0,1]}\lVert\nabla_{\xi}\mathscr{U}(\cdot,x,t)\rVert_{L^{\infty}(\mathscr{M})}<\infty\,.$

We will prove (Section 4, below) that Assumptions 2.2 and 2.4 imply that $V$
is a Lyapunov function for $P_{\kappa}^{(2)}$, and satisfies the drift
condition (2.19) with slightly larger constants. Following this, we will show
(Lemma 3.1, below) that these assumptions will also ensure $P_{\kappa}^{(2)}$
satisfies the assumptions of the Harris theorem with $\kappa$-independent
constants. As a result, a quantitative Harris theorem [HM11] will show that
$P_{\kappa}^{(2)}$ is uniformly geometrically ergodic with a
$\kappa$-independent rate (Lemma 3.2 in Section 7, below). Once this has been
established, an argument of [DKK04] will imply $X^{\kappa}$ is exponentially
mixing, and prove Lemma 3.3 (Section 8).

###  2.4. Checkable Conditions that Guarantee the Harris Conditions.

In practice it is not easy to find a Lyapunov function (Assumption 2.2). As
mentioned in Remark 1.5, recent papers [BBPS22, BCZG22] instead assume certain
conditions which are stronger than Harris conditions, but are easier to
verify. For convenience of the reader, we state these conditions here.

######  Assumption 2.5.

The transition kernels $P_{0}$, $\hat{P}_{0}$ and $P^{(2)}_{0}$ are all
Feller, and topologically irreducible.

######  Assumption 2.6.

Let $\mathscr{N}$ be one of the spaces $\mathbb{T}^{d},S\mathbb{T}^{d},$ and
$\mathbb{T}^{d,(2)}$ and $\mathscr{Y}_{k}$ be one of the corresponding maps
$\mathscr{X}_{k}$, $\hat{\mathscr{X}}_{k}$, or $\mathscr{X}^{(2)}_{k}$. For
each choice of $\mathscr{N}$ and $\mathscr{Y}$, there exist $n\geqslant 1$ and
$(\xi_{*},x_{*})\in\mathscr{M}^{n}\times\mathscr{N}$ such that the following
hold.

  1. (1)

There exists $c,\varepsilon>0$ such that for every $\xi\in\mathscr{M}^{n}$
with $\lvert\xi-\xi_{*}\rvert<\varepsilon$, we have $\rho_{n}(\xi)\geqslant
c>0$. Here $\rho_{n}$ is the density of $\underline{\omega_{n}}$ on
$\mathscr{M}^{n}$.

  2. (2)

The map $\mathscr{Y}_{n}(\cdot,x_{*})\colon\mathscr{M}^{n}\to\mathscr{N}$ is a
submersion at $\xi=\xi_{*}\,.$

######  Assumption 2.7.

For each choice of $\mathscr{N}$ and $\mathscr{Y}$ as in Assumption 2.6, there
exist $\xi_{**}\in\operatorname{supp}(\operatorname{dist}(\zeta_{1}))$ and
$y_{*}\in\mathscr{N}$ such that $\mathscr{Y}_{1}(\xi_{**},y_{*})=y_{*}$.

######  Assumption 2.8.

Let $\mathscr{N}=\mathbb{T}^{d},\mathscr{Y}=\mathscr{X}$, and let $n$,
$(\xi_{*},x_{*})$ be as in Assumption 2.6. Define a $C^{1}$-mapping
$g\colon\mathscr{M}^{n}\to\text{SL}_{d}(\mathbb{R})$ by

(2.22)
$g(\xi)=\frac{1}{\lvert\det\mathscr{A}_{n}(\xi,x_{*})\rvert^{\frac{1}{d}}}\mathscr{A}_{n}(\xi,x_{*})\,.$

Then, the restriction of the derivative $D_{\xi_{*}}g$ to $\ker
D_{\xi}\mathscr{X}_{n}(\cdot,x_{*})\subset T_{\xi}\mathscr{M}^{n}$ is
surjective onto $T_{g(\xi_{*})}\text{SL}_{d}(\mathbb{R})$.

These conditions are stronger than the Harris conditions in the following
sense.

######  Proposition 2.9.

If Assumption 2.4 and 2.5–2.8 hold, then the Assumptions 2.1–2.3 also hold.
Hence, in this case, the enhanced dissipation estimate (1.4) also holds for
some random variable $D_{\kappa}$ (depending on $u$ and $\kappa$, but
independent of $W$) satisfying (1.5).

###### Proof of Proposition 2.9.

The proof of Proposition 2.9 follows immediately from the results in [BCZG22].
Assumption 2.5 along with continuity of $\mathscr{U}$ (Assumption 2.4) and
Assumption 2.7 implies the conditions in Assumption 2.1. Assumption 2.3
follows immediately from Assumption 2.6. To obtain the Lyapunov function
(Assumption 2.2) we will apply Proposition 3.3 in [BCZG22]. In order to do
this we note that a standard Gronwall argument (see for instance Lemma 4.2,
below) and compactness imply that there exists a constant $C_{0}^{\prime}$
such that for all $\xi\in\mathscr{M}$, $x,y\in\mathbb{T}^{d}$, we have

(2.23) $\frac{1}{C_{0}^{\prime}}d(x,y)\leqslant
d(\mathscr{X}_{1}(\xi,x),\mathscr{X}_{1}(\xi,y))\leqslant
C_{0}^{\prime}d(x,y)\,.$

Along with Assumption 2.8, this allows us to apply Proposition 3.3 in
[BCZG22], and gives positivity of the top Lyapunov exponent. Now Proposition
4.5 in [BCZG22] (and Assumptions 2.4–2.8) will imply the existence of a
Lyapunov function as stated in Assumption 2.2. ∎

##  3\. Proof of Theorem 1.1

As mentioned earlier, the main idea behind the proof of Theorems 1.1, 1.3 is
to show that Assumption 2.1–2.4 imply that the Harris conditions hold for all
sufficiently small $\kappa>0$.

######  Lemma 3.1.

Suppose Assumptions 2.1–2.4 hold. Then there exist $l\in\mathbb{N}$,
$\gamma_{3}\in(0,1)$, $K>0$, $R>\frac{2K}{1-\gamma_{3}}$, $\alpha\in(0,1)$,
and a probability measure $\nu$, such that for all sufficiently small
$\kappa>0$ we have

(3.1) $\displaystyle P_{\kappa}^{(2),l}V\leqslant\gamma_{3}V+K\,,$ (3.2)
$\displaystyle\inf_{x\in\\{V\leqslant
R\\}}P_{\kappa}^{(2),l}(x,\cdot)\geqslant\alpha\nu(\cdot)\,.$

Once Lemma 3.1 is proved, the remainder of the proof can be obtained using
established methods (see for instance [BCZG22, BBPS21, DKK04]). First, a
quantitative version of the Harris theorem [HM11, MT09] combined with Lemma
3.1 will show that the two point process $X^{\kappa,(2)}$ is $V$-geometrically
ergodic.

######  Lemma 3.2 (V-geometric ergodicity).

Suppose that (3.1) and (3.2) hold. There exist constants $C>0$ and $\beta>0$
such that for all sufficiently small $\kappa\geqslant 0$, all measurable
$\varphi:\mathbb{T}^{d,(2)}\to\mathbb{R}$ such that $\|\varphi\|_{V}<\infty$,
and any $n\in\mathbb{N}$, we have

(3.3)
$\Big{\lVert}P_{\kappa}^{(2),n}\varphi-\int\varphi\,d\pi^{(2)}\Big{\rVert}_{V}\leqslant
Ce^{-\beta n}\Big{\lVert}\varphi-\int\varphi\,d\pi^{(2)}\Big{\rVert}_{V}\,.$

Combining this with Borel-Cantelli argument in [KDK05] (see also [BBPS21,
BCZG22]) will show almost sure exponential mixing of $X^{\kappa}$.

######  Lemma 3.3.

Suppose that (3.3) holds. Then, for every $\alpha>0$ and $0<q<\infty$, there
exists a random $D_{\kappa}\geqslant 1$ (which depends on $u$ but is
independent of $W$), and deterministic $\gamma>0$ (independent of $\kappa$)
such that for all sufficiently small $\kappa\geqslant 0$, every pair of mean-
zero test functions $f,g\in\dot{H}^{\alpha}$, and every $n\in\mathbb{N}$ we
have

(3.4) $\langle f,g\circ X^{\kappa}_{n}\rangle\leqslant D_{\kappa}e^{-\gamma
n}\lVert f\rVert_{H^{\alpha}}\lVert g\rVert_{H^{\alpha}}\,,\quad\text{almost
surely}\,.$

Moreover, there exists a finite constant $\bar{D}_{q}$ (independent of
$\kappa$) such that

(3.5) $\bm{E}D_{\kappa}^{q}\leqslant\bar{D}_{q}\,,$

for all sufficiently small $\kappa\geqslant 0$.

######  Remark 3.4.

By duality, (3.4) is equivalent to

(3.4′) $\lVert g\circ X^{\kappa}_{n}\rVert_{H^{-\alpha}}\leqslant
D_{\kappa}e^{-\gamma t}\lVert g\rVert_{H^{\alpha}}\,,$

for every mean-zero test function $g\in H^{-\alpha}$.

Momentarily postponing the proof of Lemma 3.3, we prove Theorems 1.1 and 1.3.

###### Proof of Theorem 1.1.

For notational convenience, we will use $\rho_{t}$ to denote $\rho(\cdot,t)$,
the slice of $\rho$ at time $t$. Fix $\alpha>0$, and let $n\in\mathbb{N}$ be a
large time. Assumption 2.4 and parabolic regularity implies

(3.6)
$\lVert\rho_{n}-\bar{\rho}\rVert_{L^{\infty}}\leqslant\frac{C}{\kappa^{\frac{2\alpha+d}{4}}}\lVert\rho_{n-1}-\bar{\rho}\rVert_{H^{-\alpha}}\,.$

To bound the right hand side, we note that the Kolmogorov backward equation
implies that for any $t\geqslant 1$ we have

(3.7) $\rho_{t+1}(x)=\bm{E}_{W}\rho_{1}\circ X^{\kappa}_{1,1+t}(x)\,.$

Here $\bm{E}_{W}$ denotes the expectation with respect to the $\bm{P}_{W}$
marginal of the product measure $\bm{P}=\bm{P}_{W}\otimes\bm{P}_{0}$, and
$X^{\kappa}_{1,\cdot}$ is the solution of the SDE

(3.8)
$dX^{\kappa}_{1,t}(x)=-u(X^{\kappa}_{1,t}(x),t)\,dt+\sqrt{2\kappa}\,dW_{t}\,,\quad
X^{\kappa}_{1,1}(x)=x\,.$

Since the distribution of $u$ is time homogeneous, we may apply Lemma 3.3 to
$X^{\kappa}_{1,\cdot}$. Thus, using (3.4′) yields

(3.9) $\displaystyle\lVert\rho_{n-1}-\bar{\rho}\rVert_{H^{-\alpha}}$
$\displaystyle=\lVert\bm{E}_{W}\rho_{1}\circ
X^{\kappa}_{1,n-1}-\bar{\rho}\rVert_{H^{-\alpha}}\leqslant\bm{E}_{W}\lVert\rho_{1}\circ
X^{\kappa}_{1,n-1}-\bar{\rho}\rVert_{H^{-\alpha}}$ (3.10)
$\displaystyle\leqslant
D_{\kappa}e^{-\gamma(n-1)}\lVert\rho_{1}-\bar{\rho}\rVert_{H^{\alpha}}\,.$

Here we used the fact that since $D_{\kappa}$ is independent of $W$ which
implies $D_{\kappa}=\bm{E}_{W}D_{\kappa}$. Finally, we note that parabolic
regularity implies

(3.11)
$\lVert\rho_{1}-\bar{\rho}\rVert_{H^{\alpha}}\leqslant\frac{C}{\kappa^{\frac{2\alpha+d}{4}}}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{1}}\,.$

Combining the above, we note

(3.12)
$\lVert\rho_{n}-\bar{\rho}\rVert_{L^{\infty}}\leqslant\frac{Ce^{-\gamma(n-1)}D_{\kappa}}{\kappa^{\alpha+d/2}}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{\infty}}\,,$

for all integer times $n$. Since the $L^{\infty}$ norm is non-increasing, we
can increase $C$ by a factor of $e^{\gamma}$ and ensure (3.12) holds for all
$t\geqslant 0$, concluding the proof. ∎

Theorem 1.3 is equivalent to Theorem 1.1 by a standard duality argument.

###### Proof of Theorem 1.3.

Let $\rho$ be a solution to (1.1) with initial data $\rho_{0}$. By the
Kolmogorov backward equation

(3.13) $\rho_{t}(x)=\bm{E}_{W}\rho_{0}\circ
X^{\kappa}_{t}(x)=\int_{\mathbb{T}^{d}}p^{\kappa}_{t}(x,y)\rho_{0}(y)\,dy\,.$

Thus

(3.14)
$\sup_{x\in\mathbb{T}^{d}}\int_{\mathbb{T}^{d}}(p^{\kappa}_{t}(x,y)-1)\rho_{0}(y)\,dy\,dx\leqslant\lVert\rho_{t}-\bar{\rho}\rVert_{L^{\infty}}\leqslant\frac{D_{\kappa}}{\kappa^{\frac{d}{2}+\alpha}}e^{-\gamma
t}\lVert\rho_{0}-\bar{\rho}\rVert_{L^{1}}$

Since $\rho_{0}\in L^{1}$ is arbitrary, this implies

(3.15) $\sup_{x\in\mathbb{T}^{d}}\lVert
p^{\kappa}_{t}(x,\cdot)-1\rVert_{L^{\infty}}\leqslant\frac{D_{\kappa}}{\kappa^{\frac{d}{2}+\alpha}}e^{-\gamma
t}\,,$

which immediately yields (1.10) as desired. ∎

The proofs of Lemmas 3.2 and 3.3 follow quickly from existing results, and the
bulk of the remainder of the paper is devoted to proving Lemma 3.1. The proof
can naturally be split into two parts – showing $V$ is a Lyapunov function,
and producing a $\kappa$-independent small set. We do each of these parts in
Section 4 and 5 respectively, and then prove Lemma 3.1 in Section 6.

##  4\. The existence of a $\kappa$-independent Lyapunov function.

The goal of this section is to produce the Lyapunov function $V$ used in Lemma
3.1 in Section 6. For this we use the function $V$ in Assumptions 2.2 and the
fact that $u\in C^{2}$ (Assumption 2.4) to show that for sufficiently small
$\kappa$, the function $V$ is still a Lyapunov function for
$P_{\kappa}^{(2)}$.

######  Lemma 4.1 (Existence of a $\kappa$-independent Lyapunov function).

Suppose Assumptions 2.2 and 2.4 hold. Then there exists $\gamma_{1}\in(0,1)$,
$b>0$ and $V:\mathbb{T}^{d,(2)}\to[1,\infty)$, all independent of $\kappa$,
such that

(4.1) $P^{(2)}_{\kappa}V\leqslant\gamma_{1}V+b$

holds for all sufficiently small $\kappa\geqslant 0$.

The main idea behind the proof of Lemma 4.1 is that the difference of four
terms $X_{1}^{\kappa}(x),X_{1}^{\kappa}(y),X_{1}^{0}(x),X_{1}^{0}(y)$ can be
estimated small when $\kappa$ is close to $0$ and $x$ and $y$ are close. To
carry out the details, we need to first lift all the processes to
$\mathbb{R}^{d}$. We identify the torus $\mathbb{T}^{d}$ with the set of
equivalence classes
$\\{[\check{x}]\nonscript\>|\nonscript\>\mathopen{}\allowbreak\check{x}\in\mathbb{R}^{d}\\}$,
where $[\check{x}]$ denotes the equivalence class of $\check{x}$ modulo
$\mathbb{Z}^{d}$. Notice that for any $\check{x},\check{y}\in\mathbb{R}^{d}$
such that $\lvert\check{x}-\check{y}\rvert<1/2$, we have

(4.2)
$d(x,y)=\lvert\check{x}-\check{y}\rvert\quad\text{and}\quad\exp_{x}^{-1}y=\check{y}-\check{x}\,,\qquad\text{where}\quad
x=[\check{x}],~{}y=[\check{y}]\,.$

We will implicitly identify periodic functions on $\mathbb{R}^{d}$ with
functions on the torus.

We also define the Brownian motion $W$ on $\mathbb{T}^{d}$ by choosing a
standard Brownian motion $\check{W}$ on $\mathbb{R}^{d}$, and setting

(4.3) $W_{t}=\check{W}_{t}\pmod{\mathbb{Z}^{d}}\,.$

Now, let $\check{X}$ be the solution to (1.7) on $\mathbb{R}^{d}$ with the
Brownian motion $\check{W}$, and notice $X_{t}=[\check{X}_{t}]$. To prove
Lemma 4.1 we need two elementary estimates on the flows $\check{X}^{\kappa}$,
which we state below.

######  Lemma 4.2.

There exists a constant
$C_{0}=C_{0}(\sup_{\xi,t}\lVert\mathscr{U}\rVert_{C^{1}})\geqslant 1$ such
that for every $\kappa\geqslant 0$ and $\check{x},\check{y}\in\mathbb{R}^{d}$,
we have

(4.4)
$\frac{\lvert\check{x}-\check{y}\rvert}{C_{0}}\leqslant\lvert\check{X}^{\kappa}_{1}(\check{x})-\check{X}^{\kappa}_{1}(\check{y})\rvert\leqslant
C_{0}\lvert\check{x}-\check{y}\rvert\,,\quad\text{almost surely}\,.$

######  Lemma 4.3.

For each $\check{x},\check{y}\in\mathbb{R}^{d}$ and $t\in[0,1]$, define
$\varrho_{t}^{\kappa}(\check{x},\check{y})$ by

(4.5)
$\varrho^{\kappa}_{t}(\check{x},\check{y})\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\big{\lvert}\check{X}_{t}^{\kappa}(\check{x})-\check{X}_{t}^{\kappa}(\check{y})-(\check{X}_{t}^{0}(\check{x})-\check{X}_{t}^{0}(\check{y}))\big{\rvert}\,.$

There exists a constant
$C_{1}=C_{1}(\sup_{\xi,t}\lVert\mathscr{U}\rVert_{C^{2}})$ such that for any
$\alpha>0$ and $\kappa>0$, we have

(4.6)
$\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}\varrho_{1}^{\kappa}(\check{x},\check{y})\leqslant
C_{1}(\alpha\sqrt{\kappa}+\lvert\check{x}-\check{y}\rvert)\big{\lvert}\check{X}_{1}^{0}(\check{x})-\check{X}_{1}^{0}(\check{y})\big{\rvert}\,.$

Here $\check{W}^{*}$ is the running maximum of $\lvert\check{W}\rvert$,
defined by

(4.7) $\check{W}^{*}_{t}=\max_{s\leqslant t}\lvert\check{W}_{s}\rvert\,.$

Momentarily postponing the proofs of Lemmas 4.2 and 4.3, we now prove Lemma
4.1.

###### Proof of Lemma 4.1.

We prove this proposition in two following steps. First, we show that for any
$c>0$, there exists a constant $\varepsilon<s_{*}$ such that

(4.8) $\limsup_{\kappa\to 0^{+}}\sup_{(x,y)\in\Delta(\varepsilon)}\frac{\lvert
P^{(2)}_{\kappa}V-P_{0}^{(2)}V\rvert}{V}<c\,.$

Then, using (2.19) and choosing $c<1-\tilde{\gamma}$ imply that for some
$\varepsilon>0$ and all sufficiently small $\kappa\geqslant 0$,

(4.9) $P_{\kappa}^{(2)}V<(\tilde{\gamma}+c)V\quad\text{on
}\Delta(\varepsilon)\,.$

Outside $\Delta(\varepsilon)$, we will show that there exists a constant $b>0$
such that

(4.10) $P_{\kappa}^{(2)}V\leqslant b\quad\text{on }\Delta(\varepsilon)^{c}\,,$

for all sufficiently small $\kappa\geqslant 0$. Using (4.9) and (4.10)
immediately implies (4.1) as desired.

In order to finish the proof we need to prove (4.8) and (4.10). To prove
(4.8), let $\varepsilon$ be a small $\kappa$-independent constant that will be
chosen later. We let $x,y\in\Delta(\varepsilon)$ and note

(4.11)
$P^{(2)}_{\kappa}V(x,y)-P^{(2)}_{0}V(x,y)=\bm{E}[V(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))-V(X_{1}^{0}(x),X_{1}^{0}(y))]\,.$

Choose $\varepsilon>0$ such that for the constant $C_{0}$ in (4.4) and $s_{*}$
from Assumption 2.2, we have $C_{0}\varepsilon<s_{*}$. Let
$\check{x},\check{y}\in\mathbb{R}^{d}$ such that $[\check{x}]=x$,
$[\check{y}]=y$, and $d(x,y)=\lvert\check{x}-\check{y}\rvert$. Then by Lemma
4.2, we have

(4.12)
$\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})\rvert<C_{0}\varepsilon<s_{*}<\frac{1}{2}\,,$

for all $\kappa\geqslant 0$, and hence

(4.13) $\frac{1}{C_{0}}d(x,y)\leqslant
d(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))=\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})\rvert\leqslant
C_{0}d(x,y)\leqslant s_{*}\,.$

Then using (2.20), we get

(4.14)
$P^{(2)}_{\kappa}V(x,y)-P^{(2)}_{0}V(x,y)=\bm{E}\Bigg{[}\frac{\psi^{\kappa}}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{\psi^{0}}{\lvert\check{Z}_{1}\rvert^{p}}\Bigg{]}\,,$

where

(4.15)
$\displaystyle\psi^{\kappa}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\psi(X_{1}^{\kappa}(x),\hat{\omega}(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y)))\,,$
(4.16)
$\displaystyle\psi^{0}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\psi(X_{1}^{0}(x),\hat{\omega}(X_{1}^{0}(x),X_{1}^{0}(y)))\,,$
(4.17) $\displaystyle
Z_{1}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\check{X}_{1}^{0}(\check{x})-\check{X}_{1}^{0}(\check{y})\,,$
(4.18)
$\displaystyle\tilde{Z}_{1}^{\kappa}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})-(\check{X}_{1}^{0}(\check{x})-\check{X}_{1}^{0}(\check{y}))\,,$
(4.19)
$\displaystyle\hat{\omega}(x^{\prime},y^{\prime})=\frac{\exp^{-1}_{x^{\prime}}(y^{\prime})}{d(x^{\prime},y^{\prime})}\,.$

In particular, we can rewrite the terms into

(4.20)
$\frac{\psi^{\kappa}}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{\psi^{0}}{\lvert\check{Z}_{1}\rvert^{p}}=\psi^{\kappa}\Big{(}\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{)}+\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}(\psi^{\kappa}-\psi^{0})\,.$

and deduce

(4.21) $\displaystyle\frac{\lvert P^{(2)}_{\kappa}V-P^{(2)}_{0}V\rvert}{V}$
$\displaystyle\leqslant\frac{d(x,y)^{p}}{\inf_{SM}\psi_{p}}\Big{(}\bm{E}\Bigl{[}\psi^{\kappa}\Big{\lvert}\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{\rvert}+\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\lvert\psi^{\kappa}-\psi^{0}\rvert\Bigr{]}\Big{)}$
(4.22)
$\displaystyle\leqslant\frac{d(x,y)^{p}}{\inf_{SM}\psi}\Bigl{(}\bm{E}\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}(F_{1}+F_{2})+\bm{P}(\check{W}_{1}^{*}\geqslant\alpha)\sup_{\Omega}(F_{1}+F_{2})\Bigr{)}\,,$

where

(4.23) $\displaystyle
F_{1}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\psi^{\kappa}\Big{\lvert}\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{\rvert}\,,\quad\text{and}\quad
F_{2}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\lvert\psi^{\kappa}-\psi^{0}\rvert\,,$

and $\alpha>0$ is a small $\kappa$-independent constant that will be chosen
shortly.

We will now bound each term on the right of (4.22). On the event
$\\{\check{W}_{1}^{*}\leqslant\alpha\\}$, we note that Lemma 4.3 implies

(4.24) $\lvert\tilde{Z}_{1}^{\kappa}\rvert\leqslant
C_{1}(\alpha\sqrt{\kappa}+\varepsilon)\lvert\check{Z}_{1}\rvert\,,$

so

(4.25)
$\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert\geqslant\lvert\check{Z}_{1}\rvert-\lvert\tilde{Z}_{1}^{\kappa}\rvert\geqslant\lvert\check{Z}_{1}\rvert(1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon))>0\,,$

for sufficiently small $\kappa,\varepsilon>0$. Moreover,

(4.26)
$\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert\leqslant\lvert\check{Z}_{1}\rvert+\lvert\tilde{Z}_{1}^{\kappa}\rvert\leqslant\lvert\check{Z}_{1}\rvert(1+C_{1}(\alpha\sqrt{\kappa}+\varepsilon))\,.$

These two inequalities imply that

(4.27)
$\displaystyle\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}\leqslant\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{(}1-\frac{1}{(1+C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}\Big{)}\,,$
(4.28)
$\displaystyle\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\leqslant\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{(}\frac{1}{(1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}-1\Big{)}\,.$

By convexity of the function $\xi\mapsto\xi^{-p}$, we have

(4.29)
$1-\frac{1}{(1+C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}\leqslant\frac{1}{(1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}-1\,.$

Combined with (4.27) and (4.28) this implies

(4.30)
$\Big{\lvert}\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{\rvert}\leqslant\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{(}\frac{1}{(1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}-1\Big{)}\,.$

Multiplying (4.30) by $\frac{d(x,y)^{p}}{\inf_{SM}\psi}\psi^{\kappa}$ and
using (4.13) gives

(4.31)
$\frac{d(x,y)^{p}}{\inf_{SM}\psi}\bm{E}\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}F_{1}\leqslant
C_{0}^{p}\frac{\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\Big{(}\frac{1}{(1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}-1\Big{)}\,.$

Next, we bound $\bm{E}\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}F_{2}$.
We note that by (4.74), (4.75), and (4.13), we can choose sufficiently small
$\kappa,\alpha,\varepsilon$ to ensure

(4.32)
$\operatorname{diam}\\{\check{X}_{1}^{\kappa}(\check{x}),\check{X}_{1}^{0}(\check{x}),\check{X}_{1}^{\kappa}(\check{y}),\check{X}_{1}^{0}(\check{y})\\}<\frac{1}{2}\,.$

In this case we have

(4.33) $\displaystyle d((X_{1}^{\kappa}(x),$
$\displaystyle\hat{\omega}(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))),(X_{1}^{0}(x),\hat{\omega}(X_{1}^{0}(x),X_{1}^{0}(y))))$
(4.34)
$\displaystyle=\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{0}(\check{x})\rvert+\Bigg{\lvert}\frac{\check{X}_{1}^{\kappa}(\check{y})-\check{X}_{1}^{\kappa}(\check{x})}{\lvert\check{X}_{1}^{\kappa}(\check{y})-\check{X}_{1}^{\kappa}(\check{x})\rvert}-\frac{\check{X}_{1}^{0}(\check{y})-\check{X}_{1}^{0}(\check{x})}{\lvert\check{X}_{1}^{0}(\check{y})-\check{X}_{1}^{0}(\check{x})\rvert}\Bigg{\rvert}$
(4.35)
$\displaystyle=\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{0}(\check{x})\rvert+\Bigg{\lvert}\frac{\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert}-\frac{\check{Z}_{1}}{\lvert\check{Z}_{1}\rvert}\Bigg{\rvert}\,.$

The inequalities (4.74) and the general bound

(4.36) $\Big{\lvert}\frac{v}{\lvert v\rvert}-\frac{v^{\prime}}{\lvert
v^{\prime}\rvert}\Big{\rvert}\leqslant\frac{2\lvert
v-v^{\prime}\rvert}{\min\\{\lvert v\rvert,\lvert v^{\prime}\rvert\\}}\,,$

immediately imply

(4.37)
$\displaystyle\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{0}(\check{x})\rvert$
$\displaystyle+\Bigg{\lvert}\frac{\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert}-\frac{\check{Z}_{1}}{\lvert\check{Z}_{1}\rvert}\Bigg{\rvert}$
(4.38) $\displaystyle\leqslant
C_{0}\alpha\sqrt{\kappa}+\frac{2\lvert\tilde{Z}_{1}^{\kappa}\rvert}{\min\\{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert,\lvert\check{Z}_{1}\rvert\\}}\,.$

Using (4.24) and (4.25) we see

(4.39)
$C_{0}\alpha\sqrt{\kappa}+2\frac{\lvert\tilde{Z}_{1}^{\kappa}\rvert}{\min\\{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert,\lvert\check{Z}_{1}\rvert\\}}\leqslant
C_{0}\alpha\sqrt{\kappa}+\frac{2C_{1}(\alpha\sqrt{\kappa}+\varepsilon)}{1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon)}\,.$

Now fix $\eta>0$ to be a $\kappa$-independent constant that will be chosen
later. Using uniform continuity of $\psi$ find $\delta>0$ such that

(4.40) $\forall z,z^{\prime}\in SM,\quad
d(z,z^{\prime})<\delta\implies\lvert\psi(z)-\psi(z^{\prime})\rvert<\eta\,.$

If $\kappa,\varepsilon$ are sufficiently small, the right hand side of (4.39)
can be made smaller than $\delta$. Using (4.37) implies

(4.41)
$d((X_{1}^{\kappa}(x),\hat{\omega}(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))),(X_{1}^{0}(x),\hat{\omega}(X_{1}^{0}(x),X_{1}^{0}(y))))<\delta\,,$

and using (4.40) implies

(4.42) $\lvert\psi^{\kappa}-\psi^{0}\rvert<\eta\,.$

Thus by (4.13), we see

(4.43)
$\frac{d(x,y)^{p}}{\inf_{SM}\psi}\bm{E}\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}F_{2}\leqslant\frac{C_{0}^{p}}{\inf_{SM}\psi}\eta\,.$

Finally, for the last term on the right of (4.22), we note

(4.44)
$\psi^{\kappa}\Big{\lvert}\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}-\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{\rvert}+\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\lvert\psi^{\kappa}-\psi^{0}\rvert\\\
\leqslant\lVert\psi\rVert_{\infty}\Big{(}\frac{1}{\lvert\check{Z}_{1}+\tilde{Z}_{1}^{\kappa}\rvert^{p}}+\frac{1}{\lvert\check{Z}_{1}\rvert^{p}}\Big{)}+\frac{2}{\lvert\check{Z}_{1}\rvert^{p}}\lVert\psi\rVert_{\infty}\,,$

so using (4.13) shows

(4.45)
$\frac{d(x,y)^{p}}{\inf_{SM}\psi}\bm{P}(\check{W}_{1}^{*}\geqslant\alpha)\sup_{\Omega}(F_{1}+F_{2})\leqslant\frac{4C_{0}^{p}\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\bm{P}[\check{W}_{1}^{*}>\alpha]\,.$

Using (4.31), (4.43) and (4.45) in (4.22) we obtain

(4.46) $\sup_{(x,y)\in\Delta(\varepsilon)}\frac{\lvert
P^{(2)}_{\kappa}V-P^{(2)}_{0}V\rvert}{V}\leqslant
C_{0}^{p}\frac{\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\Big{(}\frac{1}{(1-C_{1}(\alpha\sqrt{\kappa}+\varepsilon))^{p}}-1\Big{)}\\\
+\frac{C_{0}^{p}}{\inf_{SM}\psi}\eta+\frac{4C_{0}^{p}\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\bm{P}[W_{1}^{*}>\alpha]\,.$

Thus,

(4.47) $\limsup_{\kappa\to 0+}\sup_{(x,y)\in\Delta(\varepsilon)}\frac{\lvert
P^{(2)}_{\kappa}V-P_{0}^{(2)}V\rvert}{V}\leqslant
C_{0}^{p}\frac{\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\Big{(}\frac{1}{(1-C_{1}\varepsilon)^{p}}-1\Big{)}\\\
+\frac{C_{0}^{p}}{\inf_{SM}\psi}\eta+\frac{4C_{0}^{p}\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\bm{P}[W_{1}^{*}>\alpha]\,.$

Now, we choose $\alpha,\eta,$ and $\varepsilon$ such that

(4.48)
$\displaystyle\frac{4C_{0}^{p}\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}\bm{P}[W_{1}^{*}>\alpha]<\frac{1}{3}c\,,$
(4.49) $\displaystyle\frac{C_{0}^{p}}{\inf_{SM}\psi}\eta<\frac{1}{3}c\,,$
(4.50) $\displaystyle
C_{0}^{p}\frac{\lVert\psi\rVert_{\infty}}{\inf_{SM}\psi}(\frac{1}{(1-C_{1}\varepsilon)^{p}}-1)<\frac{1}{3}c\,,$

then using (4.47) will imply (4.8) as desired.

Finally, in order to prove (4.10), we see from the Assumption 2.2 that $V$ is
continuous on the compact set
$K\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\overline{\Delta(s_{*})}-\Delta(\frac{s_{*}}{2})$
so it can be continuously extended to the compact set
$K^{\prime}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\mathbb{T}^{d,(2)}-\Delta(\frac{s_{*}}{2})$
such that

(4.51)
$1\leqslant\inf_{K}V=\inf_{K^{\prime}}V\leqslant\sup_{K^{\prime}}V=\sup_{K}V\,.$

Now, let $(x,y)\in\Delta(\varepsilon)^{c}$. On the event
$E_{1}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\\{(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))\in\Delta(s_{*})\\}$,
there exist $\check{x},\check{y}\in\mathbb{R}^{d}$ such that $[\check{x}]=x$,
$[\check{y}]=y$, and

(4.52)
$d(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))=\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})\rvert<s_{*}\,.$

By (4.4), $\lvert\check{x}-\check{y}\rvert<\frac{s_{*}}{C_{0}}<\frac{1}{2}$ so
$\lvert\check{x}-\check{y}\rvert=d(x,y)$ and

(4.53) $d(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))\geqslant C_{0}d(x,y)\geqslant
C_{0}\varepsilon\,,$

which implies

(4.54) $\bm{E}[V(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))\bm{1}_{E_{1}}]\leqslant
C_{0}^{-p}\varepsilon^{-p}\lVert\psi\rVert_{\infty}\,.$

On the event $E_{1}^{c}$, we have

(4.55)
$\bm{E}[V(X_{1}^{\kappa}(x),X_{1}^{\kappa}(y))\bm{1}_{E_{1}^{c}}]\leqslant\sup_{K^{\prime}}V\,.$

Bounds (4.54) and (4.55) yield (4.10) with

(4.56)
$b\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}C_{0}^{-p}\varepsilon^{-p}\lVert\psi\rVert_{\infty}+\sup_{K^{\prime}}V\,.$

This completes the proof. ∎

### Gradient estimates on the stochastic flows

It remains to prove Lemmas 4.2 and 4.3. Lemma 4.2 is a direct result of
applying Grönwall’s inequality twice to the difference $\lvert
X_{t}^{\kappa}(\check{x})-X_{t}^{\kappa}(\check{y})\rvert$ near $t=0$ and
$t=1$.

###### Proof of Lemma 4.2.

For the upper bound, we see that for any $\kappa\geqslant 0$ and $0\leqslant
t\leqslant 1$,

(4.57)
$\check{X}_{t}^{\kappa}(\check{x})-\check{X}_{t}^{\kappa}(\check{y})=\check{x}-\check{y}+\int_{0}^{t}u(X_{s}^{\kappa}(\check{x}),s)-u(\check{X}_{s}^{\kappa}(\check{y}),s)ds\,,$

so

(4.58)
$\lvert\check{X}_{t}^{\kappa}(\check{x})-\check{X}_{t}^{\kappa}(\check{y})\rvert\leqslant\lvert\check{x}-\check{y}\rvert+A_{1}\int_{0}^{t}\lvert\check{X}_{s}^{\kappa}(\check{x})-\check{X}_{s}^{\kappa}(\check{y})\rvert
ds\,.$

By Grönwall’s inequality, we have

(4.59)
$\lvert\check{X}_{1}^{\kappa}(x)-\check{X}_{1}^{\kappa}(y)\rvert\leqslant
e^{A_{1}}\lvert\check{x}-\check{y}\rvert\,,$

where

(4.60)
$A_{1}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lVert\nabla_{x}\mathscr{U}\rVert_{L^{\infty}(\mathscr{M}\times\mathbb{T}^{d}\times[0,1])}$

Similarly, for the lower bound, we see that for any $\kappa\geqslant 0$ and
$0\leqslant t\leqslant 1$,

(4.61)
$\displaystyle\check{X}_{1-t}^{\kappa}(\check{x})-\check{X}_{1-t}^{\kappa}(\check{y})$
$\displaystyle=\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})-\int_{1-t}^{1}\Big{(}u(\check{X}_{s}^{\kappa}(\check{x}),s)-u(\check{X}_{s}^{\kappa}(\check{y}),s)\Big{)}ds$
(4.62)
$\displaystyle=\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})-\int_{0}^{t}\Big{(}u(\check{X}_{1-s}^{\kappa}(\check{x}),s)-u(\check{X}_{1-s}^{\kappa}(\check{y}),s)\Big{)}ds$

so

(4.63)
$\lvert\check{X}_{1-t}^{\kappa}(\check{x})-\check{X}_{1-t}^{\kappa}(\check{y})\rvert\leqslant\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})\rvert+A_{1}\int_{0}^{t}\lvert\check{X}_{1-s}^{\kappa}(\check{x})-\check{X}_{1-s}^{\kappa}(\check{y})\rvert
ds\,.$

By Grönwall’s inequality, we have

(4.64) $\lvert\check{x}-\check{y}\rvert\leqslant
e^{A_{1}}\lvert\check{X}_{1}^{\kappa}(\check{x})-\check{X}_{1}^{\kappa}(\check{y})\rvert\,.$

Thus, setting
$C_{0}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}e^{A_{1}}$
concludes the proof. ∎

To prove Lemma 4.3, we first explicitly write down the differences
$\check{X}_{t}^{\kappa}(\check{x})-\check{X}_{t}^{\kappa}(\check{y})$ and
$\check{X}_{t}^{0}(\check{x})-\check{X}_{t}^{0}(\check{y})$ by using the
differential equations they satisfy. Then, we use mean value theorem and
Grönwall’s inequality multiple times to estimate their difference.

###### Proof of Lemma 4.3.

For $0\leqslant t\leqslant 1$, we have

(4.65)
$\displaystyle\check{X}_{t}^{\kappa}(\check{x})-\check{X}_{t}^{\kappa}(\check{y})$
$\displaystyle=\check{x}-\check{y}+\int_{0}^{t}(u(\check{X}_{s}^{\kappa}(\check{x}),s)-u(\check{X}_{s}^{\kappa}(\check{y})),s)\,ds$
(4.66)
$\displaystyle=\check{x}-\check{y}+\int_{0}^{t}\nabla_{x}u(\beta_{1}(s),s)\cdot(\check{X}_{s}^{\kappa}(\check{x})-\check{X}_{s}^{\kappa}(\check{y}))\,ds$

for some $\beta_{1}(s)$ and similarly

(4.67)
$\check{X}_{t}^{0}(\check{x})-\check{X}_{t}^{0}(\check{y})=\check{x}-\check{y}+\int_{0}^{t}\nabla_{x}u(\beta_{2}(s),s)\cdot(\check{X}_{s}^{0}(\check{x})-\check{X}_{s}^{0}(\check{y}))\,ds$

for some $\beta_{2}(s)$. We define

(4.68)
$S(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\check{X}_{t}^{\kappa}(\check{x})-\check{X}_{t}^{\kappa}(\check{y})-(\check{X}_{t}^{0}(\check{x})-\check{X}_{t}^{0}(\check{y}))\,.$

Then, by taking the difference between (4.65) and (4.67), we get

(4.69) $S(t)=\int_{0}^{t}\nabla_{x}u(\beta_{1}(s),s)\cdot S(s)\\\
+(\nabla_{x}u(\beta_{1}(s),s)-\nabla_{x}u(\beta_{2}(s),s))\cdot(\check{X}_{s}^{0}(\check{x})-\check{X}_{s}^{0}(\check{y}))\,ds\,,$

which implies

(4.70) $\varrho_{t}^{\kappa}(\check{x},\check{y})\leqslant
A_{1}\int_{0}^{t}\varrho_{s}^{\kappa}(\check{x},\check{y})ds+A_{2}\int_{0}^{t}\lvert\beta_{1}(s)-\beta_{2}(s)\rvert\Big{\lvert}\check{X}_{s}^{0}(\check{x})-\check{X}_{s}^{0}(\check{y})\Big{\rvert}\,ds\,,$

where

(4.71)
$A_{2}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lVert\nabla_{x}^{2}\mathscr{U}\rVert_{L^{\infty}(\mathscr{M}\times\mathbb{T}^{d}\times[0,1])}\,.$

We see that

(4.72) $\displaystyle\lvert\beta_{1}(s)-\beta_{2}(s)\rvert\leqslant\max\\{$
$\displaystyle\lvert\check{X}_{s}^{\kappa}(\check{y})-\check{X}_{s}^{0}(\check{y})\rvert,\lvert\check{X}_{s}^{\kappa}(\check{x})-\check{X}_{s}^{0}(\check{x})\rvert,$
(4.73)
$\displaystyle\lvert\check{X}_{s}^{\kappa}(\check{y})-\check{X}_{s}^{0}(\check{x})\rvert,\lvert\check{X}_{s}^{\kappa}(\check{x})-\check{X}_{s}^{0}(\check{y})\rvert\\}$

and in particular,

(4.74)
$\displaystyle\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}\lvert\check{X}_{s}^{\kappa}(\check{x})-\check{X}_{s}^{0}(\check{x})\rvert$
$\displaystyle\leqslant\alpha\sqrt{\kappa}e^{A_{1}s}$ (4.75)
$\displaystyle\lvert\check{X}_{s}^{0}(\check{x})-\check{X}_{s}^{0}(\check{y})\rvert$
$\displaystyle\leqslant\lvert\check{x}-\check{y}\rvert e^{A_{1}s}$

by Grönwall’s inequality.

As a result, we obtain

(4.76)
$\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}\varrho_{t}^{\kappa}(\check{x},\check{y})\leqslant
A_{1}\int_{0}^{t}\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}\varrho_{s}^{\kappa}(\check{x},\check{y})ds\\\
+A_{2}(\alpha\sqrt{\kappa}+\lvert\check{x}-\check{y}\rvert)e^{A_{1}t}\int_{0}^{t}\Big{\lvert}\check{X}_{s}^{0}(\check{x})-\check{X}_{s}^{0}(\check{y})\Big{\rvert}ds\,.$

Using Grönwall’s inequality and Lemma 4.2 this gives

(4.77)
$\displaystyle\bm{1}_{\\{\check{W}_{1}^{*}\leqslant\alpha\\}}\varrho_{1}^{\kappa}(\check{x},\check{y})\leqslant
C_{1}(\alpha\sqrt{\kappa}+\lvert\check{x}-\check{y}\rvert)\Big{\lvert}\check{X}_{1}^{0}(\check{x})-\check{X}_{1}^{0}(\check{y})\Big{\rvert}\,,$

for some constant $C_{1}=C_{1}(A_{1},A_{2})$. ∎

##  5\. A stable small set.

Next, in order to obtain the minorizing condition (3.2) stated in Lemma 3.1,
we will show that Assumptions 2.3 and 2.4 imply the existence of a
$\kappa$-independent small set.

######  Lemma 5.1.

Suppose Assumptions 2.3 and 2.4 hold. Then there exist nonempty open sets
$A,B\subseteq\mathbb{T}^{d,(2)}$ and a constant $\beta>0$ such that

(5.1) $\inf_{x\in
A}P_{\kappa}^{(2),n}(x,\cdot)\geqslant\beta\operatorname{Leb}|_{B}(\cdot)\,.$

for all sufficiently small $\kappa\geqslant 0$. Here $\operatorname{Leb}|_{B}$
denotes the restriction of the Lebesgue measure to $B$.

To prove Lemma 5.1, we need the following two lemmas.

######  Lemma 5.2.

Let $f\colon B_{r}\subseteq\mathbb{R}^{N}\to\mathbb{R}^{d}$ be a $C^{1}$
function such that $f(0)=0$ and $\operatorname{rank}(Df(0))=d$. Then there
exist $\delta,s>0$ such that, for any $g\colon\mathbb{R}^{N}\to\mathbb{R}^{d}$
Lipschitz with $\|f-g\|_{L^{\infty}(B_{r})}\leqslant\delta$, we have

(5.2) $g(B_{r})\supseteq B_{s}$

Here $B_{r}$ denotes the open ball centered at $0$.

###### Proof.

By the constant rank theorem, there are diffeomorphisms $\alpha$ and $\beta$
which fix the origin such that $(\alpha\circ
f\circ\beta)(x_{1},\dots,x_{N})=(x_{1},\dots,x_{d})$. Choose $\delta>0$ so
that $\|(\alpha\circ f\circ\beta)-(\alpha\circ
g\circ\beta)\|_{L^{\infty}(B_{1})}\leqslant\frac{1}{4}$. Let
$x_{d+1},\dots,x_{N}$ be arbitrary with
$x_{d+1}^{2}+\dots+x_{N}^{2}\leqslant\frac{1}{4}$ and define
$h(x_{1},\dots,x_{d}):=(\alpha\circ
g\circ\beta)(x_{1},\dots,x_{d},x_{d+1},\dots,x_{N})$. Then $h\colon
B_{3/4}\subseteq\mathbb{R}^{d}\to\mathbb{R}^{d}$ satisfies
$|h(x)-x|\leqslant\frac{1}{4}$ at every point.

We claim that $h(B_{3/4})\supseteq B_{1/2}$. Indeed, compute the degree
$\deg(h,B_{3/4},x)=1$ for each $x\in B_{1/2}$ (the degree is a homotopy
invariant, and $h$ is homotopic to the identity).

Then $(\alpha\circ g\circ\beta)(B_{1})\supseteq B_{1/2}$, and therefore
$g(B_{r})\supseteq B_{s}$ for some $s>0$ depending only on $\alpha$ and
$\beta$. ∎

######  Lemma 5.3.

For any $n\in\mathbb{N}$, $\kappa\geqslant 0$, $x\in\mathbb{T}^{d,(2)}$,
$\xi,\xi^{\prime}\in\mathscr{M}^{n}$, and almost any realization of the noise
$W$, we have

(5.3)
$d(X_{n}^{\kappa,(2)}(\xi,x),X_{n}^{\kappa,(2)}(\xi^{\prime},x))\leqslant
A_{3}\exp(n(1+A_{1}))d_{\infty}(\xi,\xi^{\prime})\,,$

where $A_{1}$ is defined as in (4.60), and

(5.4) $\displaystyle A_{3}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lVert\nabla_{\xi}\mathscr{U}\rVert_{L^{\infty}(\mathscr{M}\times\mathbb{T}^{d}\times[0,1])}\,,$
(5.5) $\displaystyle d_{\infty}(\xi,\xi^{\prime})$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sup_{k\leqslant
n}d_{\mathscr{M}}(\xi_{k},\xi_{k}^{\prime})\,.$

###### Proof.

The proof follows immediately from Grönwall’s inequality and is very similar
to the proof of Lemma 4.2. ∎

With these lemmas, we can prove the existence of a $\kappa$-independent small
set.

###### Proof of Lemma 5.1.

Let $n,x_{*},\xi_{*},\varepsilon,c,\rho_{n}$ be defined as in Assumption 2.3.
Then, we see that $\mathscr{X}_{n}^{(2)}(\cdot,x_{*})$ satisfies the
assumption in Lemma 5.2 with $r=\varepsilon$. Let $\delta,s$ be the constants
given in Lemma 5.2 for the map $\mathscr{X}_{n}^{(2)}(\cdot,x_{*})$. Now,
choose $\alpha>0$ such that
$\bm{P}[\check{W}_{n}^{*}\leqslant\alpha]\geqslant\frac{1}{2}$ and let
$\eta,\kappa$ be small enough so that

(5.6) $(\eta+n\sqrt{\kappa}\alpha)e^{nA_{1}}<\frac{1}{\sqrt{2}}\delta\,.$

Then for each choice of $x\in B_{\eta}(x_{*})$ and a realization of Brownian
path such that $\\{\check{W}_{n}^{*}\leqslant\alpha\\}$, we have

(5.7) $d(X_{n}^{\kappa,(2)}(\xi,x),X_{n}^{0,(2)}(\xi,x_{*}))<\delta\,,$

for any $\xi\in\mathscr{M}^{n}$. Thus, we can apply Lemma 5.2 to the map
$g:\xi\mapsto X_{n}^{\kappa,(2)}(\xi,x)$ and see that

(5.8) $\displaystyle\bm{P}_{\kappa}^{(2),n}(x,A)$
$\displaystyle\geqslant\bm{E}_{W}\Big{[}\Big{(}\int_{g^{-1}(A)\cap
B_{\varepsilon}(\xi_{*})}\rho_{n}(\xi)d\xi\Big{)}\bm{1}_{\\{W_{n}^{*}\leqslant\alpha\\}}\Big{]}$
(5.9) $\displaystyle\geqslant c\bm{E}_{W}\Big{[}\lvert g^{-1}(A)\cap
B_{\varepsilon}(\xi_{*})\rvert\bm{1}_{\\{W^{*}_{n}\leqslant\alpha\\}}\Big{]}$
(5.10) $\displaystyle\geqslant\frac{c}{2}\lvert A\cap
B_{s}(\mathscr{X}(\xi_{*},x_{*}))\rvert\bm{E}_{W}\operatorname{Lip}(g)^{-\dim\mathscr{M}}\,.$

Using Lemma 5.3 we see

(5.11)
$\bm{E}_{W}\operatorname{Lip}(g)^{-\dim\mathscr{M}}\geqslant(A_{3}\exp(n(1+A_{1})))^{-\dim\mathscr{M}}\,.$

Using (5.11) in (5.10) yields (5.1) as desired. ∎

##  6\. Verification of the Harris Assumptions (Lemma 3.1).

Given the Lyapunov function $V$ from Lemma 4.1 and the small set from Lemma
5.1, we will now prove Lemma 3.1 verifying Harris conditions for the two point
chains, for all sufficiently small $\kappa\geqslant 0$.

###### Proof of Lemma 3.1.

We first note that Lemma 4.1 and induction immediately imply that for every
$l\in\mathbb{N}$,

(6.1) $P_{\kappa}^{(2),l}V\leqslant\gamma_{1}^{l}V+b_{l}\,,$

where

(6.2) $\displaystyle
b_{l}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}b\sum_{i=0}^{l-1}\gamma_{1}^{i}=b\frac{1-\gamma_{1}^{l}}{1-\gamma_{1}}\,.$

We will now show that $R$ and $l$ can be chosen so that (3.2) is also
satisfied.

We will first show that any compact set $A\subset\mathbb{T}^{d,(2)}$ is a
small set, uniformly in $\kappa$. More precisely, we will prove that for any
compact set $A\subset\mathbb{T}^{d,(2)}$, there exist $m\in\mathbb{N}$,
$\beta\in(0,1)$, and a probability measure $\mu$, all independent of $\kappa$,
such that

(6.3) $\inf_{x\in A}P_{\kappa}^{(2),m}(x,\cdot)\geqslant\beta\mu(\cdot)\,,$

for all sufficiently small $\kappa\geqslant 0$. Next, we will show that for
any $R>\frac{2b}{1-\gamma_{1}}$, there exists a compact set $S$ such that

(6.4) $\\{V\leqslant R\\}\subset S\,.$

To see why the above implies (3.2), we first fix any
$R>\frac{2b}{1-\gamma_{1}}$ and then find a compact set $S$ such that (6.4)
holds. From (6.3), we can find $\kappa$-independent constants $l,\alpha$ and a
probability measure $\nu$ such that for all sufficiently small
$\kappa\geqslant 0$, we have

(6.5) $\inf_{x\in S}P_{\kappa}^{(2),l}(x,\cdot)\geqslant\alpha\nu(\cdot)\,.$

For this particular $l\in\mathbb{N}$, we define

(6.6)
$\gamma_{3}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\gamma_{1}^{l},\quad\text{and}\quad
K\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}b_{l}\,.$

Using (6.2) we observe

(6.7)
$R>\frac{2b}{1-\gamma_{1}}=\frac{2b}{1-\gamma_{1}^{l}}\frac{1-\gamma_{1}^{l}}{1-\gamma_{1}}=\frac{2K}{1-\gamma_{3}}\,,$

which implies (3.2) as claimed.

It remains to prove (6.3) and (6.4). To prove (6.3), we first note that
Assumptions 2.1 and 2.3 imply $P_{0}^{(2)}$ is an $\psi^{(2)}$-irreducible,
aperiodic $T$-chain with the property that

(6.8) $\psi^{(2)}(\mathcal{V})>0,\quad\text{for all
}\mathcal{V}\subset\mathbb{T}^{d,(2)}\text{ open}\,.$

Assumption 2.3 and Lemma 5.1 imply that there exist open balls
$B_{r},B_{s}\subset\mathbb{T}^{d,(2)}$ and $n\in\mathbb{N}$ such that for all
sufficiently small $\kappa\geqslant 0$,

(6.9) $\inf_{x\in
B_{r}}P_{\kappa}^{(2),n}(x,\cdot)\geqslant\tilde{\mu}(\cdot)\,,$

where $B_{r}$ and $B_{s}$ are open balls with radius $r,s>0$, respectively,
and

(6.10)
$\tilde{\mu}(\cdot)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\operatorname{Leb}(\cdot\cap
B_{s})\,.$

$B_{\frac{1}{2}r}$ is small for the chain $P_{0}^{(2)}$ and
$\psi^{(2)}(B_{\frac{1}{2}r})>0$ so by Theorem 6.2.5 (ii) and Theorem 5.5.7 in
[MT09], we see that there exist some $q\in\mathbb{N}$ and $c>0$ such that

(6.11) $\inf_{x\in A}P_{0}^{(2),q}(x,B_{\frac{1}{2}r})\geqslant c>0\,.$

Then, for each $x\in A$, the measure $P_{\kappa}^{(2),q}(x,\cdot)$ converges
weak-* to the measure $P_{0}^{(2),q}(x,\cdot)$ as $\kappa\to 0$ so there
exists $\kappa_{0}(x)>0$ such that

(6.12)
$\inf_{\kappa<\kappa_{0}(x)}P_{\kappa}^{(2),q}(x,B_{\frac{1}{2}r})\geqslant\frac{1}{2}P_{0}^{(2),q}(x,B_{\frac{1}{2}r})\,.$

Also, if we assume $r<1$ without loss of generality and define $A_{1}$ as in
(4.60), then for any $x\in\mathbb{T}^{d}$ and $y\in
B(x,\frac{1}{2}\exp(-A_{1}q)r)$, we can find
$\check{x},\check{y}\in\mathbb{R}^{d}$ such that $[\check{x}]=x$,
$[\check{y}]=y$, and $d(x,y)=\lvert\check{x}-\check{y}\rvert$, and use simple
Gröwnwall bound to notice that

(6.13)
$d(X_{q}^{\kappa}(x),X_{q}^{\kappa}(y))=\lvert\check{X}_{q}^{\kappa}(\check{x})-\check{X}_{q}^{\kappa}(\check{y})\rvert\leqslant\exp(A_{1}q)\lvert\check{x}-\check{y}\rvert\leqslant\frac{1}{2}r\,.$

This immediately leads to the inequality

(6.14) $P_{\kappa}^{(2),q}(y,B_{r})\geqslant
P_{\kappa}^{(2),q}(x,B_{\frac{1}{2}r})\,.$

Now, we cover the compact set $A$ with open balls $\cup_{x\in
A}B(x,\frac{1}{2}\exp(-A_{1}q)r)$ and find a finite cover
$\bigcup_{i=1}^{N}B(x_{i},\frac{1}{2}\exp(-A_{1}q)r)$. If we let
$\kappa<\kappa_{0}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\min_{i=1}^{N}\kappa_{0}(x_{i})$
and $y\in A$, then by (6.14), (6.12), and (6.11), we see that

(6.15) $P_{\kappa}^{(2),q}(y,B_{r})\geqslant\frac{1}{2}c\,.$

This implies for any $0\leqslant\kappa<\kappa_{0}$,

(6.16) $\inf_{x\in A}P_{\kappa}^{(2),q}(x,B_{r})\geqslant\frac{1}{2}c\,.$

Defining $m\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}n+q$
and using (6.9) with (6.16) yields

(6.17) $\inf_{x\in
A}P_{\kappa}^{(2),m}(x,\cdot)\geqslant\frac{1}{2}c\tilde{\mu}(\cdot)$

for all sufficiently small $\kappa\geqslant 0$. Then, normalizing
$\frac{1}{2}c\tilde{\mu}(\cdot)$ immediately implies (6.3), as desired.

Finally, to prove (6.4), we notice that if $s^{\prime\prime}<s_{*}$ and
$(x,x^{\prime})\in\Delta(s^{\prime\prime})$, we have

(6.18)
$V(x,x^{\prime})=d(x,x^{\prime})^{-p}\psi_{p}(x,\widehat{w}(x,x^{\prime}))\geqslant(s^{\prime\prime})^{-p}\Big{(}\inf_{\text{SM}}\psi_{p}\Big{)}>0\,.$

Thus making $s^{\prime\prime}>0$ sufficiently small will ensure

(6.19) $\Delta(s^{\prime\prime})\subset\\{V>R\\}\,,$

which implies

(6.20) $\\{V\leqslant R\\}\subset\Delta(s^{\prime\prime})^{c}\,.$

Thus
$S\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\Delta(s^{\prime\prime})^{c}$
is the desired compact set, proving (6.4). This concludes the proof. ∎

##  7\. V-geometric ergodicity (Lemma 3.2).

Given Lemma 3.1, $V$-geometric ergodicity of the two point chains
$P_{\kappa}^{(2)}$ follows directly from a theorem of Harris [Har55, MT09].
The usual Harris theorem, however, isn’t quantitative enough to yield (3.3)
with $\kappa$-independent constants $C,\beta$. We will instead use the version
in [HM11] which is quantitative and can be used to prove Lemma 3.2.

For the proof, we define the metric $\rho_{\beta}$ by

(7.1)
$\rho_{\beta}(\mu_{1},\mu_{2})\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{\mathbb{T}^{d,(2)}}(1+\beta
V)\,d\lvert\mu_{1}-\mu_{2}\rvert\,,$

where $\beta\geqslant 0$, $\mu_{1},\mu_{2}$ are probability measures, and
$\lvert\mu_{1}-\mu_{2}\rvert$ denotes the variation of the signed measure
$\mu_{1}-\mu_{2}$. The quantitative Harris theorem from [HM11] shows that
$P^{(2),l}_{\kappa}$ is a contraction under $\rho_{\beta}$, and for readers
convenience we now restate this result in our context.

######  Theorem 7.1 (Theorem 1.3 in [HM11]).

Make the same assumptions as in Lemma 3.1. For any $\alpha_{0}\in(0,\alpha)$,
and any $\gamma_{0}\in(\gamma_{3}+\frac{2K}{R},1)$, define

(7.2)
$\beta\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{\alpha_{0}}{K}\,,\qquad\bar{\alpha}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\max\Big{\\{}1-(\alpha-\alpha_{0}),\frac{2+R\beta\gamma_{0}}{2+R\beta}\Big{\\}}\,.$

Then, for any two probability measures $\mu_{1},\mu_{2}$ we have

(7.3)
$\rho_{\beta}(\mu_{1}P^{(2),l}_{\kappa},\mu_{2}P^{(2),l}_{\kappa})\leqslant\bar{\alpha}\rho_{\beta}(\mu_{1},\mu_{2})\,.$

Referring to [HM11] for the proof of Theorem 7.1, we will now prove Lemma 3.2.

###### Proof of Lemma 3.2.

We will first show there exist constants $C>0$, $\gamma_{2}\in(0,1)$, and
$l\in\mathbb{N}$ such that for all sufficiently small $\kappa\geqslant 0$, any
$\varphi:\mathbb{T}^{d,(2)}\to\mathbb{R}$ such that
$\lVert\varphi\rVert_{V}<\infty$, and any $m\in\mathbb{N}$, we have

(7.4)
$\Big{\lVert}P_{\kappa}^{(2),lm}\varphi-\int\varphi\,d\pi^{(2)}\Big{\rVert}_{V}\leqslant
C\gamma_{2}^{m}\Big{\lVert}\varphi-\int\varphi\,d\pi^{(2)}\Big{\rVert}_{V}\,.$

For this, we first define a weighted norm

(7.5)
$\|\varphi\|_{\beta}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sup_{x}\frac{\lvert\varphi(x)\rvert}{1+\beta
V(x)}\,.$

Then, since $V\geqslant 1$ on $\mathbb{T}^{d,(2)}$, we see that the norms
$\|\cdot\|_{V}$ and $\|\cdot\|_{\beta}$ are equivalent for any $\beta>0$ with

(7.6)
$\frac{1}{1+\beta}\|\varphi\|_{V}\leqslant\|\varphi\|_{\beta}\leqslant\frac{1}{\beta}\|\varphi\|_{V}\,.$

We also note that given any two probability measures $\mu_{1}$ and $\mu_{2}$
on $\mathbb{T}^{d,(2)}$,

(7.7)
$\displaystyle\rho_{\beta}(\mu_{1},\mu_{2})=\sup_{\|\varphi\|_{\beta}\leqslant
1}\langle\mu_{1}-\mu_{2},\varphi\rangle$

always holds, where $\langle\mu,\varphi\rangle$ denotes the dual pairing

(7.8)
$\langle\mu,\varphi\rangle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{\mathbb{T}^{d,(2)}}\varphi\,d\mu\,.$

Now, we’re ready to prove (7.4). From here on, we set $P=P_{\kappa}^{(2),l}$
for notational simplicity. By Lemma 3.1 and Theorem 7.1 we obtain the
contraction estimate (7.3). In particular, for any $x\in\mathbb{T}^{d,(2)}$
and
$\mu_{1}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\delta_{x}$,
$\mu_{2}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\pi^{(2)}$,
we have

(7.9)
$\rho_{\beta}(\delta_{x}P^{n},\pi^{(2)})\leqslant\bar{\alpha}^{n}\rho_{\beta}(\delta_{x},\pi^{(2)})\,.$

By using (7.7) and then (7.1), we notice

(7.10)
$\displaystyle\frac{\lvert\langle\delta_{x}P^{n}-\pi^{(2)},\varphi-\langle\pi^{(2)},\varphi\rangle\rangle\rvert}{\|\varphi-\langle\pi^{(2)},\varphi\rangle\|_{\beta}}\leqslant\bar{\alpha}^{n}\rho_{\beta}(\delta_{x},\pi^{(2)})\leqslant\bar{\alpha}^{n}(1+\beta
V(x)+\langle\pi^{(2)},1+\beta V\rangle)\,,$

and hence

(7.11)
$\displaystyle\frac{\Big{\lvert}(P^{n}\varphi)(x)-\langle\pi^{(2)},\varphi\rangle\Big{\rvert}}{1+\beta
V(x)}\leqslant\bar{\alpha}^{n}(1+\langle\pi^{(2)},1+\beta
V\rangle)\|\varphi-\langle\pi^{(2)},\varphi\rangle\|_{\beta}\,.$

This holds for any $x\in\mathbb{T}^{d,(2)}$ so

(7.12)
$\Big{\lVert}P^{n}\varphi-\langle\pi^{(2)},\varphi\rangle\Big{\rVert}_{\beta}\leqslant
C_{\beta}\bar{\alpha}^{n}\|\varphi-\langle\pi^{(2)},\varphi\rangle\|_{\beta}\,.$

Finally, using (7.6) yields (7.4) where $C$ and $\gamma_{2}$ depend on
$\alpha,\gamma_{3},K,R$ but not on $\kappa$. This completes the proof for
(7.4).

The proof that (7.4) implies (3.3) is a standard argument. For any
$x\in\mathbb{T}^{d,(2)}$ and any mean $0$ function $g$ we note

$\displaystyle\Big{\lvert}\frac{(P_{\kappa}^{(2)}g)(x)}{V(x)}\Big{\rvert}=\Big{\lvert}\int_{\mathbb{T}^{d,(2)}}\frac{g(y)}{V(y)}\frac{V(y)}{V(x)}P_{\kappa}^{(2)}(x,dy)\Big{\rvert}\leqslant\|g\|_{V}\frac{(P_{\kappa}^{(2)}V)(x)}{V(x)}\leqslant(\gamma_{1}+b)\|g\|_{V}\,,$

where $\gamma_{1}$ and $b$ are the constants defined in (4.1). Thus,

(7.13)
$\big{\lVert}P_{\kappa}^{(2)}g\big{\rVert}_{V}\leqslant(\gamma_{1}+b)\|g\|_{V}\,.$

This and (7.4) imply that for any $m\in\mathbb{N}$, $0\leqslant r<l$,

(7.14) $\Big{\lVert}P_{\kappa}^{(2),lm+r}g\Big{\rVert}_{V}\leqslant
C\max(\gamma_{1}+b,1)^{r}\gamma_{2}^{m}\lVert g\rVert_{V}\leqslant
C\gamma_{2}^{m}\lVert g\rVert_{V}\,,$

which proves

(7.15) $\Big{\lVert}P_{\kappa}^{(2),n}g\Big{\rVert}_{V}\leqslant
C(\gamma_{2}^{\frac{1}{l}})^{n}\|g\|_{V}$

for general $n\in\mathbb{N}$ with possibly different constants $C$ in each
line. This completes the proof of Lemma 3.2. ∎

##  8\. Exponential Mixing of the Stochastic Flows (Lemma 3.3).

In general, the geometric ergodicity of the two-point chain implies almost
sure exponential mixing of the random dynamical system. To the best of our
knowledge this principle was introduced in [DKK04] and was also used in
[BBPS22, BCZG22]. We reproduce it here keeping track of the constants
introduced in the proof and their dependence on $\kappa$ in order to prove
that $\gamma$ in (3.4) is $\kappa$-independent.

###### Proof of Lemma 3.3.

Let
$\mathbb{Z}_{0}^{d}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\mathbb{Z}^{d}-\\{0\\}$
and denote $\\{e_{m}:m\in\mathbb{Z}^{d}\\}$ as the orthogonal basis
$e_{m}(x)=e^{im\cdot x}$ for $L^{2}({\mathbb{T}^{d}})$ and denote

(8.1) $f=\sum_{m\in\mathbb{Z}_{0}^{d}}\hat{f}_{m}e_{m}\,,\quad
g=\sum_{m\in\mathbb{Z}_{0}^{d}}\hat{g}_{m}e_{m}\,,$

as the fourier expansions of $f$ and $g$. We note that and
$\hat{f}_{0}=\hat{g}_{0}=0$ as $f$ and $g$ are mean-zero.

Now, fix $\zeta>0$ and for $m,m^{\prime}\in\mathbb{Z}_{0}^{d}$ and $\kappa>0$,
define random variables

$\displaystyle
N_{m,m^{\prime}}^{\kappa}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\max\Big{\\{}n\in\mathbb{N},~{}\Big{\lvert}\int
e_{m}(x)e_{m^{\prime}}\circ X^{\kappa}_{n}(x)\pi(dx)\Big{\rvert}>e^{-\zeta
n}\Big{\\}}\,,$ $\displaystyle
K_{\kappa}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\max\Big{\\{}\lvert
m\rvert\vee\lvert m^{\prime}\rvert:e^{\zeta N_{m,m^{\prime}}^{\kappa}}>\lvert
m\rvert\lvert m^{\prime}\rvert\Big{\\}}\,,$
$\displaystyle\widehat{D}_{\kappa}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\max_{\lvert
m\rvert,\lvert m^{\prime}\rvert\leqslant K_{\kappa}}e^{\zeta
N_{m,m^{\prime}}^{\kappa}}\,.$

Then by the definition of $N_{m,m^{\prime}}^{\kappa}$ and Chebyshev, we get

(8.2) $\displaystyle\bm{P}[N_{m,m^{\prime}}^{\kappa}>l]$
$\displaystyle\leqslant\sum_{n>l}\bm{P}\bigg{[}{~{}\Big{\lvert}\int
e_{m}(x)e_{m^{\prime}}\circ X^{\kappa}_{n}(x)\pi(dx)\Big{\rvert}>e^{-\zeta
n}}\bigg{]}$ (8.3) $\displaystyle\leqslant\sum_{n>l}e^{2\zeta
n}\bm{E}\Big{\lvert}\int e_{m}(x)e_{m^{\prime}}\circ
X_{n}^{\kappa}(x)\pi(dx)\Big{\rvert}^{2}\,.$

Observe

(8.4) $\bm{E}\Big{\lvert}\int e_{m}(x)e_{m^{\prime}}\circ
X_{n}^{\kappa}(x)\pi(dx)\Big{\rvert}^{2}=\int
e_{m^{\prime}}^{(2)}P^{(2),n}_{\kappa}e_{m}^{(2)}\,,$

where

(8.5) $\displaystyle
e_{m}^{(2)}(x,y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}e_{m}(x)\overline{e_{m}}(y)\,,$
(8.6)
$\displaystyle\pi^{(2)}(dx,dy)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\pi(dx)\pi(dy)\,.$

From (3.3), we see

(8.7) $\displaystyle\Big{\lvert}\int
e_{m^{\prime}}^{(2)}P^{(2),n}_{\kappa}e_{m}^{(2)}d\pi^{(2)}\Big{\rvert}$
$\displaystyle=\Big{\lvert}\int\Big{(}e_{m^{\prime}}^{(2)}P^{(2),n}_{\kappa}-\Big{(}\int
e_{m^{\prime}}^{(2)}d\pi^{(2)}\Big{)}\Big{)}e_{m}^{(2)}d\pi^{(2)}\Big{\rvert}$
(8.8)
$\displaystyle\leqslant\int\Big{\lvert}e_{m^{\prime}}^{(2)}P^{(2),n}_{\kappa}-\int
e_{m^{\prime}}^{(2)}d\pi^{(2)}\Big{\rvert}\,d\pi^{(2)}$ (8.9)
$\displaystyle\leqslant Ce^{-\beta n}\|e_{m^{\prime}}^{(2)}\|_{V}\int
Vd\pi^{(2)}=C_{V}e^{-\beta n}\,.$

This implies

(8.10) $\displaystyle\bm{P}[N_{m,m^{\prime}}^{\kappa}>l]\leqslant
C_{V}e^{(2\zeta-\beta)l}\,,$

provided

(8.11) $2\zeta-\beta<0\,.$

From now on, we make additional assumptions that $\zeta$ is small enough to
satisfy

(8.12) $\displaystyle d+\frac{2\zeta-\beta}{\zeta}<0\,,$ (8.13)
$\displaystyle\frac{1}{\zeta q}(2\zeta-\beta)+1<-1\,,$ (8.14)
$\displaystyle\frac{5d}{2}+\frac{2\zeta-\beta}{\zeta}<-1\,.$

Equations (8.10) and (8.11) imply that
$\bm{P}(N_{m,m^{\prime}}^{\kappa}<\infty)=1$ and we have the estimate

(8.15) $\displaystyle\Big{\lvert}\int e_{m}(x)e_{m^{\prime}}\circ
X_{n}^{\kappa}(x)\pi(dx)\Big{\rvert}\leqslant e^{\zeta
N_{m,m^{\prime}}^{\kappa}-\zeta n}\,,$ (8.16) $\displaystyle\text{ hence
}\Big{\lvert}\int f(x)g\circ X_{n}^{\kappa}(x)\pi(dx)\Big{\rvert}\leqslant
e^{-\zeta
n}\sum_{m,m^{\prime}}\lvert\hat{f}_{m}\rvert\lvert\hat{g}_{m^{\prime}}\rvert
e^{\zeta N_{m,m^{\prime}}^{\kappa}}\,.$

Moreover, using (8.10) and (8.12), we observe that

(8.17) $\displaystyle\bm{P}[K_{\kappa}>l\big{]}$ $\displaystyle\leqslant
2\sum_{m,m^{\prime}\in\mathbb{Z}_{0}^{d},\lvert
m\rvert>l}\bm{P}\bigg{[}e^{\zeta N_{m,m^{\prime}}^{\kappa}}>\lvert
m\rvert\lvert m^{\prime}\rvert\bigg{]}$ (8.18) $\displaystyle\leqslant
2\sum_{m^{\prime}\in\mathbb{Z}_{0}^{2}}\lvert
m^{\prime}\rvert^{\frac{2\zeta-\beta}{\zeta}}\sum_{m\in\mathbb{Z}_{0}^{2},\lvert
m\rvert>l}\lvert m\rvert^{\frac{2\zeta-\beta}{\zeta}}$ (8.19)
$\displaystyle\lesssim\sum_{n=1}^{\infty}n^{\frac{2\zeta-\beta}{\zeta}+d-1}\sum_{n>l}^{\infty}n^{\frac{2\zeta-\beta}{\zeta}+d-1}\lesssim
l^{d+\frac{2\zeta-\beta}{\zeta}}\,,$

where the constants in the inequalities are independent of $\kappa$. Hence
$\bm{P}(K_{\kappa}<\infty)=1$.

Noting that

$e^{\zeta N_{m,m^{\prime}}^{\kappa}}\leqslant\widehat{D}_{\kappa}\lvert
m\rvert\lvert m^{\prime}\rvert$

always holds, we conclude from (8.16) that

(8.20) $\displaystyle\Big{\lvert}\int f(x)g\circ
X_{n}^{\kappa}(x)\pi(dx)\Big{\rvert}\leqslant\widehat{D}_{\kappa}(\underline{\omega})e^{-\zeta
n}\|f\|_{H^{\frac{d}{2}+2}}\|g\|_{H^{\frac{d}{2}+2}}\,.$

Finally, the same arguments in Lemma 7.1 and Section 7.3 of [BBPS22] show that
for any $s,q>0$,

(8.21) $\displaystyle\Big{\lvert}\int f(x)g\circ
X_{n}^{\kappa}(x)\pi(dx)\Big{\rvert}\leqslant\widehat{D}_{\kappa}(\underline{\omega})e^{-(\frac{2s\zeta}{d+4})n}\|f\|_{H^{s}}\|g\|_{H^{s}}\,.$

Moreover, using (8.10) and (8.19), they show that

(8.22)
$\displaystyle\bm{E}[\widehat{D}_{\kappa}^{q}]=\sum_{l=1}^{\infty}\bm{E}\big{[}1_{\\{K_{\kappa}=l\\}}\max_{\lvert
m\rvert,\lvert m^{\prime}\rvert\leqslant l}e^{\zeta
qN_{m,m^{\prime}}^{\kappa}}\big{]}$ (8.23)
$\displaystyle\leqslant\sum_{l=1}^{\infty}\bm{P}[K_{\kappa}=l]^{\frac{1}{2}}\Big{\lVert}\max_{\lvert
m\rvert,\lvert m^{\prime}\rvert\leqslant l}e^{\zeta
qN_{m,m^{\prime}}^{\kappa}}\Big{\rVert}_{L^{2}}$ (8.24)
$\displaystyle\lesssim\sum_{l=1}^{\infty}l^{d+\frac{2\zeta-\beta}{\zeta}}\bigg{(}\sum_{\lvert
m\rvert,\lvert m^{\prime}\rvert\leqslant l}\|e^{\zeta
qN_{m,m^{\prime}}^{\kappa}}\|_{L^{2}}\bigg{)}$ (8.25)
$\displaystyle\leqslant\Big{(}1+\frac{\zeta
q}{\beta-2\zeta(1+q)}\Big{)}^{\frac{1}{2}}\sum_{l=1}^{\infty}l^{\frac{5d}{2}+\frac{2\zeta-\beta}{\zeta}}<\infty\,,$

provided (8.13) and (8.14). This completes the proof of the theorem with the
choice of
$\gamma\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{2s\zeta}{d+4}$
which can be made independent of $\kappa$ since the conditions for $\zeta$,
(8.11)–(8.14), are independent of $\kappa$. ∎

##  9\. An explicit Lyapunov function for Pierrehumbert flows.

It was proved in Section 5 of [BCZG22] that the Pierrehumbert flows defined in
Corollary 1.7 satisfy the assumptions 2.5–2.8. Then by Proposition 2.9, we see
that Assumption 2.1–2.3 (and hence Corollary 1.7) must hold. However, in the
case of Pierrehumbert flows, we can explicitly construct a simple Lyapunov
function and verify Assumption 2.2 directly.

######  Proposition 9.1.

Consider the RDS of alternating shears defined in Corollary 1.7. If the flow
amplitude $A$ (in (1.14)) is sufficiently large, then there exists $s_{*},p>0$
such that the function $V$ defined by

(9.1)
$V(x,y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}|x-y|_{\infty}^{-p}\text{
on }\Delta(s_{*})\,,$

and extended continuously to $\mathbb{T}^{d,(2)}$ is a Lyapunov function that
satisfies Assumption 2.2. (Here $\lvert z\rvert_{\infty}=\max_{i}\lvert
z_{i}\rvert$.)

###### Proof.

In this proof, we use $C$ as a generic constant that doesn’t depend on $A$ or
$\kappa$. For a given $x\in\mathbb{T}^{2}$, we write $x_{1},x_{2}$ as the
first and second coordinates of $x$, respectively. Let $u(x,t)$ be defined as
in Corollary 1.7.

First, we note that for some small $s_{*}\in(0,\frac{1}{2})$ and any
$x,y\in\Delta(s_{*})$, we can find $\check{x},\check{y}\in\mathbb{R}^{2}$ such
that $[\check{x}]=x,[\check{y}]=y$, and

(9.2)
$d(X_{2}^{0}(x),X_{2}^{0}(y))=\lvert\check{X}_{2}^{0}(\check{x})-\check{X}_{2}^{0}(\check{y})\rvert\leqslant
CA^{2}\lvert\check{x}-\check{y}\rvert=CA^{2}d(x,y)<\frac{1}{2}\,,$

so with slight abuse of notation, we still write $x,y,X_{2}^{0}(x),$ and
$X_{2}^{0}(y)$ for $\check{x},\check{y},\check{X}_{2}^{0}(\check{x}),$ and
$\check{X}_{2}^{0}(\check{y})$, respectively.

We aim to show that there is a constant $0<\beta<1$ such that, for any
$(x,y)\in\Delta(s_{*})$, we have

(9.3) $\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))]\leqslant\beta V(x,y),$

where $\Phi_{t}$ denotes the flow map induced by $u$ at time $t$.

Fix $(x,y)\in\Delta(s_{*})$. First, note that we have $|x-y|_{\infty}\leqslant
CA^{2}|\Phi_{2}(x)-\Phi_{2}(y)|_{\infty}$ for some large $A$ and therefore
$V(\Phi_{2}(x),\Phi_{2}(y))\leqslant{(CA^{2})}^{p}V(x,y)$ almost surely. We
break into two cases.

_Case I: $|x_{2}-y_{2}|\geqslant|x_{1}-y_{1}|$._ Let $E_{0}$ be the event

(9.4)
$E_{0}=\\{\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert<2|x_{2}-y_{2}|\\}\,.$

Since
$\lvert\Phi_{2}(x)-\Phi_{2}(y)\rvert_{\infty}\geqslant\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert_{\infty}$,
we have

(9.5) $\displaystyle\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))]$
$\displaystyle\leqslant\bm{E}[\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert^{-p}]$
(9.6)
$\displaystyle\leqslant\bm{E}[\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert^{-p}\bm{1}_{E_{0}}]+\bm{E}[\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert^{-p}\bm{1}_{E_{0}^{c}}]$
(9.7) $\displaystyle\leqslant{(CA^{2})}^{p}V(x,y)P(E_{0})+2^{-p}V(x,y)\,.$

We will now estimate $P(E_{0})$. For this, we use the explicit form of the
vector field $u$ to write

$\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}=x_{1}-y_{1}+A\sin(2\pi(x_{2}-\zeta_{0}))-A\sin(2\pi(y_{2}-\zeta_{0})).$

Thus $E_{0}$ is contained in the event that

$|x_{1}-y_{1}|+|A\sin(2\pi(x_{2}-\zeta_{0}))-A\sin(2\pi(y_{2}-\zeta_{0}))|<2|x_{2}-y_{2}|.$

In view of the assumption $\lvert x_{2}-y_{2}\rvert\geqslant\lvert
x_{1}-y_{1}\rvert$, the above inequality is implied by

(9.8)
$|A\sin(2\pi(x_{2}-\zeta_{0}))-A\sin(2\pi(y_{2}-\zeta_{0}))|<|x_{2}-y_{2}|.$

Using the fundamental theorem of calculus, the left-hand side above can be
written as the convolution $\lvert\left(2\pi
A\cos(2\pi\cdot)\ast\mathbf{1}_{[x_{2},y_{2}]}\right)(\zeta_{0})\rvert$. Here
$[x_{2},y_{2}]$ denotes the smallest interval (mod $\mathbb{Z}$) with $x_{2}$
and $y_{2}$ as endpoints. Since the derivative of $\cos(2\pi\cdot)$ is bounded
away from zero near the zeros of $\cos(2\pi\cdot)$, the same is true for the
convolution (rescaling by $|x_{2}-y_{2}|$). This implies that the set of
$\zeta_{0}$ which satisfy (9.8) has measure at most $CA^{-1}$. This in turn
implies $\bm{P}(E_{0}\leqslant C/A)$.

Using this in (9.7) implies

(9.9) $\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))]\leqslant
CA^{2p-1}V(x,y)+2^{-p}V(x,y)\,.$

Choosing $p\in(0,\frac{1}{2})$ and $A>0$ sufficiently large, we can ensure

(9.10) $\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))]\leqslant 2^{-p/2}V(x,y)\,,$

as desired.

_Case II: $|x_{2}-y_{2}|<|x_{1}-y_{1}|$._ Let $E_{1}$ be the event that
$\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert\leqslant A^{-1/2}\lvert
x_{1}-y_{1}\rvert$. Then $\bm{P}[E_{1}]\leqslant CA^{-1/2}$ by the same
argument as in the previous case. On the other hand, if $E_{2}$ is the event
that $\lvert\Phi_{2}(x)_{2}-\Phi_{2}(y)_{2}\rvert\leqslant
2A^{1/2}\lvert\Phi_{1}(x)_{1}-\Phi_{1}(y)_{1}\rvert$, then we similarly have
$\bm{P}[E_{2}]\leqslant CA^{-1/2}$. Putting these together, we conclude

$\displaystyle\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))]$
$\displaystyle=\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))\bm{1}_{E_{1}\cup
E_{2}}]+\bm{E}[V(\Phi_{2}(x),\Phi_{2}(y))\bm{1}_{(E_{1}\cup E_{2})^{c}}]$
$\displaystyle\leqslant{(CA^{2})}^{p}V(x,y)(CA^{-1/2})+2^{-p}V(x,y),$

so for $p\in(0,\frac{1}{4})$ we can choose $A>0$ large enough to conclude as
in the previous case. ∎

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