toltxlabel

# Phase space contraction of degenerately damped random splittings

David P. Herzog1 and Jonathan C. Mattingly2 1 Department of Mathematics, Iowa
State University, Ames, Iowa, USA<EMAIL_ADDRESS>2 Department of
Mathematics, Duke University, Durham, North Carolina, USA
<EMAIL_ADDRESS>
###### Abstract.

When studying out-of-equilibrium systems, one often excites the dynamics in
some degrees of freedom while removing the excitation in others through
damping. In order for the system to converge to a statistical steady state,
the dynamics must transfer the energy from the excited modes to the
dissipative directions. The precise mechanisms underlying this transfer are of
particular interest and are the topic of this paper. We explore a class of
randomly switched models introduced in [2, 3] and provide some of the first
results showing that minimal damping is sufficient to stabilize the system in
a fluids model.

##  1\. Introduction and Motivation

Many dynamical systems of scientific importance are expected to be ergodic,
where time averages of observables with respect to the system converge in the
long-run to spatial averages against an invariant measure. We are interested
in settings where such statistical steady states are not in local equilibrium,
but rather have a nontrivial flux through the system. Particular examples
include chains of coupled oscillators with ends connected to heat baths at
different temperatures [8, 15, 17] and Euler equations with forcing and
dissipation acting on limited degrees of freedom [6, 23]. See also [10] for
graphs of coupled oscillators with only some oscillators containing damping
and random forcing. In such examples, energy is injected in some directions in
space and then transferred by the conservative part of the dynamics to the
dissapative degrees of freedom. This process typically produces an equilibrium
steady state with a flux from the forced degrees of freedom to the dissipated
ones. We will refer to such forced-dissipative systems where the dissipation
only acts on a subset of the degrees of freedom as partially dissipative.

In partially dissipative systems, establishing the existence of a statistical
steady state can be highly nontrivial. One of the main goals of this paper is
to prove the existence of such steady states under a randomly modified
dynamics, in settings such as finite-dimensional projections of the partially
damped-driven Euler equations.

To make the discussion more concrete, fix an ordinary differential equation
(ODE)

(1.1) $\displaystyle\dot{x}=V(x)$

on a Hilbert space $\mathbf{X}$ which is nonexplosive for all initial
conditions $x_{0}\in\mathbf{X}$ and conserves a quantity
$H\colon\mathbf{X}\rightarrow[0,\infty)$. Note that conservation simply means
that $H(x_{t})=H(x_{0})$ for all $t\geq 0$. Often, the vector field $V$ is the
Hamiltonian flow associated with Hamiltonian $H$. To the dynamics (1.1), one
can then add dissipation and forcing to arrive at the forced-damped system

$\displaystyle\dot{x}=V(x)-\Lambda x+F\,$

where $\Lambda:\mathbf{X}\rightarrow\mathbf{X}$ is symmetric, non-negative
definite and $F:[0,\infty)\rightarrow\mathbf{X}$ is time-dependent. When
$\Lambda$ is furthermore assumed strictly positive-definite, in many scenarios
it is not hard to show that that there exist constants $\alpha\in(0,1)$ and
$f\geq 0$ so that for some fixed time $t$

(1.2) $\displaystyle H(x_{t})\leq\alpha H(x_{0})+f\,.$

If the sublevel sets $\\{x:H(x)\leq c\\}$, $c\geq 0$, are furthermore compact
in $\mathbf{X}$, then (1.2) quickly implies the existence of a stationary
measure. In random systems with sufficient noise to ensure that the stationary
measure is unique, the bound (1.2) in expectation is critical in proving that
the system converges to the equilibrium at a prescribed rate.

In scenarios where $\Lambda$ has a nontrivial kernel, proving estimates like
(1.2) can be difficult because it requires detailed knowledge of the
conservative dynamics (1.1). Even in this context, however, there are examples
where the forcing allows the conservative part (1.1) to transport energy to
the dissipative directions, keeping it from building up in the undamped
degrees of freedom. When this phenomenon happens at a sufficiently fast rate,
one can ensure the existence of a stationary state and its uniqueness if
additional conditions are satisfied. We refer to [6, 9, 23] for concrete
examples in fluid-like models on Euclidean space and to [8, 10, 15, 17] for
Langevin-like systems. Note that, different from the models in [6, 9, 23]
where a significant number of directions need damping, one needs far less
dissipative directions in the situation of oscillators provided the
interaction potentials are strong enough relative to the pinning potentials
[8, 10, 15]. However, when the interactions are weak, even in the case of a
short chain of oscillators, the propagation of energy to dissipative modes can
be highly nontrivial [17].

In contexts similar to [6, 9, 23], this paper will present some of the first
results establishing the existence of steady states where the dissipative
directions are restricted to a small number of degrees of freedom, even when
the system’s dimension is large. This will be done for _randomly split_
versions of equations such as Galerkin projections of two-dimensional Euler
and the Lorenz ’96 model. Random splittings are simplified models which afford
stronger results at the cost of disrupting the original dynamics. However,
they are insightful as they help build intuition about the full dynamics and
the finer mechanisms involving the transport of energy to these isolated
dissipative directions. For example, our main results establish the existence
of statistical steady states for the randomly split Galerkin 2D Euler
equations and the randomly split Lorenz ’96 models under the hypothesis that
the dissipative operator is restricted to a finite number of well-placed
modes. Furthermore, the number of dissipative modes does not grow with the
dimension of the system, which is new in these settings.

Random splittings can be thought of as a “randomly teased” version of the
original dynamics at every time step, but in way that is natural to the
dynamics. On average, the resulting random dynamics is very close to the
deterministic dynamics and all of the fluctuations are built from the base
vector fields that define the dynamics, but in a different mixture than the
deterministic system. To build the random splittings, we choose elements of
the dynamics that are thought to be fundamental to the structure of the flow
of the base dynamics. For example, for the 2D Euler equation, we use the
three-mode resonant interactions as our dynamical building blocks which
maintain many of the invariant structures of the original system. This is in
contrast to more traditional stochastic agitation methods which use additive
Brownian forcing.

Although some unlikely stochastic realizations are leveraged in the proof of
the existence of a stationary measure, we will see that the arguments leading
to energy dissipation align well with the scenarios people often invoke in
heuristic explanations of why such systems with partial dissipation should be
stable. In particular, we will see how energy transports via the building
blocks, e.g. the three-mode resonant interactions in Euler, to the dissipative
directions. Additionally, the probabilities of the identified dissipative
events scale sufficiently well with the energy that the implied convergence
rates would be geometric for the Lorenz ’96 system and stretched exponential
for the Euler system. The stretched exponential convergence is not completely
surprising given fact that the fixed points of Euler and their nearby slow
dynamics persist in the random splitting. It is also worth noting that we make
no use of any additional stochastic forcing. Consequently, we do not deduce
any convergence-to-equilibrium theorems. Rather, we only give existence of
stationary measures. In existing, related works [2, 3, 23], the uniqueness of
the invariant measure for a stochastically switched Euler system was shown
within the class of measures absolutely continuous with respect to Lebesgue
measure on the constraint surface. While these results could almost certainly
be adapted to this setting, they do not preclude the system converging to one
of the fixed points preserved by the random splitting. Random forcing is
important to show that the system leaves the neighborhood of the fixed points
in an efficient way. In future work, it would be interesting to combine random
switching and random forcing, though we do not explore that here.

Paper Organization:  In Section 2, we introduce some general results for
proving the tightness needed to guarantee the existence of an invariant
measure for the system. We also make some comments on the return times to
compact sets that give an indication what the rates of convergence to
equilibrium might be. In Section 3, we introduce randomly split systems and
the first of our two examples; namely, the Lorenz ’96 model. We demonstrate
most of the ideas introduced in this section by proving the needed assumptions
for the Lorenz ’96 model. In Section 4, we establish the required uniform
lower bound on the probability of entering into a dissipative region in the
Lorenz ’96 model. In Section 5, we turn to the Euler equations and prove all
of the needed estimates in that setting by adapting the calculations used for
the Lorenz ’96. In many ways, however, the Euler setting is significantly more
complicated.

Acknowledgments: We graciously acknowledge support from National Science
Foundation grants DMS-2246491 and DMS-2038056. D.P.H. acknowledges support
from the Simons Foundation through grant MP-TSM-00002755. The authors also
acknowledge fruitful conversations with Andrea Agazzi, Kyle Liss, Theo Drivas,
and Omar Melikechi. The authors are grateful for useful feedback from the
participants at the workshop “Singularity and Prediction in Fluids” during the
summer of 2022 where an early version of these results was first presented.

##  2\. General Results for Partially Dissipative Systems

In this section, $(\mathbf{X},\rho)$ denotes a complete, separable metric
space and $\mathcal{B}$ its associated Borel $\sigma$-algebra of subsets. We
let $(X_{n})$ be a discrete-time Markov chain on the state space $\mathbf{X}$
with transition semigroup $(\mathcal{P}_{n})$ defined over the probability
space $(\Omega,\mathcal{F},\mathbf{P},\mathbf{E})$. The operator
$\mathcal{P}_{n}$ acts on bounded measurable functions
$\phi:\mathbf{X}\rightarrow\mathbf{R}$ and finite $\mathcal{B}$ measures $\mu$
via

(2.1)
$\displaystyle(\mathcal{P}_{n}\phi)(x):=\mathbf{E}_{x}\phi(X_{n})\qquad\text{
and }\qquad(\mu\mathcal{P}_{n})(B):=\int_{\mathbf{X}}\mathbf{P}_{x}(X_{n}\in
B)\mu(dx)$

for $x\in\mathbf{X}$ and $B\in\mathcal{B}$. In (2.1), the notation
$\mathbf{E}_{x}$ and $\mathbf{P}_{x}$ means $\mathbf{E}$ and $\mathbf{P}$, but
indicates that the Markov chain has initial state $X_{0}=x$. For any $n\geq
0$, $B\in\mathcal{B}$ and $x\in\mathbf{X}$, we let

(2.2)
$\displaystyle\mathcal{P}_{n}(x,B)=(\mathcal{P}_{n}\mathbf{1}_{B})(x)=\mathbf{P}_{x}(X_{n}\in
B)$

denote the Markov transition probabilities. Throughout, $\mathcal{P}_{1}$ will
be denoted by $\mathcal{P}$.

The following elementary result is the beginning of our analysis.

######  Proposition 2.3.

Let $H\colon\mathbf{X}\rightarrow[1,\infty)$ and
$p\colon[1,\infty)\rightarrow(0,1]$ be measurable, and let $\alpha\in(0,1)$
and $f_{1},f_{2}\geq 0$ be constants. Suppose for any $x\in\mathbf{X}$ there
exists an event $A=A(x)\in\mathcal{F}$ such that $\mathbf{P}_{x}(A)\geq
p(H(x))$ and such that the following bounds hold

(2.4) $\displaystyle\mathbf{E}_{x}\big{[}H(X_{1})\,\mathbf{1}_{A}\big{]}$
$\displaystyle\leq\alpha H(x)\mathbf{P}_{x}(A)+f_{1},$ (2.5)
$\displaystyle\mathbf{E}_{x}\big{[}H(X_{1})\,\mathbf{1}_{A^{c}}\big{]}$
$\displaystyle\leq H(x)\mathbf{P}_{x}(A^{c})+f_{2}$

for every $x\in\mathbf{X}$. Then the global estimate

(2.6) $\displaystyle\mathcal{P}H-H$
$\displaystyle\leq-(1-\alpha)p(H)\,H\,+f_{1}+f_{2}$

holds on $\mathbf{X}$.

###### Proof.

Observe for any $x\in\mathbf{X}$ we have

$\displaystyle\mathcal{P}H(x)$
$\displaystyle=\mathbf{E}_{x}\big{[}H\big{(}X_{1}\big{)}\,\mathbf{1}_{A}\big{]}+\mathbf{E}_{x}\big{[}H\big{(}X_{1}\big{)}\,\mathbf{1}_{A^{c}}\big{]}$
$\displaystyle\leq[\alpha\mathbf{P}_{x}(A)+(1-\mathbf{P}_{x}(A))]H(x)+f_{1}+f_{2}$
$\displaystyle\leq[1-\mathbf{P}_{x}(A)(1-\alpha)]H(x)+f_{1}+f_{2}$
$\displaystyle\leq[1-(1-\alpha)p(H(x))]H(x)+f_{1}+f_{2}.$

This concludes the proof. ∎

The next lemma shows that the conclusion Proposition 2.3 guaranties the
existence of (at least one) stationary distribution for the Markov chain
provided $x\mapsto p(H(x))H(x)$ has compact sublevel sets and the semigroup
$(\mathcal{P}_{n})$ is _Feller_ ; that is, $x\mapsto\mathcal{P}\phi(x)$ is
continuous whenever $\phi:\mathbf{X}\rightarrow\mathbf{R}$ is bounded,
continuous.

######  Lemma 2.7.

Assume $(\mathcal{P}_{n})$ is Feller. Suppose there exist measurable
$H\colon\mathbf{X}\rightarrow[1,\infty)$ and
$G\colon[1,\infty)\rightarrow(0,\infty)$ such that
$\\{x\in\mathbf{X}\,:\,G(H(x))\leq\alpha\\}$ is compact for every $\alpha>0$,
and for some constant $f\geq 0$ we have the following estimate on
$\mathbf{X}$:

(2.8) $\displaystyle\mathcal{P}H-H\leq-G(H)+f.$

Then the Markov chain $(X_{n})$ has at least one stationary distribution.

###### Proof.

This result is essentially the well-known Krylov-Bogolyubov theorem for Feller
Markov semigroups, so we sketch the proof and refer the reader to [12, 18, 21]
for further details. Fix $x\in\mathbf{X}$ and define the sequence of
$\mathcal{B}$ measures $(\mu_{n})$ by

$\displaystyle\mu_{n}(B)=\frac{1}{n}\sum_{\ell=0}^{n-1}\mathbf{P}_{x}(X_{\ell}\in
B),\,\,n\geq 1,\,B\in\mathcal{B}.$

Using (2.8) and the Markov property we find that

$\displaystyle\sum_{\ell=0}^{n-1}\mathbf{E}_{x}G(H(X_{\ell}))\leq
H(x)-\mathcal{P}_{n}H(x)+nf\leq H(x)+nf\,.$

Hence for any $R>0$,

$\displaystyle\mu_{n}(x:G(H(x))>R)=\frac{1}{n}\sum_{\ell=0}^{n-1}\mathbf{P}_{x}(G(H(X_{\ell}))>R)\leq\frac{1}{n}\sum_{\ell=0}^{n-1}\frac{\mathbf{E}_{x}G(H(X_{\ell}))}{R}\leq\frac{H(x)+f}{R},$

which shows that the family of measures $(\mu_{n})$ is tight since
$\\{x\,:\,G(H(x))\leq R\\}$ is compact by hypothesis. Let $\mu$ be a weak
limit point of this family. To see that $\mu$ is a stationary, observe that
for any $B\in\mathcal{B}$

$\displaystyle|\mu_{n}\mathcal{P}(B)-\mu_{n}(B)|=\frac{1}{n}\big{|}\mathbf{P}_{x}(X_{n}\in
B)-\mathbf{1}_{B}(x)\big{|}\leq\frac{1}{n}.$

Hence if $(\mu_{n_{k}})$ is a subsequence converging weakly to $\mu$, then for
any $B\in\mathcal{B}$

$\displaystyle\mu\mathcal{P}(B)=\lim_{k\rightarrow\infty}\mu_{n_{k}}\mathcal{P}(B)=\lim_{k\rightarrow\infty}(\mu_{n_{k}}\mathcal{P}(B)-\mu_{n_{k}}(B))+\lim_{k\rightarrow\infty}\mu_{n_{k}}(B)=\mu(B).$

∎

######  Remark 2.9.

Suppose that (2.8) is satisfied and $G(t)\rightarrow\infty$ as
$t\rightarrow\infty$. Then for any $\epsilon\in(0,1)$ there exists $R>0$ large
enough so that the global bound holds

$\displaystyle\mathcal{P}H-H\leq-(1-\epsilon)G(H)+(1+\epsilon)f\mathbf{1}_{\\{H\leq
R\\}}.$

Thus under this hypothesis on $G$, the bound (2.8) is equivalent to the bound

(2.10) $\displaystyle\mathcal{P}H-H\leq-G(H)+f\mathbf{1}_{\\{H\leq R\\}}$

up to a relabeling of the righthand sides.

###  2.1. Return Times and Convergence

Let $R,f\geq 0$ and suppose that (2.10) is satisfied for some
$H:\mathbf{X}\rightarrow[1,\infty)$ measurable and some concave nondecreasing
differentiable function $G:[1,\infty)\rightarrow(0,\infty)$. We next consider
implications of this estimate for return times of the Markov chain $(X_{n})$
to the set

(2.11) $\displaystyle H_{\leq R}=\\{x\in\mathbf{X}\,:\,H(x)\leq R\\}.$

Let

(2.12) $\displaystyle T_{R}=\min\\{n\geq 0\,:\,H(X_{n})\leq R\\}$

and note that in the special case when $G(t)=\alpha t$ for some constant
$\alpha>0$, relation (2.10) implies

(2.13) $\displaystyle\mathbf{E}_{x}e^{\alpha T_{R}}<\infty\,\,\text{ for all
}\,\,x\in\mathbf{X}.$

This fact is well-known. However, when $G$ is a general concave, nondecreasing
function, the corresponding form of the moment bound changes.

To describe how the estimates change in this more general context, define
$K:[1,\infty)\rightarrow[0,\infty)$ by

(2.14) $\displaystyle K(t)=\int_{1}^{t}\frac{1}{G(s)}\,ds.$

Since $G$ is assumed concave, we have that

$\displaystyle G(t)\leq G(1)+G^{\prime}(1)(t-1).$

This fact, along with the condition that $G$ is nondecreasing and positive,
implies that $K$ is strictly increasing and $K(t)\rightarrow\infty$ as
$t\rightarrow\infty$. In particular, $K$ has inverse function
$K^{-1}:[0,\infty)\rightarrow[1,\infty)$ with $K^{-1}(0)=1$ which is also
strictly increasing. Furthermore, $K$ itself is concave as $G$ is
nondecreasing. Define $r:[0,\infty)\rightarrow(0,\infty)$ by

(2.15) $\displaystyle r(t):=(K^{-1})^{\prime}(t)=G(K^{-1}(t)).$

It is worth noting that the properties of $r$ and $K^{-1}$ imply that
$K^{-1}(n)$, $n\in\mathbf{Z}_{\geq 1}$, is comparable to $\sum_{k=1}^{n}r(k)$.
In particular, it holds that

(2.16) $\displaystyle K^{-1}(n)-1=\int_{0}^{n}r(s)\,ds\leq\sum_{k=1}^{n}r(k).$

The following result also holds:

######  Theorem 2.17.

Let $R,f\geq 0$ and suppose that the estimate (2.10) holds for some measurable
$H:\mathbf{X}\rightarrow[1,\infty)$ and some concave nondecreasing
differentiable function $G:[1,\infty)\rightarrow(0,\infty)$. Then for all
$x\notin H_{\leq R}$

(2.18)
$\displaystyle\mathbf{E}_{x}\bigg{[}\sum_{k=0}^{T_{R}-1}r(k)\bigg{]}\leq H(x)$

and for all $x\notin H_{\leq R}$, $n\geq 0$

(2.19) $\displaystyle\mathbf{P}_{x}(T_{R}\geq
n+1)\leq\frac{H(x)+1}{K^{-1}(n)}.$

The bound (2.18) follows directly from [11, 22] while relation (2.19) follows
from (2.18) using (2.16). Furthermore, if in addition to the conclusions of
Theorem 2.17 one has a minorization condition of the form

$\displaystyle\inf_{x\in H_{\leq R^{\prime}}}\mathcal{P}(x,\;\cdot\;)\geq
c\lambda(\;\cdot\;)$

for some constant $c>0$, $R^{\prime}>2R$ and $\mathcal{B}$-probability measure
$\lambda$, then one can prove that (see [11, 22])
$\mathcal{P}_{n}(x,\;\cdot\;)$ converges in total variation to the unique
stationary distribution of the chain as $n\rightarrow\infty$ at a rate
proportional to $1/r(n)$. In Example 2.20 and Example 2.21 below, this will in
turn correspond to exponential, polynomial and stretched exponential rates of
convergence to equilibrium, depending on the structure of $G$. This fits with
the intuition that one mechanism slowing convergence to a subexponential rate
is slow return times to the minorizing set $H_{\leq R}$. See also [14, 16] for
subgeometric rates of convergence for Markov processes in the continuous-time
setting.

######  Example 2.20.

Suppose that $G(t)=\alpha t^{a}$ for some $\alpha>0$ and $a\in(0,1]$. Thus $G$
is strictly increasing and concave on $[1,\infty)$ and $K$, $K^{-1}$ are given
by

$\displaystyle K(t)=\begin{cases}\frac{1}{\alpha}\log(t)&\text{ if }a=1\\\
\frac{t^{1-a}-1}{\alpha(1-a)}&\text{ if }a\in(0,1)\end{cases}\quad\text{ and
}\quad K^{-1}(t)=\begin{cases}e^{\alpha t}&\text{ if }a=1\\\
(\alpha(1-a)t+1)^{1/(1-a)}&\text{ if }a\in(0,1)\end{cases}.$

Furthermore,

$\displaystyle r(t)=\begin{cases}\alpha e^{\alpha t}&\text{ if }\alpha=1\\\
\alpha(\alpha(1-a)t+1)^{\frac{a}{1-a}}&\text{ if }a\in(0,1)\end{cases}.$

In particular, in the case when $a=1$, the bound (2.18) recovers (2.13).

The next example will be particularly important in this paper.

######  Example 2.21.

Let $\alpha>0$ and $G_{0}:(1,\infty)\rightarrow(0,\infty)$ be given by

$\displaystyle G_{0}(t)=\frac{\alpha t}{\log(t)}$

and note that $G_{0}$ is nondecreasing and concave for $t\geq e^{2}$. For
$t\geq 1$, we define $G(t)=G_{0}(t+e^{2})$ and observe that

$\displaystyle
K(t)=\int_{1}^{t}\frac{1}{G(s)}\,ds=\int_{1+e^{2}}^{t+e^{2}}\frac{1}{G_{0}(s)}\,ds=\frac{1}{2\alpha}\bigg{[}(\log(t+e^{2}))^{2}-(\log(1+e^{2}))^{2}\bigg{]},$

and it follows that

$\displaystyle K^{-1}(t)$ $\displaystyle=\exp\big{(}(2\alpha
t+[\log(1+e^{2})]^{2})^{\frac{1}{2}}\big{)}-e^{2},$ $\displaystyle r(t)$
$\displaystyle=\alpha\frac{\exp\big{(}(2\alpha
t+[\log(1+e^{2})]^{2})^{\frac{1}{2}}\big{)}}{(2\alpha
t+[\log(1+e^{2})]^{2})^{\frac{1}{2}}}.$

##  3\. Random Splittings

We now specialize to the setting of a discrete-time Markov chain $(X_{n})$
with state space $(\mathbf{X},\rho)=(\mathbf{R}^{d},|\cdot|)$ where $|\cdot|$
is the usual Euclidean distance. In order to describe the specific Markov
chains to be studied in the rest of the paper, we first fix some notation
convenient to this setting.

###  3.1. Notation

For $x,y\in\mathbf{R}^{d}$, we use $\langle x,y\rangle$ to denote the standard
inner product between $x$ and $y$ so that $|x|=\sqrt{\langle x,x\rangle}$. The
symbol $\mathscr{V}_{d}$ denotes the set of complete, $C^{\infty}$ vector
fields on $\mathbf{R}^{d}$. For any $V\in\mathscr{V}_{d}$, we let
$(t,x)\mapsto\varphi_{t}^{V}(x):\mathbf{R}\times\mathbf{R}^{d}\rightarrow\mathbf{R}^{d}$
denote the flow map associated to the differential equation

(3.1) $\displaystyle\dot{x}_{t}=V(x_{t}).$

That is, for $(t,x)\in\mathbf{R}\times\mathbf{R}^{d}$, $\varphi_{t}^{V}(x)$
denotes the solution of (3.1) at time $t$ with initial condition
$x_{0}=x\in\mathbf{R}^{d}$. For any $V\in\mathscr{V}_{d}$, we say that a
finite collection
$\mathscr{S}=\\{V_{1},V_{2},\ldots,V_{m}\\}\subset\mathscr{V}_{d}$ is a
_splitting_ of $V$ provided

$\displaystyle\sum_{i=1}^{m}V_{i}=V.$

We will use $S_{m}$ to denote the set of permutations of $\\{1,2,\ldots,m\\}$.
Given a splitting $\mathscr{S}=\\{V_{1},\ldots,V_{m}\\}\subset\mathscr{V}_{d}$
of $V\in\mathscr{V}_{d}$, we will often use $\varphi_{t}^{i}(x)$,
$x\in\mathbf{R}^{d}$, to denote $\varphi_{t}^{V_{i}}(x)$. In this setting, for
any $\sigma=(\sigma_{1},\ldots,\sigma_{m})\in S_{m}$, $x\in\mathbf{R}^{d}$ and
$t=(t_{1},\ldots,t_{m})\in[0,\infty)^{m}$ we let

(3.2)
$\displaystyle\Phi^{\sigma}_{t}(x):=\varphi_{t_{m}}^{\sigma_{m}}\circ\cdots\circ\varphi_{t_{1}}^{\sigma_{1}}(x)$

where $\circ$ denotes composition. In words, $\Phi^{\sigma}_{t}(x)$ is the
result of flowing along the vector fields in the splitting $\mathscr{S}$ in
the order prescribed by the permutation $\sigma$ starting from
$x\in\mathbf{R}^{d}$, with corresponding times spent along each trajectory
given by $t=(t_{1},t_{2},\ldots,t_{m})$. For $m\in\mathbf{N}$ as in
$\mathscr{S}=\\{V_{1},V_{2},\ldots,V_{m}\\}$, we let

(3.3) $\displaystyle
u_{n}=(u_{1n},u_{2n},\ldots,u_{mn}),\,\,\,n\in\mathbf{Z}_{\geq 0},$

denote a countably infinite collection of independent, uniformly distributed
random variables on $S_{m}$. Also, let

(3.4)
$\displaystyle\tau_{n}=(\tau_{1n},\tau_{2n},\ldots,\tau_{mn}),\,\,\,n\in\mathbf{Z}_{\geq
0},$

be an independent collection of random vectors such that for each $i$ and $j$,
$\tau_{ij}\sim\exp(1/h)$. We furthermore suppose that the collection
$\\{\tau_{n}\,:\,n\in\mathbf{Z}_{\geq 0}\\}$ is independent of the collection
$\\{u_{n}\,:\,n\in\mathbf{Z}_{\geq 0}\\}$.

###  3.2. Random Splittings

We can construct a discrete-time Markov chain $(X_{n})$ on with state space
$\mathbf{R}^{d}$ associated to the splitting $\mathscr{S}$ inductively by

(3.5) $\displaystyle X_{n+1}=\Phi^{u_{n}}_{\tau_{n}}\circ
X_{n}=\varphi^{u_{mn}}_{\tau_{mn}}\circ\cdots\circ\varphi^{u_{1n}}_{\tau_{1n}}\circ
X_{n}.$

That is, starting from state $X_{n}$, the state $X_{n+1}$ is determined by
selecting a uniformly distributed ordering of the vector fields in
$\mathscr{S}$ and then flowing along each of these in the selected order for
an independent, $\exp(1/h)$ amount of time. We call the Markov chain $(X_{n})$
the _random splitting_ associated to $\mathscr{S}$. For notational
convenience, we will also make use of the random variables $X_{n,k}$, $n\geq
0$, $k=0,1,2,\ldots,m$, defined by

$\displaystyle X_{n,k}=\begin{cases}X_{n}&\text{ if }k=0,n\geq 0\\\
\varphi^{u_{kn}}_{\tau_{kn}}\circ\cdots\circ\varphi^{u_{1n}}_{\tau_{1n}}\circ
X_{n},&\text{ if }k=1,2,\ldots,m,\,\,n\geq 0.\end{cases}$

Note that the stochastic sequence $(X_{n,k})_{k=0}^{m}$ keeps track of the
dynamics between steps $n$ and $n+1$ of the random splitting. The sequence
$(X_{n,k})$ will be referred to as the _random dynamics_ associated to the
splitting $\mathscr{S}$. Dynamics generated by a random splitting can be
understood as a _random dynamical system_ [4] or a collection of random maps
[19]. We use the term “random splitting” to emphasize the origin of the random
maps and because we will vary the time step parameter $h$. The concept of a
random splitting is also closely related to piecewise deterministic Markov
processes (PDMPs) [5, 7, 13].

####  3.2.1. Random splitting of Lorenz ’96

As a non-trivial, illustrative example in this section, we consider a
splitting of the forced and partially damped Lorenz ’96 equation on
$\mathbf{R}^{d}$

(3.6)
$\displaystyle\dot{x}_{i}=(x_{i+1}-x_{i-2})x_{i-1}-\delta_{1i}x_{i}+\beta_{i}$

where $i=1,\ldots,d$, $d\geq 4$. The indices in equation (3.6) live on the
discrete circle so that, in particular, $x_{d+1}\equiv x_{1}$, $x_{0}\equiv
x_{d}$, and $x_{-1}\equiv x_{d-1}$. We assume that the system (3.6) is forced
externally by real-valued constants $\beta_{i}\neq 0$, $i=1,\ldots,d$, and
that dissipation enters through the first mode only; that is, $\delta_{1i}=1$
if $i=1\,(\text{mod}\,d)$ and $\delta_{1i}=0$ otherwise.

To define the splitting, let $\\{e_{i}\\}_{i=1}^{d}$ denote the standard
orthornormal basis of $\mathbf{R}^{d}$ and introduce the following vector
fields on $\mathbf{R}^{d}$

(3.7) $\displaystyle
V_{i}(x)=x_{i+1}x_{i-1}e_{i}-x_{i}x_{i-1}e_{i+1},\,\,i=1,\ldots,d,\qquad
V_{\star}(x)=-x_{1}e_{1}+\sum_{i=1}^{d}\beta_{i}e_{i},$ (3.8)
$\displaystyle\qquad\qquad\qquad
V(x)=\sum_{i=1}^{d}[(x_{i+1}-x_{i-2})x_{i-1}-\delta_{1i}x_{i}+\beta_{i}]e_{i}.$

One can show that each of the vector fields in (3.7)-(3.8) belongs to
$\mathscr{V}_{d}$ since the flows along each $V_{i}$, $i=1,2,\ldots,d$,
conserve the Euclidean length $|\cdot|$. Furthermore,
$\mathscr{S}:=\\{V_{1},\ldots,V_{d},V_{\star}\\}$ forms a splitting of
$V\in\mathscr{V}_{d}$ since

(3.9) $\displaystyle V(x)=V_{\star}(x)+\sum_{i=1}^{d}V_{i}(x).$

Let $(X_{n})$ denote the random splitting associated to $\mathscr{S}$ and
$(\mathcal{P}_{n})$ be the associated Markov semigroup. If
$H:\mathbf{R}^{d}\rightarrow[1,\infty)$ is given by

(3.10) $\displaystyle H(x)=|x|+1,$

one of our main goals below will be to prove the following result.

######  Theorem 3.11.

For every $h>0$ small enough but independent of the dimension $d$, there exist
constants $\alpha=\alpha(d,h)\in(0,1)$ and $\beta=\beta(d,h)>0$ such that the
bound is satisfied

(3.12) $\displaystyle\mathcal{P}_{d+1}H(x)\leq\alpha H(x)+\beta$

for all $x\in\mathbf{R}^{d}$.

The proof of Theorem 3.11 will be established partly here in Section 3 and
partly in Section 4 by validating the hypotheses of Proposition 2.3 with a
constant function $p\equiv p_{*}>0$. Surprisingly, even though damping is
externally present only on the first mode, it propagates through the random
dynamics to produce the globally contractive-type estimate (3.12) for the
random splitting.

###  3.3. Section overview

Returning to the setting of a general random splitting, in Section 3.4,
Section 3.5 and Section 3.6 we describe the important structural features that
produce the bounds required by Proposition 2.3, and in particular those that
lead to estimates like (3.12). Namely:

  * •

A quantity approximately conserved under each vector field in the splitting,
i.e. the function $H$ in this discussion. This will be further elaborated in
Section 3.4 using the notion of _subconservation_. Importantly,
subconservation produces the bound (2.5) for some constant $f_{2}\geq 0$, for
any event $A\in\mathcal{F}$.

  * •

Dissipation present in at least one vector field in the splitting
$\mathscr{S}$. This will be discussed in Section 3.5 below. This structure
allows one to establish a bound of the form (2.4) for a collection of events
$A=A(x)$. These events correspond to certain entrances of the random dynamics
to a “dissipative region” $D$ in space.

  * •

Such entrances to the dissipative region $D$ must have sufficient probability
so that the estimate $\mathbf{P}_{x}(A)\geq p(H(x))$ in Proposition 2.3 is
satisfied for all $x\in\mathbf{R}^{d}$ for some “reasonable” function
$p:[1,\infty)\rightarrow(0,1]$. In the case of the splitting for the Lorenz
’96 equation above, we need to be able to take $p$ in Proposition 2.3 to be a
constant function, i.e. $p\equiv p_{*}>0$. This point will be discussed in
more detail in Section 3.6.

Unless otherwise specified, throughout the remainder of this section we let
$\tau\sim\exp(1/h)$,
$\mathscr{S}=\\{V_{1},V_{2},\ldots,V_{m}\\}\subset\mathscr{V}_{d}$ be a
splitting of $V\in\mathscr{V}_{d}$, and let
$H:\mathbf{R}^{d}\rightarrow[1,\infty)$ be measureable.

###  3.4. Subconservation

Perhaps the most basic structural feature of the splittings considered in this
paper is that each vector field in $\mathscr{S}$ “approximately conserves” the
function $H$. To be more precise:

######  Definition 3.13.

We call a splitting $\mathscr{S}$ _subconservative_ with respect to $H$ if for
every $i=1,2,\ldots,m$ there exists measurable
$F_{i}:[0,\infty)\rightarrow[0,\infty)$ such that
$f_{i}:=\mathbf{E}F_{i}(\tau)<\infty$ and

(3.14) $\displaystyle H(\varphi_{t}^{i}(x))\leq H(x)+F_{i}(t)$

for all $x\in\mathbf{R}^{d}$, $t\geq 0$.

######  Remark 3.15.

Observe that if a vector field $W\in\mathscr{V}_{d}$ conserves $H$, then
$H(\varphi_{t}^{W}(x))=H(x)$ for all $x\in\mathbf{R}^{d}$, $t\geq 0$. Thus
subconservation is a slight deviation from this concept where the function $H$
along all of the vector fields in the splitting does not grow “too fast”.

######  Example 3.16.

Consider the splitting $\mathscr{S}=\\{V_{1},\ldots,V_{d},V_{\star}\\}$ of the
Lorenz ’96 equation (3.6) introduced in (3.7), and let $H$ be as in (3.10). We
claim that $\mathscr{S}$ is subconservative with respect to $H$. To see why,
note first that for $i=1,2,\ldots,d$ and $t\geq 0$

$\displaystyle H(\varphi^{i}_{t}(x))=H(x)$

since each of the vector fields $V_{1},V_{2},\ldots,V_{d}$ conserves length
$|x|$. In particular, we can take $F_{i}\equiv 0$ in (3.14). On the other
hand, letting $\varphi_{t}^{\star}(x):=\varphi_{t}^{V_{\star}}(x)$ and using
the triangle inequality we have

$\displaystyle H(\varphi_{t}^{\star}(x))$
$\displaystyle=|(e^{-t}x_{1}+(1-e^{-t})\beta_{1},x_{2}+\beta_{2}t,\ldots,x_{d}+\beta_{d}t)|+1$
$\displaystyle\leq H(x)+|(\beta_{1},\beta_{2},\ldots,\beta_{d})|t,\quad t\geq
0.$

Hence we can take $F_{\star}(t)=|(\beta_{1},\ldots,\beta_{d})|t$ as in (3.14)
and the claim is established.

As a basic consequence of subconservation, we have the following result. See
also (2.5).

######  Proposition 3.17.

Suppose that $H$ is subconservative for $\mathscr{S}$ and let $(X_{n})$ denote
the random splitting associated to $\mathscr{S}$. Then

(3.18)
$\displaystyle\mathbf{E}_{x}\Big{[}\mathbf{1}_{A}H(X_{n})\Big{]}\leq\mathbf{P}_{x}(A)H(x)+n\sum_{i=1}^{m}f_{i}$

for all $x\in\mathbf{R}^{d}$, $n\geq 0$ and $A\in\mathcal{F}$. In the above,
$f_{i}=\mathbf{E}F_{i}(\tau)\geq 0$ is as in (3.14).

###### Proof.

The bound (3.18) is trivially satisfied when $n=0$. Thus suppose $n\geq 1$.
Since $\mathscr{S}$ is subconservative with respect to $H$, let $F_{i}$ and
$f_{i}$ be as in (3.14) and observe that for any $i\in\\{1,2,\ldots,m-1\\}$ we
have by definition of the random splitting and subconservation

(3.19) $\displaystyle H(X_{n})=H(X_{n-1,m})$ $\displaystyle\leq
H(X_{n-1,m-i})+\textstyle{\sum_{k=m-i+1}^{m}}F_{u_{k(n-1)}}(\tau_{k(n-1)})$
$\displaystyle\leq
H(X_{n-1})+\textstyle{\sum}_{k=1}^{m}F_{u_{k(n-1)}}(\tau_{k(n-1)}).$

Hence using independence it follows for any $A\in\mathcal{F}$

(3.20)
$\displaystyle\mathbf{E}_{x}[\mathbf{1}_{A}H(X_{n})]\leq\mathbf{E}_{x}[\mathbf{1}_{A}H(X_{n-1})]+\textstyle{\sum}_{i=1}^{m}f_{i}.$

The bound (3.18) follows by iterating (3.20). ∎

###  3.5. Partial dissipation

We next explore the basic structural feature of the random splittings in this
paper which give rise to estimates of the form (2.4). Recall that
$\mathcal{B}$ denotes the Borel measurable subsets of the state space
$\mathbf{X}$, which in this section is $\mathbf{R}^{d}$.

######  Definition 3.21.

Let $D\in\mathcal{B}$ be non-empty. We call the splitting $\mathscr{S}$
_partially dissipative for_ $H$ _on the set_ $D$ _with index_
$\ell\in\\{1,2,\ldots,m\\}$ if there exists a measurable function
$G_{\ell}:[0,\infty)\rightarrow[0,\infty)$ with
$g_{\ell}:=\mathbf{E}G_{\ell}(\tau)<\infty$ and a strictly decreasing function
$\alpha_{\ell}:[0,\infty)\rightarrow(0,1]$ for which the bound holds

(3.22) $\displaystyle
H(\varphi^{\ell}_{t}(x))\leq\alpha_{\ell}(t)H(x)+G_{\ell}(t),\,\,\,x\in
D,\,\,t\geq 0.$

Notationally, we set $a_{\ell}=\mathbf{E}\alpha_{\ell}(\tau)\in(0,1)$. The set
$D$ will be called a _dissipative region_ for the random splitting $(X_{n})$.
The index $\ell$ will be called a _dissipative direction_.

######  Example 3.23.

Consider again the splitting $\mathscr{S}=\\{V_{1},\ldots,V_{d},V_{\star}\\}$
of the Lorenz ’96 equation (3.6) introduced in (3.7), and let $H$ be as in
(3.10). Fixing $\eta\in(0,1)$, we claim that $\mathscr{S}$ is partially
dissipative with respect to $H$ on the set

(3.24) $\displaystyle
D_{\eta}=\\{x\in\mathbf{R}^{n}\,:\,|x_{1}|^{2}\geq\eta|x|^{2}\\}$

with index $\ell=\star$. Indeed, if
$F_{\star}(t)=|(\beta_{1},\ldots,\beta_{d})|t$, then for $t\geq 0$ we have

$\displaystyle H(\varphi_{t}^{\star}(x))$
$\displaystyle=|(e^{-t}x_{1}+(1-e^{-t})\beta_{1},x_{2}+\beta_{2}t,\ldots,x_{d}+\beta_{d}t)|+1$
$\displaystyle\leq H(e^{-t}x_{1},x_{2},\ldots,x_{d})+F_{\star}(t)$
$\displaystyle=\sqrt{(e^{-2t}-1)x_{1}^{2}+|x|^{2}}+1+F_{\star}(t)$
$\displaystyle\leq\sqrt{1-\eta^{2}(1-e^{-2t})}|x|+1+F_{\star}(t)\leq\alpha_{\star}(t)H(x)+F_{\star}(t)+1$

where $\alpha_{\star}(t)=\sqrt{1-\eta^{2}(1-e^{-2t})}$ is strictly decreasing
on $[0,\infty)$ and maps $[0,\infty)$ into $(0,1]$. This finishes the proof of
the claim.

Our next result relates partial dissipation and subconservation to an estimate
of the form (2.4). To state it, recalling that $(X_{n,k})$ denotes the random
dynamics associated to the splitting $\mathscr{S}$, for any $D\in\mathcal{B}$,
$\ell\in\\{1,2,\ldots,m\\}$ and $n\geq 0$, let

$\displaystyle T_{n,D}=\min\big{\\{}k\in\\{0,1,\ldots,m-1\\}\,:\,X_{n,k}\in
D\big{\\}}$

and

(3.25) $\displaystyle A_{n}^{\ell}(D)$
$\displaystyle=\bigcup_{k=0}^{m-1}\Big{\\{}T_{n,D}=k\Big{\\}}\cap\Big{\\{}\ell=u_{(k+1)n}\Big{\\}}=:\bigcup_{k=0}^{m-1}A_{n,k}^{\ell}(D).$

Note that $A_{n,k}^{\ell}(D)$ is the event that the random sequence
$(X_{n,j})_{j=0}^{m-1}$ first enters the set $D$ after the $k$th flow and the
next flow, namely the $(k+1)$st flow, follows the vector field $V_{\ell}$.

######  Proposition 3.26.

Let $D\in\mathcal{B}$ be nonempty and suppose that the splitting $\mathscr{S}$
is subconservative for $H$ and partially dissipative for $H$ on the set $D$
with index $\ell$, and let $a_{\ell}\in(0,1)$ be as in Definition 3.21. Then
there exists constants $f_{n,k}\geq 0$ such that

(3.27)
$\displaystyle\mathbf{E}_{x}\big{[}\mathbf{1}_{A_{n}^{\ell}(D)}H(X_{k})\big{]}\leq
a_{\ell}\mathbf{P}_{x}(A_{n}^{\ell}(D))H(x)+f_{n,k}$

for all $x\in\mathbf{R}^{d}$, $k\geq n+1$.

###### Proof.

Let $D\in\mathcal{B}$ be nonempty and suppose that the bound (3.22) holds for
some index $\ell\in\\{1,2,\ldots,m\\}$, measurable
$G_{\ell}:[0,\infty)\rightarrow[0,\infty)$ with
$g_{\ell}=\mathbf{E}G_{\ell}(\tau)<\infty$ and strictly decreasing function
$\alpha_{\ell}:[0,\infty)\rightarrow(0,1]$ with
$a_{\ell}=\mathbf{E}\alpha_{\ell}(\tau)$. Since $\mathscr{S}$ is
subconservative, suppose the bounds (3.14) hold for some
$F_{i}:[0,\infty)\rightarrow[0,\infty)$ measurable with
$f_{i}=\mathbf{E}F_{i}(\tau)<\infty$. For simplicity in the arguments below,
set $A=A_{n}^{\ell}(D)$ and $A_{j}=A_{n,j}^{\ell}(D)$.

Since $k\geq n+1$ and the union (3.25) is disjoint, relation (3.20) gives

$\displaystyle\mathbf{E}_{x}\mathbf{1}_{A}H(X_{k})$
$\displaystyle\leq\mathbf{E}_{x}\mathbf{1}_{A}H(X_{n+1})+(k-n-1)\textstyle{\sum_{i=1}^{m}f_{i}}$
$\displaystyle=\textstyle{\sum_{j=0}^{m-1}}\mathbf{E}_{x}\mathbf{1}_{A_{j}}H(X_{n+1})+(k-n-1)\textstyle{\sum_{i=1}^{m}f_{i}}$
(3.28)
$\displaystyle\leq\textstyle{\sum_{j=0}^{m-1}}\mathbf{E}_{x}\mathbf{1}_{A_{j}}H(X_{n,j+1})+(k-n)\textstyle{\sum_{i=1}^{m}f_{i}}.$

Next, observe that

$\displaystyle\mathbf{1}_{A_{j}}H(X_{n,j+1})\leq\mathbf{1}_{A_{j}}\alpha_{\ell}(\tau_{(j+1)n})H(X_{n,j})+\mathbf{1}_{A_{j}}G_{\ell}(\tau_{(j+1)n}).$

Using independence of $\tau_{(j+1)n}$ and $\mathbf{1}_{A_{j}}H(X_{n,j})$, and
independence of $\tau_{(j+1)n}$ and $\mathbf{1}_{A_{j}}$, we thus obtain

$\displaystyle\mathbf{E}_{x}\mathbf{1}_{A_{j}}H(X_{n,j+1})\leq
a_{\ell}\mathbf{E}_{x}[\mathbf{1}_{A_{j}}H(X_{n,j})]+g_{\ell}\mathbf{P}(A_{j}).$

Hence, this estimate with subconservation implies

$\displaystyle\mathbf{E}_{x}\mathbf{1}_{A_{j}}H(X_{n,j+1})$ $\displaystyle\leq
a_{\ell}\mathbf{E}_{x}[\mathbf{1}_{A_{j}}H(X_{n,j})]+g_{\ell}\mathbf{P}(A_{j})$
(3.29) $\displaystyle\leq
a_{\ell}\mathbf{P}_{x}(A_{j})H(x)+(n+1)\textstyle{\sum_{j=1}^{m}}f_{i}+g_{\ell}\mathbf{P}(A_{j}).$

Combining (3.28) with (3.29), we obtain the claimed estimate (3.27) with
$f_{n,k}$ given by

(3.30) $\displaystyle
f_{n,k}=g_{\ell}+((m-1)n+m+k)\textstyle{\sum_{i=1}^{m}f_{i}}.$

∎

###  3.6. Entrances to a dissipative region

As we have seen in the case of the random splitting of the Lorenz ’96 equation
(3.6), checking subconservation and partial dissipation for the random
splitting is relatively straightforward. See Example 3.16 and Example 3.23 for
further details of this point. By way of Proposition 2.3, the central
difficulty in showing a bound like (2.6) is estimating
$\mathbf{P}_{x}(A_{n}^{\ell}(D))$ for a dissipative set $D$ and a dissipative
direction $\ell\in\\{1,2,\ldots,m\\}$. Such an estimate comes from the
specific nature of the dynamics of the splitting, as the system transports an
initial condition $x\in\mathbf{R}^{d}$ to $D$ in a certain number of steps
$n$. The way in which this transport happens inherently depends on $x$. In
this section, we provide the condition (Theorem 3.32 below) we will check in
subsequent sections to establish the needed lower bound on this probability.

To state the result, define for $R>0$

(3.31) $\displaystyle H_{>R}=\\{x\in\mathbf{R}^{d}\,:\,H(x)>R\\}.$

######  Theorem 3.32.

Let $D\in\mathcal{B}$ be nonempty and suppose that the splitting $\mathscr{S}$
is subconservative with respect to $H$ and partially dissipative with respect
to $H$ on the set $D$ with index $\ell$. Suppose there exist
$n\in\mathbf{Z}_{\geq 1}$, $R>0$ and $p:[1,\infty)\rightarrow(0,1]$ such that

(3.33) $\displaystyle\mathbf{P}_{x}(A_{j(x)}^{\ell}(D))\geq
p(H(x))\,\,\,\text{ for all }\,\,\,x\in H_{>R},$

for some index $j(x)\leq n-1$. Then the Markov chain
$(\tilde{X}_{j}):=(X_{jn})$ satisfies the hypotheses of Proposition 2.3 with
the same choice of $p$.

######  Remark 3.34.

We emphasize that the index $j(x)$ in (3.33) depends on the initial state $x$
but has the uniform upper bound $j(x)\leq n-1$ over all $x\in H_{>R}$.

###### Proof of Theorem 3.32.

Since $\mathscr{S}$ is subconservative with respect to $H$, pick $F_{i}$ and
$f_{i}=\mathbf{E}F_{i}(\tau)<\infty$ so that the bounds (3.14) hold for
$i=1,2,\ldots,m$. Since $\mathscr{S}$ is partially dissipative with respect to
$H$ on $D$ with index $\ell$, suppose that the bound (3.22) is satisfied for
some measurable $G_{\ell}:[0,\infty)\rightarrow[0,\infty)$ and
$\alpha_{\ell}:[0,\infty)\rightarrow(0,1]$ with $\alpha_{\ell}$ strictly
decreasing and $g_{\ell}=\mathbf{E}G_{\ell}(\tau)<\infty$ and
$a_{\ell}=\mathbf{E}\alpha_{\ell}(\tau)\in(0,1)$. By hypothesis, pick
$n\in\mathbf{Z}_{\geq 1}$ and $R>0$ such that for every $x\in H_{>R}$ we have
(3.33). We claim that the Markov chain $(\tilde{X}_{j}):=(X_{jn})$ satisfies
the hypothesis of Proposition 2.3.

For $x\in H_{>R}$, define the event $A=A(x)$ in the statement of Proposition
2.3 by $A(x)=A_{j(x)}^{\ell}(D)$. Then

$\displaystyle\mathbf{P}_{x}(A)=\mathbf{P}(A(x))\geq p(H(x))\,\,\text{ for all
}x\in H_{>R}.$

On the other hand, for $x\in H_{\leq R}$ we define $A=A(x)=\Omega$, so that

$\mathbf{P}_{x}(A)=\mathbf{P}(A(x))=1\geq p(H(x))\,\,\text{ for all }\,\,x\in
H_{\leq R}.$

Next, by Proposition 3.26, for any $x\in H_{>R}$ we have

$\displaystyle\mathbf{E}_{x}\mathbf{1}_{A}H(\tilde{X}_{1})=\mathbf{E}_{x}\mathbf{1}_{A}H(X_{n})$
$\displaystyle\leq a_{\ell}\mathbf{P}_{x}(A)H(x)+c_{n}$

for some constant $c_{n}\geq 0$. By Proposition 3.17, we have for any
$B\in\mathcal{F}$ and $x\in\mathbf{R}^{d}$

$\displaystyle\mathbf{E}_{x}\mathbf{1}_{B}H(\tilde{X}_{1})=\mathbf{E}_{x}\mathbf{1}_{B}H(X_{n})$
$\displaystyle\leq\mathbf{P}_{x}(B)H(x)+n\textstyle{\sum_{i=1}^{m}f_{i}}.$

In particular, for any $x\in H_{\leq R}$ we also have the estimate

$\displaystyle\mathbf{E}_{x}\mathbf{1}_{A}H(\tilde{X}_{1})\leq
H(x)+n\textstyle{\sum_{i=1}^{m}f_{i}}\leq
a_{\ell}\mathbf{P}_{x}(A)H(x)+(1-a_{\ell})R+n\textstyle{\sum_{i=1}^{m}f_{i}}.$

∎

##  4\. Entrance probabilities to the dissipative region: the Lorenz ’96
Splitting

Recall the random splitting $(X_{n})$ of the Lorenz ’96 equation (3.6)
introduced in Section 3.2.1 and discussed in Example 3.16 and Example 3.23. In
this section we finish proving Theorem 3.11 by way of Theorem 3.32. In
particular, we have left to show the required lower bound on the entrance
probabilities as in (3.33).

To setup the precise statement of the result to be proven in this section,
pick $h_{*}\in(0,\pi/12)$ small enough so that for $h\leq h_{*}$ each of the
following conditions are met:

(4.1) $\displaystyle|\sin(|y|)|\leq h\,\,\text{ implies
}\,\,||y|-j\pi|<3h\,\,\text{ for some }\,\,j\in\mathbf{Z}_{\geq 0},$
$\displaystyle\qquad e^{-6h}\geq\tfrac{3}{4},\,\,\,\text{ and
}\,\,\,1-e^{-5h}-3he^{-5h}\geq h.$

Throughout this section, we fix $h\in(0,h_{*}]$ and recall that $h$
corresponds to the exponential clocks $\tau_{ij}\sim\exp(1/h)$. Also recall
the set $D_{\eta}$ introduced in (3.24). Using Example 3.16 and Example 3.23,
to show Theorem 3.11 it suffices to show the following result by Theorem 3.32.

######  Theorem 4.2.

There exist $\eta\in(0,1)$, $R>0$ and a constant $p_{*}>0$ such that for every
$x\in H_{>R}$

(4.3) $\displaystyle\mathbf{P}_{x}(A_{j(x)}^{\ell}(D_{\eta}))\geq p_{*}$

for some index $j(x)\leq d$.

To prove Theorem 4.2, we first deduce some key lemmata below in Section 4.1.
These results explain how “energy” is transferred and/or maintained along each
of the vector fields in the splitting $\mathscr{S}$ under certain assumptions
on the initial data. Afterwards, we will use the lemmata to conclude Theorem
4.2.

###  4.1. The key lemmata

In what follows, we let $\pi_{j}:\mathbf{R}^{d}\rightarrow\mathbf{R}$ denote
the projection onto the $j$th coordinate. The first lemma describes how energy
is transferred downward from mode $j$ to $j-1$ along $V_{j-1}$ as in (3.7). An
analogous result holds going upward from $j-1$ to $j$, but we provide only the
result below for expeditiousness.

Because of the cyclical nature of the vector fields
$V_{1},V_{2},\ldots,V_{d}$, we adopt the convention that their indices are
defined modulo $d$, e.g. $V_{d+1}=V_{1}$, with the appropriate shifts in the
definition of the vector field. Recall also the notation

$\displaystyle\varphi^{\star}_{t}(x)=\varphi_{t}^{V_{\star}}(x),\,\,t\geq
0,\,x\in\mathbf{R}^{d}.$

######  Lemma 4.4 (Downward energy transfer).

Let $x\in\mathbf{R}^{d}_{\neq 0}$, and suppose that for some $j=1,\ldots,d$
_(mod $d$)_, there exists constants $c_{1}>c_{2}>0$ such that $|x_{j}|\geq
c_{1}|x|$ and $|x_{j-1}|\leq c_{2}h|x|$. If $|x_{j-2}|\geq c_{3}>0$ for some
constant $c_{3}>0$, then with probability at least $p_{1}=p_{1}(c_{3})>0$
depending only on $c_{3}$ we have

$\displaystyle|\pi_{j-1}(\varphi_{\tau}^{j-1}(x))|\geq(c_{1}-c_{2})h|x|=(c_{1}-c_{2})h|\varphi_{\tau}^{j-1}(x)|.$

###### Proof.

Note that

(4.5) $\displaystyle
z_{1}:=\pi_{j-1}(\varphi_{\tau}^{j-1}(x))=x_{j-1}\cos(|x_{j-2}|\tau)+\text{sgn}(x_{j-2})x_{j}\sin(|x_{j-2}|\tau).$

Observe that on the event $B_{1}=\\{|z_{1}|\leq(c_{1}-c_{2})h|x|\\}$ we have
under our assumptions on the initial data that

(4.6) $\displaystyle|\sin(|x_{j-2}|\tau)|\leq h.$

By the choice of $h_{*}>0$ in (4.1), on the event $B_{1}=\\{|z_{1}|\leq h\\}$
for $h\leq h_{*}$ the event

(4.7)
$\displaystyle\tilde{B}_{1}(x_{j-2}):=\bigcup_{\ell=0}^{\infty}\big{\\{}||x_{j-2}|\tau-\ell\pi|\leq
3h\big{\\}}$

must occur. Furthermore, by the choice of $h_{*}$, the union above is
disjoint. We will estimate
$\mathbf{P}(\tilde{B}_{1}(x_{j-1})^{c})\leq\mathbf{P}(B_{1}^{c})$. To this
end, setting $a=|x_{j-2}|\geq c_{3}>0$ for simplicity we obtain

$\displaystyle\mathbf{P}(\tilde{B}_{1}(|x_{j-2}|)^{c})=\sum_{\ell=0}^{\infty}\mathbf{P}\big{(}a\tau\in(3h+\pi\ell,-3h+\pi(\ell+1))\big{)}$
$\displaystyle=\sum_{\ell=0}^{\infty}e^{-\frac{\pi\ell}{ha}}\bigg{\\{}e^{-\frac{3}{a}}-e^{\frac{3}{a}-\frac{\pi}{ha}}\bigg{\\}}$
$\displaystyle=\frac{e^{-\frac{3}{a}}-e^{\frac{3}{a}-\frac{\pi}{ha}}}{1-e^{-\frac{\pi}{ha}}}.$

Since $h_{*}<\pi/12$ we note that $\pi/h-6\geq\pi/(2h)$. Hence

$\displaystyle\mathbf{P}(\tilde{B}_{1}(x_{j-1})^{c})=\frac{e^{-\frac{3}{a}}-e^{\frac{3}{a}-\frac{\pi}{ha}}}{1-e^{-\frac{\pi}{ha}}}=e^{-\frac{3}{a}}\frac{1-e^{-(\frac{\pi}{ha}-\frac{6}{a})}}{1-e^{-\frac{\pi}{ha}}}$
$\displaystyle\geq
e^{-\frac{3}{a}}\frac{1-e^{-\frac{\pi}{2ha}}}{1-e^{-\frac{\pi}{ha}}}$
$\displaystyle=\frac{e^{-\frac{3}{a}}}{1+e^{-\frac{\pi}{2ha}}}\geq\frac{e^{-\frac{3}{c_{3}}}}{2}>0.$

This finishes the proof.

∎

The next result is complementary to the previous. Notice how in Lemma 4.4 we
use the fact that the speed of rotation along $V_{j-1}$, namely $|x_{j-2}|$,
is positive. The following shows how to make $|x_{j-2}|$ sufficiently large
using the forcing terms $\beta_{i}$ while maintaining the hypotheses of Lemma
4.4.

######  Lemma 4.8 (Speed generation).

Let $\beta:=|(\beta_{1},\ldots,\beta_{d})|$ and suppose that $2\leq j\leq d$
and that there exists a constant $c\in(0,1/4)$ such that $x\in\mathbf{R}^{d}$
satisfies

(4.9) $\displaystyle|x|\geq\frac{36\beta}{c},\qquad|x_{j}|\geq
4c|x|,\qquad\text{ and }\qquad|x_{j-1}|\leq ch|x|.$

If $y=\varphi_{\tau}^{\star}(x)$, then with probability at least $p_{2}>0$
independent of $c,h$ we have

(4.10) $\displaystyle|y|\geq\frac{|x|}{2},\quad|y_{j}|\geq
3c|y|,\qquad|y_{j-1}|\leq 2ch|y|,\qquad\text{ and }\quad|y_{j-2}|\geq
h|\beta_{j-2}|.$

###### Proof.

First note the almost sure bound

(4.11) $\displaystyle e^{-\tau}|x|-\beta\tau\leq|y|\leq|x|+\beta\tau.$

We will prove the result using cases depending on the value of $j$.

_Case 1_ ($4\leq j\leq d$). First suppose that $|x_{j-2}|\geq
2|\beta_{j-2}|h$. Then using the choice of $h_{*}$ in (4.1) and $|x|$ in the
statement, on the event $\\{\tau\leq h\\}$ we have that

(4.12) $\displaystyle|y_{j}|$
$\displaystyle=|x_{j}+\beta_{j}\tau|\geq|x_{j}|-|\beta_{j}|\tau\geq
4c|x|-\beta\tau\geq 3c|y|,$ (4.13) $\displaystyle|y_{j-1}|$
$\displaystyle=|x_{j-1}+\beta_{j-1}\tau|\leq|x_{j-1}|+|\beta_{j-1}|\tau\leq
ch|x|+\beta\tau\leq 2ch|y|,$ (4.14) $\displaystyle|y_{j-2}|$
$\displaystyle=|x_{j-2}+\beta_{j-2}\tau|\geq|x_{j-2}|-|\beta_{j-2}|\tau\geq|\beta_{j-2}|h,$
(4.15) $\displaystyle|y|$ $\displaystyle\geq
e^{-\tau}|x|-\beta\tau\geq\frac{3}{4}|x|-\beta\tau\geq\tfrac{|x|}{2}.$

Furthermore, $\mathbf{P}\\{\tau\leq h\\}=1-e^{-1}$. On the other hand, if
$|x_{j-2}|\leq 2|\beta_{j-2}|h$, then on the event $\\{3h\leq\tau\leq 4h\\}$,
which has probability $e^{-3}-e^{-4}$, we have (4.12), (4.13), (4.15) and

(4.16) $\displaystyle|y_{j-2}|$
$\displaystyle=|x_{j-2}+\beta_{j-2}\tau|\geq|\beta_{j-2}|\tau-|x_{j-2}|\geq|\beta_{j-2}|h.$

This finishes the proof in Case 1.

_Case 2_ $(j=3)$. First suppose that $|x_{j-2}|\geq 3|\beta_{j-2}|h$. Using
the choice of $h_{*}$ as in (4.1), on the event $\\{\tau\leq h\\}$ we have
(4.12), (4.13), (4.15) and

$\displaystyle|y_{j-2}|$
$\displaystyle=|e^{-\tau}x_{j-2}+(1-e^{-\tau})\beta_{j-2}|\geq|\beta_{j-2}|h\\{e^{-h}3-1\\}\geq|\beta_{j-2}|h$

where we used the bound $1-e^{-h}\leq h$. On the other hand, if $|x_{j-2}|\leq
3|\beta_{j-2}|h$, then on the event $\\{5h\leq\tau\leq 6h\\}$, which has
probability $e^{-5}-e^{-6}$, we have (4.12), (4.13), (4.15) and

$\displaystyle|y_{j-2}|$
$\displaystyle=|e^{-\tau}x_{j-2}+(1-e^{-\tau})\beta_{j-2}|\geq|\beta_{j-2}|\\{1-e^{-5h}-3he^{-5h}\\}\geq|\beta_{j-2}|h$

where again we used the choice of $h_{*}$ as in (4.1).

_Case 3_ $(j=2)$. Suppose first that $|x_{j-2}|\geq 2|\beta_{j-2}|h$. Then on
the event $\\{\tau\leq h\\}$ we have (4.12), (4.14), (4.15) and

(4.17) $\displaystyle|y_{j-1}|$
$\displaystyle=|e^{-\tau}x_{j-1}+(1-e^{-\tau})\beta_{j-1}\tau|\leq|x_{j-1}|+|\beta_{j-1}|\tau\leq
2ch|y|.$

On the other hand, if $|x_{j-2}|\leq 2|\beta_{j-2}|h$, then on the event
$\\{3h\leq\tau\leq 4h\\}$ we have (4.12), (4.15), (4.16), and (4.17). This
finishes the proof of the result.

∎

Finally, the last lemma below shows how we can keep energy previously stored
while going through the entire splitting.

######  Lemma 4.18 (Energy maintenance).

Let $r>0$ and suppose $x\in\mathbf{R}^{d}$ has $|x|\geq r$. Suppose that for
some $j=2,3,\ldots,d$ and constant $c>0$ we have $|x_{j}|\geq c|x|$. Then with
probability at least $p_{3}=p_{3}(r,c,\beta)>0$ depending only on $r,c,\beta$
we have that

(4.19)
$\displaystyle|\pi_{j}\varphi_{\tau}^{\ell}(x)|\geq\frac{c}{2}|\varphi^{\ell}_{\tau}(x)|,\,\,\ell=j-1,j,\qquad\text{
and
}\qquad|\pi_{j}\varphi^{\star}_{\tau}(x)|\geq\frac{c}{2}|\varphi^{\star}_{\tau}(x)|.$

######  Remark 4.20.

Observe that if $\ell\neq j-1,j$, then by conservation we have

(4.21) $\displaystyle|\pi_{j}\varphi_{\tau}^{\ell}(x)|=|x_{j}|\geq
c|x|=c|\varphi^{\ell}_{\tau}(x)|$

almost surely.

###### Proof of Lemma 4.18.

We omit the argument for $\ell=j$ since it is similar to the case when
$\ell=j-1$. If $\ell=j-1$, observe that

$\displaystyle\pi_{j}\varphi^{\ell}_{\tau}(x)=x_{j}\cos(x_{j-2}\tau)-x_{j-1}\sin(x_{j-2}\tau).$

Thus we consider the event

$B_{2}(j):=\big{\\{}|\pi_{j}\varphi^{\ell}_{\tau}(x)|\geq\tfrac{c}{2}|\varphi_{\tau}^{\ell}(x)|\big{\\}}=\big{\\{}|x_{j}\cos(x_{j-2}\tau)-x_{j-1}\sin(x_{j-2}\tau)|\geq\tfrac{c}{2}|x|\big{\\}}.$

Now for $\delta>0$ to be determined, if $|x_{j-2}|\leq\delta$, then on the
event $\\{\tau\leq h\\}$ we can pick $\delta>0$ small enough depending only on
$c$ so that $B_{2}(j)$ occurs. On the other hand, if $|x_{j-2}|\geq\delta>0$,
then there exists $\epsilon\in(0,\pi/2)$ small enough depending only on $c$ so
that

(4.22) $\displaystyle
B_{3}(x_{j-2}):=\bigcup_{k=0}^{\infty}\big{\\{}||x_{j-2}|\tau-\pi
k|\leq\epsilon\big{\\}}\subset B_{2}(j)$

and the union in (4.22) above is disjoint. Now, for any
$a=|x_{j-2}|\geq\delta$ we have

$\displaystyle\mathbf{P}(B_{3}(x_{j-2}))$
$\displaystyle=1-e^{-\frac{\epsilon}{ha}}+\sum_{k=1}^{\infty}e^{-\frac{\pi
k-\epsilon}{ha}}-e^{-\frac{\pi k+\epsilon}{ha}}$
$\displaystyle=1-e^{-\frac{\epsilon}{ha}}+\frac{e^{-\frac{\pi}{ha}}}{1-e^{-\frac{\pi}{ha}}}\bigg{[}e^{\frac{\epsilon}{ha}}-e^{\frac{-\epsilon}{ha}}\bigg{]}:=g(ha)\in(0,1].$

Note that $g(x)\rightarrow 2\epsilon/\pi$ as $x\rightarrow\infty$ and
$g(x)\rightarrow 1$ as $x\rightarrow 0^{+}$. Thus there exists a constant
$p_{3}>0$ depending only on $\epsilon,\delta$ so that

$\displaystyle\inf_{|x_{j-2}|\geq\delta}\mathbf{P}(B_{3}(x_{j-2}))\geq p_{3}.$

This finishes the proof in the case when $\ell=j-1$.

Lastly, recall that $j\neq 1$. Thus observe that on the event
$\\{\tau\leq\tfrac{r}{(1+2/c)\beta}\\}$

$\displaystyle|\pi_{j}\varphi^{\star}(x)|=|x_{j}+\beta_{j}\tau|\geq|x_{j}|-|\beta_{j}|\tau\geq
c|x|-|\beta_{j}|\tau\geq\tfrac{c}{2}|\varphi_{\tau}^{\star}(x)|,$

finishing the proof.

∎

###  4.2. Proof of Theorem 4.2

Given the previous lemmata, we can now conclude Theorem 4.2. The needed bound
will be verified region-by-region in space, essentially depending on how “far”
away the process is from the dissipative region.

To define the relevant regions, let $R_{0}=1/\sqrt{d}$, define constants
$R_{j}$, $j=1,2,\ldots,d-1$, inductively by

(4.23) $\displaystyle R_{j}=\frac{R_{j-1}h}{16}$

and let

(4.24) $\displaystyle R=\frac{2^{d+5}(\beta\vee 1)}{R_{d-1}}.$

Let $U_{j}\subset\mathbf{R}^{d}$, $j=1,2,\ldots,d$, be defined by

(4.25) $\displaystyle U_{1}$ $\displaystyle=\bigg{\\{}|x|\geq
R\,:\,|x_{1}|\geq R_{d-1}|x|\bigg{\\}},$ (4.26) $\displaystyle U_{j}$
$\displaystyle=\bigg{\\{}|x|\geq 2^{j-1}R\,:\,|x_{\ell}|\geq
R_{d-j}|x|,\,\,\,|x_{\ell-1}|<R_{d-j+1}|x|\bigg{\\}},\qquad 2\leq j\leq d.$

We first note the following.

######  Lemma 4.27.

Let $R_{*}=2^{d-1}R+1$. Then we have

$\displaystyle H_{>R_{*}}\subset\bigcup_{j=1}^{d}U_{j}.$

###### Proof.

Suppose $x\in H_{>R_{*}}$ so that $|x|\geq 2^{d-1}R$. Hence there must exist
$j=1,2,\ldots,d$ such that $|x_{j}|\geq R_{d-j}|x|$. We will show inductively
that if $|x_{j}|\geq R_{d-j}|x|$ with $|x|\geq R_{*}$, then

(4.28) $\displaystyle x\in\bigcup_{\ell=1}^{j}U_{\ell}.$

If $j=1$, then $|x_{1}|\geq R_{d-1}|x|$ and $|x|\geq R_{*}\geq R$ so that
$x\in U_{1}$. Inductively if $|x_{j}|\geq R_{d-j}|x|$ with $|x|\geq R_{*}\geq
2^{j-1}R$, then either $|x_{j-1}|<R_{d-j+1}|x|$, in which case $x\in U_{j}$,
or $|x_{j-1}|\geq R_{d-j+1}|x|$, in which case the inductive hypothesis
implies (4.28). ∎

###### Proof of Theorem 4.2.

Note that if $x\in U_{1}$, then $x\in D_{R_{d-1}}$. Thus we have

$\displaystyle\inf_{x\in
U_{1}}\mathbf{P}_{x}(A_{0}^{\star}(D_{R_{d-1}}))\geq\mathbf{P}(u_{10}=\star)=\frac{1}{d+1}.$

Using the Markov property, it therefore suffices to show that for all
$j\in\\{2,\ldots,d\\}$

(4.29) $\displaystyle\inf_{x\in U_{j}}\mathbf{P}_{x}(X_{1}\in U_{j-1})>0.$

To this end, let $j\in\\{2,\ldots,d\\}$ and assume that $x\in U_{j}$ so that

(4.30) $\displaystyle|x|\geq 2^{j-1}R,\,\,\,|x_{j}|\geq R_{d-j}|x|\,\,\,\text{
and }\,\,\,|x_{j-1}|<R_{d-j+1}|x|.$

Let $\sigma$ be any permutation of $\\{1,\ldots,d,\star\\}$ with
$\sigma_{1}=\star$ and $\sigma_{2}=1$. First applying Lemma 4.8 and then
applying Lemma 4.4, we see that with probability $q_{0}(j,d,h,\beta)>0$ the
point
$y=\varphi_{\tau_{20}}^{\sigma_{2}}\circ\varphi_{\tau_{10}}^{\sigma_{1}}(x)$
satisfies

(4.31) $\displaystyle|\pi_{j-1}y|\geq\frac{R_{d-j}}{4}h|y|\qquad\text{ and
}\qquad|y|\geq\frac{|x|}{2}\geq 2^{j-2}R.$

Applying Lemma 4.18 and setting
$z=\varphi_{\tau_{(d+1)0}}^{\sigma_{d+1}}\circ\cdots\circ\varphi_{\tau_{30}}^{\sigma_{3}}\circ
y$, we find that

(4.32) $\displaystyle|\pi_{j-1}z|\geq\frac{R_{d-j}}{8}h|z|\geq
R_{d-j+1}|z|\qquad\text{ and }\qquad|z|\geq\frac{|x|}{2}\geq 2^{j-2}R.$

with probability $q(U_{2},d,h,\beta)>0$. That is, with probability
$q(U_{j},d,h,\beta)$, $\Phi_{\tau_{0}}^{\sigma}(x)\in U_{j-1}$ so that, by
independence,

$\displaystyle\inf_{x\in U_{j}}\mathbf{P}_{x}(X_{1}\in U_{j-1})$
$\displaystyle\geq\inf_{x\in
U_{j}}\mathbf{P}_{x}(\Phi_{\tau_{0}}^{u_{0}}(x)\in
U_{j-1},u_{10}=\star,u_{20}=1)$
$\displaystyle\geq\frac{q(U_{j},d,h,\beta)}{d(d+1)}>0.$

This finishes the proof.

∎

##  5\. Random Splittings of Galerkin Truncations of 2D Euler

In this section, we study random splittings of finite-dimensional projections
of the two-dimensional partially forced and damped Euler equation on the
periodic box $\mathbf{T}^{2}=[0,2\pi]^{2}$. A full derivation of the splitting
below, starting from the infinite-dimensional partial differential equation,
can be found in the Appendix.

Let $N\geq 4$, $d=2N(N+2)$ and define

(5.1)
$\displaystyle\mathscr{Z}^{2}:=\big{\\{}\mathbf{j}=(j_{1},j_{2})\in\mathbf{Z}^{2}_{\neq
0}\,:\,\,0\leq j_{1}\leq N,\,0\leq j_{2}\leq N\big{\\}}.$

Our phase space for the splitting below is $\mathbf{R}^{d}$ where points
$q\in\mathbf{R}^{d}$ are represented by $q=(a,b)$ where
$a,b\in\mathbf{R}^{d/2}$ are each indexed by $\mathscr{Z}^{2}$; that is,

(5.2) $\displaystyle
a=(a_{\mathbf{j}})_{\mathbf{j}\in\mathscr{Z}^{2}}\qquad\text{ and }\qquad
b=(b_{\mathbf{j}})_{\mathbf{j}\in\mathscr{Z}^{2}}.$

Using this convention, for $\mathbf{j}\in\mathscr{Z}^{2}$ we let
$e_{a_{\mathbf{j}}}$, respectively $e_{b_{\mathbf{j}}}$, denote the vector on
$\mathbf{R}^{d}$ which is $1$ in the $a_{\mathbf{j}}$th entry, respectively
$1$ in the $b_{\mathbf{j}}$th entry, and $0$ elsewhere. For any
$\mathbf{k},\mathbf{l}\in\mathscr{Z}^{2}$, define real-valued constants
$\theta_{\mathbf{k}\mathbf{l}}$ by

(5.3)
$\displaystyle\theta_{\mathbf{k}\mathbf{l}}:=\frac{\mathbf{k}\cdot\mathbf{l}^{\perp}}{4\pi}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)},$

where $\cdot$ denotes the dot product on $\mathbf{R}^{2}$ while
$\mathbf{l}^{\perp}=(l_{1},l_{2})^{\perp}:=(l_{2},-l_{1})$. For any
$\mathbf{j},\mathbf{k},\mathbf{l}\in\mathscr{Z}^{2}$ with
$\mathbf{j}+\mathbf{k}=\mathbf{l}$ and $q=(a,b)\in\mathbf{R}^{d}$, we define
smooth vector fields $V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}$,
$V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}$,
$V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}$,
$V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}$ on $\mathbf{R}^{d}$ by

(5.4) $\displaystyle V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(a,b)$
$\displaystyle=\theta_{\mathbf{k}\mathbf{l}}a_{\mathbf{k}}a_{\mathbf{l}}e_{a_{\mathbf{j}}}+\theta_{\mathbf{j}\mathbf{l}}a_{\mathbf{j}}a_{\mathbf{l}}e_{a_{\mathbf{k}}}-\theta_{\mathbf{j}\mathbf{k}}a_{\mathbf{j}}a_{\mathbf{k}}e_{a_{\mathbf{l}}},$
$\displaystyle V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}(a,b)$
$\displaystyle=\theta_{\mathbf{k}\mathbf{l}}b_{\mathbf{k}}b_{\mathbf{l}}e_{a_{\mathbf{j}}}+\theta_{\mathbf{j}\mathbf{l}}a_{\mathbf{j}}b_{\mathbf{l}}e_{b_{\mathbf{k}}}-\theta_{\mathbf{j}\mathbf{k}}a_{\mathbf{j}}b_{\mathbf{k}}e_{b_{\mathbf{l}}},$
$\displaystyle V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}(a,b)$
$\displaystyle=\theta_{\mathbf{k}\mathbf{l}}a_{\mathbf{k}}b_{\mathbf{l}}e_{b_{\mathbf{j}}}+\theta_{\mathbf{j}\mathbf{l}}b_{\mathbf{j}}b_{\mathbf{l}}e_{a_{\mathbf{k}}}-\theta_{\mathbf{j}\mathbf{k}}b_{\mathbf{j}}a_{\mathbf{k}}e_{b_{\mathbf{l}}},$
$\displaystyle V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}(a,b)$
$\displaystyle=-\theta_{\mathbf{k}\mathbf{l}}b_{\mathbf{k}}a_{\mathbf{l}}e_{b_{\mathbf{j}}}-\theta_{\mathbf{j}\mathbf{l}}b_{\mathbf{j}}a_{\mathbf{l}}e_{b_{\mathbf{k}}}+\theta_{\mathbf{j}\mathbf{k}}b_{\mathbf{j}}b_{\mathbf{k}}e_{a_{\mathbf{l}}}.$

As a simple consequence of conservation laws (see Section 5.4 below), the
vector fields above are moreover complete, i.e. belong to $\mathscr{V}_{d}$.

Let $\mathfrak{m}\in\mathbf{N}$. For $\mathbf{j}\in\mathscr{Z}^{2}$ and
$\ell=1,2,\ldots,\mathfrak{m}$, fix constants

$\displaystyle\lambda_{\mathbf{j}}\geq 0\quad\text{ and
}\quad\beta_{a_{\mathbf{j}}}^{\ell},\beta_{b_{\mathbf{j}}}^{\ell}\in\mathbf{R}.$

We also define vector fields $V_{\text{damp}},V_{\ell}\in\mathscr{V}_{d}$,
$\ell=1,2,\ldots,\mathfrak{m}$, for $q=(a,b)\in\mathbf{R}^{d}$ by

(5.5) $\displaystyle V_{\text{damp}}(a,b)$
$\displaystyle:=-\sum_{\mathbf{j}\in\mathscr{Z}^{2}}\lambda_{\mathbf{j}}(a_{\mathbf{j}}e_{a_{\mathbf{j}}}+b_{\mathbf{j}}e_{b_{\mathbf{j}}})$
(5.6) $\displaystyle V_{\ell}(a,b)$
$\displaystyle:=\sum_{\mathbf{j}\in\mathscr{Z}^{2}}\beta_{a_{\mathbf{j}}}^{\ell}e_{a_{\mathbf{j}}}+\beta_{b_{\mathbf{j}}}^{\ell}e_{b_{\mathbf{j}}}.$

Letting

(5.7)
$\displaystyle\mathscr{I}=\\{(\mathbf{j},\mathbf{k},\mathbf{l})\in(\mathcal{Z}^{2})^{3}\,:\,\mathbf{l}=\mathbf{j}+\mathbf{k},\,\mathbf{j}\cdot\mathbf{k}^{\perp}\neq
0\\},$

our sought after splitting in this section is given by

(5.8)
$\displaystyle\mathscr{S}:=\big{\\{}V_{\text{damp}},V_{1},\ldots,V_{\mathfrak{m}}\big{\\}}\cup\bigcup_{(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}}\big{\\{}V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}},V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}},V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}},V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}\big{\\}}.$

Clearly, $\mathscr{S}$ is a splitting of the sum of its vector fields. This
sum corresponds to Galerkin projections of the two-dimensional, damped and
forced, Euler equation on $\mathbf{T}^{2}$, as derived in the Appendix.

Throughout, we let

(5.9)
$\displaystyle\mathscr{D}=\\{\mathbf{j}\in\mathscr{Z}^{2}\,:\,\lambda_{\mathbf{j}}>0\\}$

denote the collection of _damped_ modes.

#### Notation

Below, for $q\in\mathbf{R}^{d}$, $t\geq 0$, and
$(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$, we will use the notation

$\displaystyle\varphi_{t}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q):=\varphi_{t}^{V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}}(q),\quad\varphi_{t}^{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}(q):=\varphi_{t}^{V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}}(q),\quad\varphi_{t}^{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}(q):=\varphi_{t}^{V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}}(q),$
$\displaystyle\varphi_{t}^{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}(q):=\varphi_{t}^{V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}}(q),\quad\varphi_{t}^{\text{damp}}(q):=\varphi_{t}^{V_{\text{damp}}}(q),\quad\varphi_{t}^{\ell}(q)=\varphi_{t}^{V_{\ell}}(q),$

where $\ell=1,2,\ldots,\mathfrak{m}.$ We offer the convenient abuse of
notation by letting $S_{|\mathscr{S}|}$ in this section denote the set of
permutations of the set of symbols

(5.10)
$\displaystyle\big{\\{}\text{damp},1,\ldots,\mathfrak{m}\big{\\}}\cup\bigcup_{(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}}\big{\\{}a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},\,a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}},\,b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}},\,b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}\big{\\}}.$

With this abuse of notation, $\sigma_{1},\sigma_{2},\ldots,\sigma_{j},\ldots$
will denote elements of $S_{|\mathscr{S}|}$ with $u_{1},u_{2},\ldots,\\\
u_{j},\ldots$ being independent, uniformly distributed elements on
$S_{|\mathscr{S}|}$. We also offer slight abuses of notation (see the
$\beta^{\ell}$ and $\Lambda$ in (5.93) in the appendix) by letting
$\beta^{\ell}\in\mathbf{R}^{d}$ be the vector

(5.11)
$\displaystyle\beta^{\ell}=\sum_{j\in\mathscr{Z}^{2}}\beta_{a_{\mathbf{j}}}^{\ell}e_{a_{\mathbf{j}}}+\beta_{b_{\mathbf{j}}}^{\ell}e_{b_{\mathbf{j}}}$

and letting $\Lambda$ be the $d\times d$ diagonal matrix defined by

(5.12) $\displaystyle\Lambda e_{a_{\mathbf{j}}}\cdot
e_{a_{\mathbf{j}}}=\Lambda e_{b_{\mathbf{j}}}\cdot
e_{b_{\mathbf{j}}}=\lambda_{\mathbf{j}},\,\,\,\mathbf{j}\in\mathscr{Z}^{2}.$

###  5.1. Statement of the main result

In order to connect with the setup in Section 3 and state the main result to
be proven in this section, let $H:\mathbf{R}^{d}\rightarrow[1,\infty)$ be
given by

(5.13) $\displaystyle
H(q)=\sqrt{\sum_{\mathbf{j}\in\mathscr{Z}^{2}}a_{\mathbf{j}}^{2}+b_{\mathbf{j}}^{2}}+1,\,\,\,\,q=(a,b)\in\mathbf{R}^{d}.$

We first establish subconservation and partial dissipation of the splitting.

######  Lemma 5.14.

Suppose that the set of damped modes $\mathscr{D}$ as in (5.9) is nonempty,
and let $\eta\in(0,1)$. Then the splitting $\mathscr{S}$ as in (5.8) is:

  * (i)

subconservative with respect to $H$ with corresponding measurable functions as
in (3.14) given by

$\displaystyle
F_{\ell}(t)=|\beta^{\ell}|t,\,\,\ell=1,\ldots,\mathfrak{m},\quad\text{ and }$
$\displaystyle
F_{\text{\emph{damp}}}(t)=F_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(t)=F_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}(t)=F_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}(t)=F_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}(t)\equiv
0,$

for all $(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$.

  * (ii)

partially dissipative with respect to $H$ with index damp on the set

(5.15) $\displaystyle
D_{\eta}=\Big{\\{}q\in\mathbf{R}^{d}\,:\,\sum_{\mathbf{j}\in\mathscr{D}}|q_{\mathbf{j}}|^{2}\geq\eta|q|^{2}\Big{\\}}.$

###### Proof.

For part (i), note that for any
$(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$, the dynamics along the
vector fields $V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}$,
$V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}$,
$V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}$, and
$V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}$ conserves $H(q)$, implying

$\displaystyle
F_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(t)=F_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}(t)=F_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}(t)=F_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}(t)=0$

for all $t\geq 0$. Also, for any $q\in\mathbf{R}^{d}$,
$\ell=1,2,\ldots,\mathfrak{m}$ and $t\geq 0$ we have

$\displaystyle H(\varphi_{t}^{\text{damp}}(q))=\big{|}e^{-\Lambda
t}q\big{|}+1\leq H(q)\quad\text{ and }\quad
H(\varphi_{t}^{\ell}(q))=|q+t\beta^{\ell}|\leq H(q)+t|\beta^{\ell}|.$

This establishes part (i).

To establish (ii), observe that for any $t\geq 0$ and any $q\in\mathbf{R}^{d}$
with $q\in D_{\eta}$

$\displaystyle H(\varphi_{t}^{\text{damp}}(q))=|e^{-\Lambda t}q|+1$
$\displaystyle=\sqrt{\sum_{\mathbf{j}\in\mathscr{D}}(e^{-\lambda_{\mathbf{j}}t}-1)|q_{\mathbf{j}}|^{2}+|q|^{2}}+1$
$\displaystyle\leq\sqrt{\sum_{\mathbf{j}\in\mathscr{D}}(e^{-\lambda
t}-1)|q_{\mathbf{j}}|^{2}+|q|^{2}}+1$ $\displaystyle\leq|q|\sqrt{(e^{-\lambda
t}-1)\eta^{2}+1}+1\leq\alpha(t)H(x)+1$

where $\alpha(t)=\sqrt{(e^{-\lambda t}-1)\eta^{2}+1}$. ∎

In order to avoid certain invariant sets in the splitting, we need further
structure in our forcing terms. Specifically, for any
$\mathbf{j},\mathbf{k},\mathbf{l}\in\mathscr{Z}^{2}$ and any
$\beta\in\mathbf{R}^{d}$ let

(5.16) $\displaystyle\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta)$
$\displaystyle:=(\beta_{a_{\mathbf{j}}})^{2}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-(\beta_{a_{\mathbf{l}}})^{2}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)},$
$\displaystyle\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}(\beta)$
$\displaystyle:=(\beta_{a_{\mathbf{j}}})^{2}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-(\beta_{b_{\mathbf{l}}})^{2}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}.$

######  Definition 5.17.

We say that the forcing in the splitting $\mathscr{S}$ is _non-resonant for_
$\mathbf{j}\in\mathscr{Z}^{2}$, denoted by $\mathbf{j}\in\mathscr{F}$, if
there exists $\ell=1,2,\ldots,\mathfrak{m}$ such that for any
$\mathbf{k},\mathbf{l}\in\mathscr{Z}$ with $\mathbf{j}+\mathbf{k}=\mathbf{l}$
and $|\mathbf{j}|\neq|\mathbf{k}|$

(5.18) $\displaystyle\beta^{\ell}_{a_{\mathbf{j}}}\neq
0,\quad\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{\ell})\neq
0,\quad\text{ and
}\quad\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}(\beta^{\ell})\neq 0.$

Although there are many other examples of forcings which are non-resonant for
$\mathbf{j}\in\mathscr{Z}^{2}$, we now provide a couple of simple examples for
reference.

######  Example 5.19.

If, for $\mathbf{j}\in\mathscr{Z}^{2}$ fixed,
$\beta^{1}=\beta_{a_{\mathbf{j}}}^{1}e_{a_{\mathbf{j}}}$ for some
$\beta_{a_{\mathbf{j}}}^{1}\neq 0$, then $\mathbf{j}\in\mathscr{F}$.

######  Example 5.20.

Note that if $\mathbf{j},\mathbf{k}\in\mathscr{Z}^{2}$ are fixed with
$|\mathbf{j}|=|\mathbf{k}|$, $\mathbf{j}\neq\mathbf{k}$, and

$\displaystyle\beta^{1}=\beta_{a_{\mathbf{j}}}^{1}e_{a_{\mathbf{j}}}+\beta^{1}_{a_{\mathbf{k}}}e_{a_{\mathbf{k}}}$

for some $\beta_{a_{\mathbf{j}}}^{1}\neq 0$ and
$\beta_{a_{\mathbf{k}}}^{1}\neq 0$, then
$\mathbf{j},\mathbf{k}\in\mathscr{F}$.

Throughout this section, we will employ the following assumptions on
$\mathscr{D}$ and $\mathscr{F}$.

######  Assumption 5.21.

One of the following conditions is met:

  * (DF1)

$\\{(0,1),(1,0)\\}\subset\mathscr{F}$ and
$\\{(1,0),(0,1),(N,N)\\}\subset\mathscr{D}.$

  * (DF2)

$\\{(0,1),(1,0),(1,1)\\}\subset\mathscr{F}$ and either
$\\{(1,0),(N,N)\\}\subset\mathscr{D}$ or
$\\{(0,1),(N,N)\\}\subset\mathscr{D}$.

######  Theorem 5.22.

Let $h\in(0,1)$. Suppose that Assumption 5.21 holds and let $\psi(x):=\log x$
for $x\geq 2$ and $\psi(x):=\log 2$ for $x\leq 2$. Then there exist constants
$c=c(N,h)>0,C=C(N,h)>0$ such that the global estimate holds

(5.23) $\displaystyle\mathcal{P}H-H\leq-\frac{c}{\psi(H)}H+C.$

In order to prove Theorem 5.22, combining Lemma 5.14 with Theorem 3.32 it
suffices to show the following result.

######  Theorem 5.24.

Let $h\in(0,1)$ and suppose that Assumption 5.21 is satisfied. Then there
exists constants $p_{*}>0,R\geq 2$ and $\eta\in(0,1)$ such that

(5.25)
$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{\emph{damp}}}(D_{\eta}))\geq\frac{p_{*}}{\log(H(q))}$

for all $q\in H_{>R}$.

In order to prove Theorem 5.24, we need to first establish some key properties
of the dynamics defined in (5.4). Similar to the case of Lorenz ’96 in Section
4, we will provide conditions under which energy is transferred among the
directions in these vector fields. All of this analysis is carried out in
Section 5.2, Section 5.3 and Section 5.4. Finally, given the facts deduced in
these sections, in Section 5.5 we conclude Theorem 5.24.

###  5.2. Preliminary observations

Note first that the dynamics along every vector field
$V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}$,
$V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}$,
$V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}$,
$V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}$ with
$(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$ can be written in the form

(5.26) $\displaystyle\dot{X}$
$\displaystyle=\theta_{\mathbf{k}\mathbf{l}}YZ,\qquad\dot{Y}=\theta_{\mathbf{j}\mathbf{l}}XZ,\qquad\dot{Z}=-\theta_{\mathbf{j}\mathbf{k}}XY.$

This is clear for $V_{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}$,
$V_{a_{\mathbf{j}}b_{\mathbf{k}}b_{\mathbf{l}}}$ and
$V_{b_{\mathbf{j}}a_{\mathbf{k}}b_{\mathbf{l}}}$. For
$V_{b_{\mathbf{j}}b_{\mathbf{k}}a_{\mathbf{l}}}$, we simply let
$X=b_{\mathbf{j}},Y=b_{\mathbf{k}}$ and $Z=-a_{\mathbf{l}}$ to arrive at
(5.26). It will be useful to rescale (5.26) by $H=H(q)=H(a,b)$, and we assume
below that $H\geq 3$. Because $H$ is conserved by the dynamics along every
vector field in
$\mathscr{S}-\\{V_{\text{damp}},V_{1},\ldots,V_{\mathfrak{m}}\\}$, we treat it
as a constant when analyzing (5.26). For simplicity, we define $\delta=1/H$
and set $x=\delta X,y=\delta Y,z=\delta Z$ where $X,Y,Z$ are as in (5.26).
Note then that the triple $(x,y,z)$ satisfies the equation

(5.27)
$\displaystyle\dot{x}=\frac{\theta_{\mathbf{k}\mathbf{l}}}{\delta}yz,\qquad\dot{y}=\frac{\theta_{\mathbf{j}\mathbf{l}}}{\delta}xz,\qquad\dot{z}=-\frac{\theta_{\mathbf{j}\mathbf{k}}}{\delta}xy.$

Under certain assumptions on the initial data as well as the indices
$\mathbf{j},\mathbf{k}$, throughout this section we study the behavior of
equation (5.27) as $\delta\rightarrow 0$, i.e. as $H\rightarrow\infty$.

We break the analysis apart according to two cases; namely, when
$|\mathbf{k}|=|\mathbf{j}|$ or $|\mathbf{k}|>|\mathbf{j}|$. Both cases will
always be under the hypotheses that $\mathbf{l}=\mathbf{j}+\mathbf{k}$ and
$\mathbf{j}\cdot\mathbf{k}^{\perp}\neq 0$.

###  5.3. Case 1: $(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$ with
$|\mathbf{k}|=|\mathbf{j}|$

In this case, $\theta_{\mathbf{j}\mathbf{k}}=0$ and
$\theta_{\mathbf{k}\mathbf{l}}=-\theta_{\mathbf{j}\mathbf{l}}\neq 0$. Thus
equation (5.27) reduces to

(5.28)
$\displaystyle\dot{x}=-\frac{\theta_{\mathbf{j}\mathbf{l}}}{\delta}yz,\qquad\dot{y}=\frac{\theta_{\mathbf{j}\mathbf{l}}}{\delta}xz,\qquad\dot{z}=0.$

The unique solution of (5.28) with initial condition
$(x_{0},y_{0},z_{0})\in\mathbf{R}^{3}$ is explicitly given by

(5.29) $\displaystyle
x_{t}=\rho_{0}\cos\big{(}\tfrac{\theta_{\mathbf{j}\mathbf{l}}z_{0}}{\delta}t+\theta_{*}\big{)},\qquad
y_{t}=\rho_{0}\sin\big{(}\tfrac{\theta_{\mathbf{j}\mathbf{l}}z_{0}}{\delta}t+\theta_{*}\big{)}\qquad\text{
and }\qquad z_{t}=z_{0},\,\,\,t\geq 0,$

where $\rho_{0}=\sqrt{x_{0}^{2}+y_{0}^{2}}$ and $\theta_{*}\in[0,2\pi)$ is the
unique point for which

$\displaystyle(x_{0},y_{0})=\rho_{0}(\cos\theta_{*},\sin\theta_{*}).$

Recalling that $\tau\sim\exp(1/h)$ with $h\in(0,1)$, using these expressions
we will prove the following lemma.

######  Lemma 5.30.

Suppose that $(x_{t},y_{t},z_{t})$ solves (5.28) with initial condition
$(x_{0},y_{0},z_{0})$ satisfying $|(x_{0},y_{0},z_{0})|\leq 1$. If
$\max\\{|x_{0}|,|y_{0}|\\}\geq c_{0}$ and $|z_{0}|\geq c_{1}\delta$ for some
constants $c_{0}>0,c_{1}>0$ independent of $\delta$, then with probability
$q=q(c_{0},h)>0$ depending only on $c_{0},h$ we have

$\displaystyle\min\\{|x_{\tau}|,|y_{\tau}|\\}\geq c_{0}/2.$

###### Proof.

We suppose without loss of generality that
$\theta_{\mathbf{j}\mathbf{l}}z_{0}>0$ as the other case is similar. Consider
the event

(5.31) $\displaystyle
B_{4}=B_{4}(\mathbf{j},\mathbf{l}):=\bigcup_{k=1}^{\infty}\bigg{\\{}2\pi
k+\frac{\pi}{6}\leq\frac{\theta_{\mathbf{j}\mathbf{l}}z_{0}\tau}{\delta}+\theta_{*}\leq\frac{\pi}{4}+2\pi
k\bigg{\\}}.$

On $B_{4}$, we have the desired bounds

$\displaystyle|x_{\tau}|=\rho_{0}\bigg{|}\cos\bigg{(}\frac{\theta_{\mathbf{j}\mathbf{l}}z_{0}\tau}{\delta}+\theta_{*}\bigg{)}\bigg{|}\geq\frac{\rho_{0}}{\sqrt{2}}\geq\frac{c_{0}}{2}\,\,\,\text{
and
}\,\,\,|y_{\tau}|=\rho_{0}\bigg{|}\sin\bigg{(}\frac{\theta_{\mathbf{j}\mathbf{l}}z_{0}\tau}{\delta}+\theta_{*}\bigg{)}\bigg{|}\geq\frac{\rho_{0}}{2}\geq\frac{c_{0}}{2}.$

We have left to estimate $\mathbf{P}(B_{4})$. To this end, note that if
$w:=\tfrac{\delta}{\theta_{\mathbf{j}\mathbf{l}}z_{0}h}$, we have that

$\displaystyle\mathbf{P}(B_{4})$
$\displaystyle=\sum_{k=1}^{\infty}\mathbf{P}\bigg{\\{}\frac{(2\pi
k+\pi/6-\theta_{*})\delta}{\theta_{\mathbf{j}\mathbf{l}}z_{0}}\leq\tau\leq\frac{(\pi/4+2\pi
k-\theta_{*})\delta}{\theta_{\mathbf{j}\mathbf{l}}z_{0}}\bigg{\\}}$
$\displaystyle=\sum_{k=1}^{\infty}e^{-\frac{(2\pi
k+\pi/6-\theta_{*})\delta}{\theta_{\mathbf{j}\mathbf{l}}z_{0}h}}-e^{-\frac{(\pi/4+2\pi
k-\theta_{*})\delta}{\theta_{\mathbf{j}\mathbf{l}}z_{0}h}}$
$\displaystyle=e^{\theta_{*}w}\frac{e^{-2\pi w}}{1-e^{-2\pi
w}}(e^{-\frac{\pi}{6}w}-e^{-\frac{\pi}{4}w}).$

By our assumption, $\theta_{\mathbf{j}\mathbf{l}}z_{0}\geq
c_{1}|\theta_{\mathbf{j}\mathbf{l}}|\delta$. Hence, $0<w\leq
1/(c_{1}|\theta_{\mathbf{j}\mathbf{l}}|h)$. In particular,
$\mathbf{P}(B_{4})>0$ for all such $w$. On the other hand, as $w\rightarrow 0$

$\displaystyle e^{\theta_{*}w}\frac{e^{-2\pi w}}{1-e^{-2\pi
w}}(e^{-\frac{\pi}{6}w}-e^{-\frac{\pi}{4}w})\rightarrow\frac{1}{24}>0.$

In particular,

(5.32)
$\displaystyle\inf_{0<w\leq\frac{1}{c_{1}|\theta_{\mathbf{j}\mathbf{l}}|h}}\mathbf{P}(B_{4})>0,$

which finishes the proof. ∎

###  5.4. Case 2: $(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$ with
$|\mathbf{k}|>|\mathbf{j}|$

To see when energy is transferred in (5.27) in this case, first observe that
equation (5.27) has two conserved quantities; namely, the _relative energy_
and the _relative enstrophy_ , which are respectively given by

(5.33) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)$
$\displaystyle=\frac{x^{2}}{|\mathbf{j}|^{2}}+\frac{y^{2}}{|\mathbf{k}|^{2}}+\frac{z^{2}}{|\mathbf{l}|^{2}},$
(5.34) $\displaystyle\mathcal{E}(x,y,z)$ $\displaystyle=x^{2}+y^{2}+z^{2}.$

For simplicity, in this section we will use the shorthand notations

(5.35) $\displaystyle E=E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\quad\text{
and }\quad\mathcal{E}=\mathcal{E}(x,y,z).$

Define a constant $\zeta_{0}(\mathbf{j},\mathbf{k})$ by

(5.36)
$\displaystyle\zeta_{0}(\mathbf{j},\mathbf{k}):=\min\bigg{\\{}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}\frac{1}{2d},\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\frac{1}{2d}\bigg{\\}}>0$

we will apply the following assumption on the initial data and the indices.

######  Assumption 5.37.

We have that $(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$ with
$|\mathbf{k}|>|\mathbf{j}|$ and either _(A1)_ or _(A2)_ below holds:

  * (A1)

There exists constants $\xi\in(0,1)$ and
$\zeta\in(0,\zeta_{0}(\mathbf{j},\mathbf{k})d)$ independent of $\delta$ such
that

$\displaystyle\xi\leq\mathcal{E}\leq 1\qquad\text{ and
}\qquad\frac{\mathcal{E}}{|\mathbf{l}|^{2}}+\zeta\leq
E\leq\frac{\mathcal{E}}{|\mathbf{k}|^{2}}-\zeta\delta^{2}.$

  * (A2)

There exists constants
$\xi\in(0,1),\zeta\in(0,\zeta_{0}(\mathbf{j},\mathbf{k})d)$ independent of
$\delta$ such that

$\displaystyle\xi\leq\mathcal{E}\leq 1\qquad\text{ and
}\qquad\frac{\mathcal{E}}{|\mathbf{k}|^{2}}+\zeta\delta^{2}\leq
E\leq\frac{\mathcal{E}}{|\mathbf{j}|^{2}}-\zeta.$

######  Remark 5.38.

Although it is suggestive, Assumption 5.37 is _not_ equivalent to the
condition that there exists a constant $c>0$ small, independent of
$\delta\in(0,1)$ for which

(5.39) $\displaystyle c\leq y^{2}_{0}\leq\mathcal{E}-c\delta^{2}.$

The above relation (5.39) means that, initially, the middle mode $y$ in (5.27)
contains some, but not all of the relative enstrophy $\mathcal{E}$. To see why
they are not equivalent, note that the condition
$E=\mathcal{E}/|\mathbf{k}|^{2}$ is equivalent to the initial data $x_{0}$ and
$z_{0}$ in (5.27) satisfying

(5.40) $\displaystyle
x^{2}_{0}=z^{2}_{0}\frac{\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}}.$

Note that if $y_{0}$ satisfies (5.39), then using the fact that
$|\mathbf{l}|^{2}>|\mathbf{k}|^{2}>|\mathbf{j}|^{2}$ we can find $x_{0},z_{0}$
satisfying (5.40). When returning back to the randomly switched system, in
order to deal with this family of _bad_ initial conditions, we will employ the
assumption that $\mathbf{j}\in\mathscr{F}$. In particular, this will allow us
to ensure, with positive probability, that we can initially perturb the
dynamics away from the surface (5.40) so that for some constant
$\zeta^{\prime}>0$ independent of $\delta$ the resulting coordinates
$(x_{0},y_{0},z_{0})$ satisfy either

$\displaystyle
x^{2}_{0}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}\leq
z^{2}_{0}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}-\zeta^{\prime}\delta^{2},$

which is equivalent to
$E\leq\tfrac{\mathcal{E}}{|\mathbf{k}|^{2}}-\zeta^{\prime}\delta^{2}$, or

$\displaystyle
x^{2}_{0}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}\geq
z^{2}_{0}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}+\zeta^{\prime}\delta^{2},$

which is equivalent to
$E\geq\frac{\mathcal{E}}{|k|^{2}}+\zeta^{\prime}\delta^{2}.$ This will then
imply Assumption 5.37.

In order to state our main result of this section, for $\eta\in(0,1/2)$ and
$\delta\in(0,1)$, consider the following set

(5.41) $\displaystyle B_{\eta,\delta}=\\{s\geq
0\,:\,\min\\{|x_{s}|,|y_{s}|,|z_{s}|\\}\geq\eta\\}$

where $(x_{s},y_{s},z_{s})$ denotes the solution of (5.27). Of course, the
solution $(x_{s},y_{s},z_{s})$ depends on $\delta$, but we suppress this
dependence for notational simplicity.

######  Theorem 5.42 (Thermalization).

Suppose that Assumption 5.37 is satisfied. Then there exist constants
$\eta\in(0,1/4),c>0$ and $\delta_{*}\in(0,1)$ depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,\zeta,\xi,h$ so that

(5.43) $\displaystyle\mathbf{P}\\{\tau\in
B_{\eta,\delta}\\}\geq\frac{c}{|\log\delta|}\,\,\,\text{ for all
}\delta\in(0,\delta_{*}).$

######  Remark 5.44.

Note that conclusion (5.43) is _thermalization_ of the triple
$(x_{s},y_{s},z_{s})$ with probability $\geq c/|\log\delta|$.

In order to prove Theorem 5.42, we first introduce and deduce properties of
the so-called _Jacobi elliptic functions_. As we will see below, these
functions can be used to arrive at explicit expressions for solutions of
equation (5.27).

To begin, for $(\mathbf{j},\mathbf{k},\mathbf{l})\in\mathscr{I}$ with
$|\mathbf{k}|>|\mathbf{j}|$, we first define the following parameters

(5.45)
$\displaystyle\kappa_{1}=\frac{E-\frac{\mathcal{E}}{|\mathbf{l}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}},\qquad\gamma_{1}=\frac{\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}{E-\frac{\mathcal{E}}{|\mathbf{l}|^{2}}},\qquad\kappa_{2}=\frac{\frac{\mathcal{E}}{|\mathbf{j}|^{2}}-E}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}},\qquad\gamma_{2}=\frac{{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}}}{\frac{\mathcal{E}}{|\mathbf{j}|^{2}}-E}$

where $E,\mathcal{E}$ are the conserved quantities in (5.33). Observe that
each of these parameters is strictly positive under Assumption 5.37.
Furthermore, these parameters arise from relations (5.33)-(5.34) by solving
for $x$ and $z$ in terms of $y$. Indeed, we see that (5.33)-(5.34) together
imply

(5.46) $\displaystyle x^{2}=\kappa_{1}(1-\gamma_{1}y^{2})\qquad\text{ and
}\qquad z^{2}=\kappa_{2}(1-\gamma_{2}y^{2}).$

Next, define

(5.47)
$\displaystyle\gamma_{\text{min}}=\gamma_{1}\wedge\gamma_{2},\qquad\gamma_{\text{max}}=\gamma_{1}\vee\gamma_{2},\qquad\text{
and }\qquad\rho=\frac{\gamma_{\text{min}}}{\gamma_{\text{max}}}.$

One can check that

$\displaystyle\frac{\mathcal{E}}{|\mathbf{l}|^{2}}<E<\frac{\mathcal{E}}{|\mathbf{k}|^{2}}\,\,\,\text{
implies }\,\,\,\gamma_{1}>\mathcal{E}^{-1}\text{ and
}\gamma_{2}<\mathcal{E}^{-1},\text{ so }\rho=\frac{\gamma_{2}}{\gamma_{1}}<1.$

Similarly,

$\displaystyle\frac{\mathcal{E}}{|\mathbf{k}|^{2}}<E<\frac{\mathcal{E}}{|\mathbf{j}|^{2}}\,\,\,\text{
implies }\,\,\,\gamma_{2}>\mathcal{E}^{-1}\text{ and
}\gamma_{1}<\mathcal{E}^{-1},\text{ so }\rho=\frac{\gamma_{1}}{\gamma_{2}}<1.$

######  Remark 5.48.

To motivate the use of Jacobi elliptic functions with the parameters above, we
observe that, formally, if $b:=y\sqrt{\gamma_{\text{max}}}$ where $y$ is as in
(5.27), then

$\displaystyle\dot{b}=\frac{\sqrt{\gamma_{\text{max}}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}\sqrt{1-\rho
b^{2}}\sqrt{1-b^{2}}.$

Since $\rho\in(0,1)$, the above system is explicitly (locally) integrable but
we need to be careful about the signs in the square roots. These signs depend
on the initial data as well as the periodicity in the system (5.27). A similar
line of reasoning can also be used to arrive at formal explicitly integrable
equations for $x$ and $z$ in (5.27).

Following the previous remark, define a function $T_{\rho}$ on $[0,1]$ and a
number $K_{\rho}$ by

(5.49) $\displaystyle T_{\rho}(s)=\int_{0}^{s}\frac{db}{\sqrt{1-\rho
b^{2}}\sqrt{1-b^{2}}}\qquad\text{ and }\qquad K_{\rho}:=T_{\rho}(1).$

Observe that since $\rho<1$, $T_{\rho}$ is a well-defined, strictly increasing
function on $[0,1]$ with range $[0,K_{\rho}]$. Let
$\text{sn}_{\rho}:[0,K_{\rho}]\rightarrow[0,1]$ denote the inverse of
$T_{\rho}$. We extend $\text{sn}_{\rho}$ to the larger interval $[0,4K_{r}]$
by defining

(5.50) $\displaystyle\text{sn}_{\rho}(x):=\text{sn}_{\rho}(2K_{\rho}-x),\qquad
x\in[K_{\rho},2K_{\rho}],$
$\displaystyle\text{sn}_{\rho}(x):=-\text{sn}_{\rho}(4K_{\rho}-x)\qquad
x\in[2K_{\rho},4K_{\rho}].$

We extend $\text{sn}_{\rho}$ periodically to all of $\mathbf{R}$ with period
$4K_{\rho}$. Similarly, the function $\text{cn}_{\rho}$ is defined initially
on the interval $[0,4K_{r}]$ by

(5.51) $\displaystyle\text{cn}_{\rho}(x)$
$\displaystyle:=\sqrt{1-\text{sn}_{\rho}^{2}(x)},\qquad x\in[0,K_{\rho}],$
$\displaystyle\text{cn}_{\rho}(x)$
$\displaystyle:=-\text{cn}_{\rho}(2K_{\rho}-x),\quad
x\in[K_{\rho},2K_{\rho}],$ $\displaystyle\text{cn}_{\rho}(x)$
$\displaystyle:=\text{cn}_{\rho}(4K_{\rho}-x),\quad x\in[2K_{\rho},4K_{\rho}]$

and then extended to all of $\mathbf{R}$ to a function which is periodic with
period $4K_{\rho}$. The function $\text{dn}_{\rho}$ is defined by

(5.52)
$\displaystyle\text{dn}_{\rho}(x):=\sqrt{1-\rho\text{sn}_{\rho}(x)^{2}},\qquad
x\in\mathbf{R}.$

Note that because $\rho<1$, $\text{dn}_{\rho}$ remains positive for all values
of $x$ whereas $\text{sn}_{\rho}$ and $\text{cn}_{\rho}$ oscillate between $1$
and $-1$, similar to the usual sine and cosine. The function
$\text{sn}_{\rho}$, $\text{cn}_{\rho}$, and $\text{dn}_{\rho}$ are called
_Jacobi elliptic functions_. One can check that (see [1]) the functions
$\text{sn}_{\rho}$, $\text{cn}_{\rho}$ and $\text{dn}_{\rho}$ are
differentiable and satisfy the following differential identities

(5.53)
$\displaystyle\text{sn}_{\rho}^{\prime}(x)=\text{cn}_{\rho}(x)\text{dn}_{\rho}(x),\quad\text{cn}_{\rho}^{\prime}(x)=-\text{sn}_{\rho}(x)\text{dn}_{\rho}(x),\quad\text{dn}_{\rho}^{\prime}(x)=-\rho\text{sn}_{\rho}(x)\text{cn}_{\rho}(x).$

Using the above facts, we can now deduce explicit expressions involving the
Jacobi elliptic functions for solutions of (5.27). The explicit expressions
vary slightly depending on the value of the relative energy $\mathcal{E}$ and
the initial data, as we now note.

#### Assumption _(A1)_

Suppose that Assumption 5.37 (A1) is satisfied. First observe that
$\mathcal{E}/|\mathbf{l}|^{2}<E<\mathcal{E}/|\mathbf{k}|^{2}$ implies that the
initial condition $z_{0}$ in (5.27) is nonzero. Indeed, if $z_{0}=0$, then
using $|\mathbf{l}|>|\mathbf{k}|>|\mathbf{j}|$ gives

$\displaystyle
E=\frac{x^{2}_{0}}{|\mathbf{j}|^{2}}+\frac{y^{2}_{0}}{|\mathbf{k}|^{2}}\geq\frac{\mathcal{E}}{|\mathbf{k}|^{2}}.$

Now, if $z_{0}>0$, then we can express $x$, $y$ and $z$ as

$\displaystyle x_{t}=\sqrt{\kappa_{1}}\text{cn}_{\rho}\bigg{[}$
$\displaystyle\frac{\sqrt{\gamma_{1}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},\quad
y_{t}=\frac{1}{\sqrt{\gamma_{1}}}\text{sn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{1}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$
$\displaystyle
z_{t}=\sqrt{\kappa_{2}}\text{dn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{1}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$

where $\theta_{0}\in[0,4K_{\rho})$ is the unique point so that

(5.54) $\displaystyle
x_{0}=\sqrt{\kappa_{1}}\text{cn}_{\rho}(\theta_{0})\qquad\text{ and }\qquad
y_{0}=\frac{1}{\sqrt{\gamma_{1}}}\text{sn}_{\rho}(\theta_{0}).$

On the other hand, if $z_{0}<0$, then the solution of (5.27) can be written as

$\displaystyle x_{t}=\sqrt{\kappa_{1}}\text{cn}_{\rho}\bigg{[}$
$\displaystyle\frac{\sqrt{\gamma_{1}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},\quad
y_{t}=-\frac{1}{\sqrt{\gamma_{1}}}\text{sn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{1}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$
$\displaystyle
z_{t}=-\sqrt{\kappa_{2}}\text{dn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{1}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$

where $\theta_{0}\in[0,4K_{\rho})$ is the unique point so that

(5.55) $\displaystyle
x_{0}=\sqrt{\kappa_{1}}\text{cn}_{\rho}(\theta_{0})\qquad\text{ and }\qquad
y_{0}=-\frac{1}{\sqrt{\gamma_{1}}}\text{sn}_{\rho}(\theta_{0}).$

Additionally, it is not hard to check that the conditions
$\mathcal{E}\in[\xi,1]$ and $\mathcal{E}/|\mathbf{l}|^{2}+\zeta\leq
E\leq\mathcal{E}/|\mathbf{k}|^{2}-\zeta\delta^{2}$ imply that the parameters
$\kappa_{1},\gamma_{1},\kappa_{2},\gamma_{2}$ are bounded above and below by
positive constants that do not depend on $\delta$. Indeed, we have the
following bounds

(5.56)
$\displaystyle\frac{\zeta}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}$
$\displaystyle\leq\kappa_{1}\leq\frac{\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}},\qquad\,\,\,1\leq\gamma_{1}\leq\frac{\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}{\zeta},$
$\displaystyle\frac{\frac{\xi}{|\mathbf{j}|^{2}}-\frac{\xi}{|\mathbf{k}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}$
$\displaystyle\leq\kappa_{2}\leq\frac{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}-\zeta}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}},\qquad\frac{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}-\zeta}\leq\gamma_{2}\leq
1/\xi.$

#### Assumption _(A2)_

Suppose now that Assumption 5.37 (A2) is satisfied. Then a similar argument to
the one used above implies that $x_{0}\neq 0$. If $x_{0}>0$, then the unique
solution of (5.27) is given by

$\displaystyle x_{t}=\sqrt{\kappa}_{1}\text{dn}_{\rho}\bigg{[}$
$\displaystyle\frac{\sqrt{\gamma_{2}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},\,\,\,\,y_{t}=\frac{1}{\sqrt{\gamma_{2}}}\text{sn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{2}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$
$\displaystyle
z_{t}=\sqrt{\kappa_{2}}\text{cn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{2}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$

where $\theta_{0}\in[0,4K_{\rho})$ is defined by

(5.57) $\displaystyle
y_{0}=\frac{1}{\sqrt{\gamma_{2}}}\text{sn}_{\rho}(\theta_{0})\qquad\text{ and
}\qquad z_{0}=\sqrt{\kappa_{2}}\text{cn}_{\rho}(\theta_{0}).$

On the other hand if $x_{0}<0$, then the unique solution of (5.27) is given by

$\displaystyle x_{t}=-\sqrt{\kappa}_{1}\text{dn}_{\rho}\bigg{[}$
$\displaystyle\frac{\sqrt{\gamma_{2}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},\,\,\,\,y_{t}=-\frac{1}{\sqrt{\gamma_{2}}}\text{sn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{2}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$
$\displaystyle
z_{t}=\sqrt{\kappa_{2}}\text{cn}_{\rho}\bigg{[}\frac{\sqrt{\gamma_{2}\kappa_{1}\kappa_{2}}\theta_{\mathbf{j}\mathbf{l}}}{\delta}t+\theta_{0}\bigg{]},$

where $\theta_{0}\in[0,4K_{\rho})$ is defined by

(5.58) $\displaystyle
y_{0}=-\frac{1}{\sqrt{\gamma_{2}}}\text{sn}_{\rho}(\theta_{0})\qquad\text{ and
}\qquad z_{0}=\sqrt{\kappa_{2}}\text{cn}_{\rho}(\theta_{0}).$

In this case, we also have bounds for the positive parameters
$\kappa_{1},\gamma_{1},\kappa_{2},\gamma_{2}$ analogous to those in (5.56):

(5.59)
$\displaystyle\frac{\frac{\xi}{|\mathbf{k}|^{2}}-\frac{\xi}{|\mathbf{l}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}$
$\displaystyle\leq\kappa_{1}\leq\frac{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}-\zeta}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}},\qquad\frac{\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}-\zeta}\leq\gamma_{1}\leq\frac{1}{\xi},$
$\displaystyle\frac{\zeta}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}}$
$\displaystyle\leq\kappa_{2}\leq\frac{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}}{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}},\qquad\frac{\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}}{\zeta}\leq\gamma_{2}\leq\frac{1}{\xi}.$

Given the expressions above for solutions of (5.27) as well as the bounds
(5.56) and (5.59), we are almost ready to prove Theorem 5.42. The last
ingredient needed is a lower asymptotic bound on the quantity $K_{\rho}$ in
certain regimes in Assumption 5.37. We recall that $K_{\rho}$ is $1/4$ of the
period of the Jacobi elliptic functions $\text{sn}_{\rho}$ and
$\text{cn}_{\rho}$. We do this in the following lemma.

######  Lemma 5.60.

Suppose that Assumption 5.37 is satisfied and, furthermore,

(5.61) $\displaystyle
E=\frac{\mathcal{E}}{|\mathbf{k}|^{2}}+\epsilon(\delta)\qquad\text{ or }\qquad
E=\frac{\mathcal{E}}{|\mathbf{k}|^{2}}-\epsilon(\delta)$

where $\epsilon(\delta)>0$ and $\epsilon(\delta)\rightarrow 0$ as
$\delta\rightarrow 0$. Then as $\delta\rightarrow 0$

(5.62) $\displaystyle K_{\rho}=-\frac{\log\epsilon(\delta)}{2}+O(1).$

######  Remark 5.63.

If Assumption 5.37 is satisfied and $E$ is uniformly bounded away from
$\mathcal{E}/|\mathbf{k}|^{2}$ as $\delta\rightarrow 0$, then the parameter
$\rho=\gamma_{\text{min}}/\gamma_{\text{max}}<1$ is then uniformly bounded
away from $1$ as $\delta\rightarrow 0$. Hence, $K_{\rho}$ is order one,
positive as $\delta\rightarrow 0$. Thus the analysis in the lemma above
focuses on the case when $E\rightarrow\mathcal{E}/|\mathbf{k}|^{2}$ as
$\delta\rightarrow 0$.

###### Proof of Lemma 5.60.

Let $E=\mathcal{E}/|\mathbf{k}|^{2}\pm\epsilon(\delta)$ where
$\epsilon(\delta)>0$ satisfies $\epsilon(\delta)\rightarrow 0$ as
$\delta\rightarrow 0$. Without loss of generality, we study the case when
$E=\mathcal{E}/|k|^{2}-\epsilon(\delta)$ as the other case is nearly
identical. Observe that as $\delta\rightarrow 0$

$\displaystyle\rho=\frac{\gamma_{\text{min}}}{\gamma_{\text{max}}}=\frac{\frac{1}{1+\frac{\epsilon(\delta)}{\frac{\mathcal{E}}{|\mathbf{j}|^{2}}-\frac{\mathcal{E}}{|\mathbf{k}|^{2}}}}}{\frac{1}{1-\frac{\epsilon(\delta)}{\frac{\mathcal{E}}{|\mathbf{k}|^{2}}-\frac{\mathcal{E}}{|\mathbf{l}|^{2}}}}}=1-\bigg{(}\frac{1}{\mathcal{E}(\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}})}+\frac{1}{\mathcal{E}(\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}})}\bigg{)}\epsilon(\delta)+O(\epsilon^{2}(\delta)).$

Using the asymptotics in [20], we thus obtain

$\displaystyle K_{\rho}=\int_{0}^{1}\frac{db}{\sqrt{1-\rho
b^{2}}\sqrt{1-b^{2}}}\,db=-\frac{\log\epsilon(\delta)}{2}+O(1)$

as $\delta\rightarrow 0$. ∎

Given the previous result, we now turn to the proof of Theorem 5.42.

###### Proof of Theorem 5.42.

Set
$C=\theta_{\mathbf{j}\mathbf{l}}\sqrt{\gamma_{\text{max}}\kappa_{1}\kappa_{2}}$
and assume without loss of generality that $C>0$. Observe that for $\eta>0$,
$\delta_{*}>0$ small enough depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,\zeta,\xi$, there exists
$\eta^{\prime}>0$ with $\eta^{\prime}<1-\eta^{\prime}$ depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,\zeta,\xi$ such that
$\delta\in(0,\delta_{*})$ implies

$\displaystyle\mathbf{P}\\{\tau\in
B_{\eta,\delta}\\}=\int_{B_{\eta,\delta}}\frac{1}{h}e^{-t/h}\,dt$
$\displaystyle\geq\int_{0}^{\infty}\mathbf{1}\\{\eta^{\prime}\leq|\text{sn}_{\rho}(Ct/\delta+\theta_{0})|\leq
1-\eta^{\prime}\\}\frac{1}{h}e^{-t/h}\,dt$
$\displaystyle=\frac{\delta}{C}e^{\theta_{0}\frac{\delta}{Ch}}\int_{\theta_{0}}^{\infty}\mathbf{1}\\{\eta^{\prime}\leq|\text{sn}_{\rho}(t)|\leq
1-\eta^{\prime}\\}\frac{1}{h}e^{-\frac{\delta t}{Ch}}\,dt$
$\displaystyle\geq\frac{\delta}{C}e^{\theta_{0}\frac{\delta}{Ch}}\sum_{j=1}^{\infty}\int_{4jK_{\rho}}^{(4j+1)K_{\rho}}\mathbf{1}\\{\eta^{\prime}\leq\text{sn}_{\rho}(t)\leq
1-\eta^{\prime}\\}\frac{1}{h}e^{-\frac{\delta t}{Ch}}\,dt$
$\displaystyle\geq\frac{\delta}{C}e^{\theta_{0}\frac{\delta}{Ch}}\sum_{j=1}^{\infty}e^{-\frac{\delta
K_{\rho}}{Ch}(4j+1)}\int_{4jK_{\rho}}^{(4j+1)K_{\rho}}\mathbf{1}\\{\eta^{\prime}\leq\text{sn}_{\rho}(t)\leq
1-\eta^{\prime}\\}dt$
$\displaystyle=\frac{\delta}{C}\exp\Big{(}\frac{\theta_{0}\delta}{Ch}\Big{)}(T_{\rho}(1-\eta^{\prime})-T_{\rho}(\eta^{\prime}))\sum_{j=1}^{\infty}e^{-\frac{\delta
K_{\rho}}{Ch}(4j+1)}.$

Next, we note that

$\displaystyle\frac{\delta}{C}\exp\Big{(}\frac{\theta_{0}\delta}{Ch}\Big{)}(T_{\rho}(1-\eta^{\prime})-T_{\rho}(\eta^{\prime}))\sum_{j=1}^{\infty}e^{-\frac{\delta
K_{\rho}}{Ch}(4j+1)}$ $\displaystyle=\frac{\frac{\delta}{C}}{1-e^{-4\delta
K_{\rho}/Ch}}\exp\Big{(}\frac{\theta_{0}\delta}{Ch}-\frac{5\delta
K_{\rho}}{Ch}\Big{)}[T_{\rho}(1-\eta^{\prime})-T_{\rho}(\eta^{\prime})].$

Using Lemma 5.60 and the definition of $T_{\rho}(s)$ we see that there exists
$c_{1}=c_{1}(h)>0$ independent of $\delta\in(0,\delta_{*})$ for which

(5.64) $\displaystyle\exp\Big{(}\frac{\theta_{0}\delta}{Ch}-\frac{t\delta
K_{\rho}}{Ch}\Big{)}[T_{\rho}(1-\eta^{\prime})-T_{\rho}(\eta^{\prime})]\geq
c_{1}.$

Also, using Lemma 5.60, we find that as $\delta\rightarrow 0$,

$\displaystyle 1-e^{-4\delta K_{\rho}/Ch}=\frac{4\delta
K_{\rho}}{Ch}+O(\delta^{2}K_{\rho}^{2}).$

Hence, there exists $\delta_{*}>0$ small enough depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,\zeta,\xi,h$ and a constant
$c_{2}=c_{2}(h)>0$ independent of $\delta\in(0,\delta_{*})$ for which

(5.65) $\displaystyle\frac{\frac{\delta}{C}}{1-e^{-4\delta
K_{\rho}/Ch}}\geq\frac{c_{2}}{|\log\delta|}\,\,\,\text{ for all
}\delta\in(0,\delta_{*}).$

Setting $c=c_{1}c_{2}$ and combining (5.64) with (5.65) completes the proof. ∎

###  5.5. Proof of Theorem 5.24

Given the analysis of the transfer of energy in the triples (5.100) of the
previous section, we are now ready to prove the main result of this section,
Theorem 5.24. Due to its length, the proof will be split into two parts
depending on what assumption on the forced and damped modes, either (DF1) or
(DF2), is satisfied.

###### Proof of Theorem 5.24 under Assumption 5.21 (DF1).

Suppose that Assumption 5.21 (DF1) is satisfied and let $R>3$ and $q=(a,b)\in
H_{>R+1}$ so that $|q|>R$. Then there exists $\mathbf{k}\in\mathcal{Z}^{2}$
such that either

$\displaystyle|a_{\mathbf{k}}|\geq\frac{|q|}{\sqrt{d}}\,\,\,\,\text{ or
}\,\,\,\,|b_{\mathbf{k}}|\geq\frac{|q|}{\sqrt{d}}.$

Without loss of generality, we suppose that $|a_{\mathbf{k}}|\geq|q|/\sqrt{d}$
and let $S=S_{|\mathscr{S}|}$ for simplicity.

If $\mathbf{k}\in\mathscr{D}$, then $q\in D_{\frac{1}{d}}$ and hence

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{1/d}))\geq\mathbf{P}(u_{10}=\text{damp})=\frac{1}{|S|}>0.$

On the other hand, if $\mathbf{k}\notin\mathscr{D}$ then
$\mathbf{k}\notin\\{(1,0),(0,1),(N,N)\\}$ and hence $|\mathbf{k}|>1$. In this
case, either $(1,0)+\mathbf{k}\in\mathcal{Z}^{2}$ or
$(0,1)+\mathbf{k}\in\mathcal{Z}^{2}$. Let $\mathbf{j}\in\\{(1,0),(0,1)\\}$ be
such that $\mathbf{l}=\mathbf{j}+\mathbf{k}\in\mathcal{Z}^{2}$.

For notational simplicity in what follows, let $H_{0}=H(q)$,
$\delta_{0}=1/H_{0}$, $x_{0}=a_{\mathbf{j}}/H_{0}$,
$y_{0}=a_{\mathbf{k}}/H_{0}$ and $z_{0}=a_{\mathbf{l}}/H_{0}$. Observe that

(5.66)
$\displaystyle\xi_{0}:=\frac{1}{2d}\leq\mathcal{E}(x_{0},y_{0},z_{0})\leq 1.$

Furthermore,

(5.67) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})$
$\displaystyle\geq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{l}|^{2}}+\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}y^{2}_{0}$
$\displaystyle\geq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{l}|^{2}}+\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\xi_{0}$

and

(5.68) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})$
$\displaystyle\leq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{j}|^{2}}-\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}y^{2}_{0}$
$\displaystyle\leq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{j}|^{2}}-\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}\xi_{0}.$

In particular, if $\zeta_{0}=\zeta_{0}(\mathbf{j},\mathbf{k})$ as in (5.36) we
have that

(5.69)
$\displaystyle\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{l}|^{2}}+\zeta_{0}\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})\leq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{j}|^{2}}-\zeta_{0}.$

_Case 1_ : The quantity
$E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})$ does _not_ belong to
the interval

(5.70)
$\displaystyle\bigg{[}\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}-\zeta_{0}\delta_{0},\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}+\zeta_{0}\delta_{0}\bigg{]}.$

In particular, in this case Assumption 5.37 is satisfied, where in the
statements $(x_{0},y_{0},z_{0})$ replaces the triple $(x,y,z)$ and
$\delta_{0}$ replaces $\delta$. Applying Theorem 5.42 and using the fact that
$\mathbf{j}\in\mathscr{D}$, we find that there exists constants
$\eta_{1}=\eta_{1}(|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h)>0$,
$c_{1}=c_{1}(|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h)>0$,
$\delta_{1}(|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h)\in(0,1/4)$ such that
$R\geq 1/\delta_{1}$ implies

(5.71)
$\displaystyle\mathbf{P}(\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)\in
D_{\eta_{1}})\geq\frac{c_{1}}{|\log H(q)|}.$

In particular, using independence we obtain

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{\eta_{1}}))$
$\displaystyle\geq\mathbf{P}\Big{(}\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)\in
D_{\eta_{1}},u_{10}=a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},u_{20}=\text{damp}\Big{)}$
$\displaystyle\geq\frac{c_{1}}{\log|H(q)|}\frac{1}{|S|(|S|-1)}.$

_Case 2_ : The quantity
$E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})$ belongs to the
interval

(5.72)
$\displaystyle\bigg{[}\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}-\zeta_{0}\delta_{0},\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}+\zeta_{0}\delta_{0}\bigg{]}.$

This part of the argument is more involved because we need to first force the
random dynamics out of this interval. Since $\mathbf{j}\in\mathscr{F}$, we may
suppose without loss of generality that $\beta^{1}$ has
$\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})>0$ where
$\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$ was defined in (5.16). The
argument when $\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})<0$ is
similar. Let

$\displaystyle
q^{\prime}=(a^{\prime},b^{\prime}):=\varphi_{\tau_{10}}^{1}(q),$

and notice that by definition of $V_{1}$

$\displaystyle
q^{\prime}=q+\beta^{1}\tau_{10}=(a_{\mathbf{n}}+\beta_{a_{\mathbf{n}}}^{1}\tau_{10},b_{\mathbf{n}}+\beta_{b_{\mathbf{n}}}^{1}\tau_{10})_{\mathbf{n}\in\mathscr{Z}^{2}}.$

In this proof, to connect with the notation used in the analysis of (5.27)
above, we offer a convenient abuse of notation and set $H:=H(q^{\prime})$,
$\delta=1/H$, $x=a_{\mathbf{j}}^{\prime}/H$, $y=a_{\mathbf{k}}^{\prime}/H$,
$z=a_{\mathbf{l}}^{\prime}/H$.

First, suppose furthermore that for some $C>0$

(5.73) $\displaystyle
a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\geq-C.$

Then we have

(5.74) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)$
$\displaystyle=\frac{H_{0}^{2}}{H^{2}}\bigg{\\{}E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})+2\delta_{0}^{2}\tau_{10}\bigg{(}\frac{a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}}{|\mathbf{j}|^{2}}+\frac{a_{\mathbf{k}}\beta_{a_{\mathbf{k}}}^{1}}{|\mathbf{k}|^{2}}+\frac{a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}}{|\mathbf{l}|^{2}}\bigg{)}$
$\displaystyle\qquad\qquad\qquad+\delta_{0}^{2}\tau_{10}^{2}E_{\mathbf{j}\mathbf{k}\mathbf{l}}(\beta_{a_{\mathbf{j}}}^{1},\beta_{a_{\mathbf{k}}}^{1},\beta_{a_{\mathbf{l}}}^{1})\bigg{\\}}$
$\displaystyle\geq\frac{H_{0}^{2}}{H^{2}}\bigg{\\{}\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}-\zeta_{0}\delta_{0}^{2}+2\delta_{0}^{2}\tau_{10}\bigg{(}\frac{a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}}{|\mathbf{j}|^{2}}+\frac{a_{\mathbf{k}}\beta_{a_{\mathbf{k}}}^{1}}{|\mathbf{k}|^{2}}+\frac{a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}}{|\mathbf{l}|^{2}}\bigg{)}$
$\displaystyle\qquad\qquad+\delta_{0}^{2}\tau_{10}^{2}E_{\mathbf{j}\mathbf{k}\mathbf{l}}(\beta_{a_{\mathbf{j}}}^{1},\beta_{a_{\mathbf{k}}}^{1},\beta_{a_{\mathbf{l}}}^{1})\bigg{\\}}.$

Hence on the event

$\displaystyle
B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}:=\left\\{1+\frac{2C+2\zeta_{0}}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\leq\tau_{10}\leq
2+\frac{2C+2\zeta_{0}}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\right\\},$

we have

(5.75) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)$
$\displaystyle\geq\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}-\zeta_{0}\delta^{2}+2\delta^{2}\tau_{10}\bigg{\\{}a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\bigg{\\}}$
$\displaystyle\qquad+\delta^{2}\tau^{2}_{10}\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})$
$\displaystyle\geq\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}-\zeta_{0}\delta^{2}-2C\delta^{2}\tau_{10}+\delta^{2}\tau^{2}_{10}\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})\geq\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{0}\delta^{2}.$

Also, provided we choose $R>0$ large enough so that

(5.76) $\displaystyle|q|\geq
R\geq\frac{4|\beta^{1}|}{\xi_{0}}\bigg{(}2+\frac{2C+2\zeta_{0}}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\bigg{)}+1,$

we have on $B^{1}_{\mathbf{j}\mathbf{k}\mathbf{l}}$

(5.77)
$\displaystyle\frac{H_{0}}{H}=\frac{|q|+1}{|q+\beta^{1}\tau_{10}|+1}\geq\frac{|q|+1}{|q|+|\beta^{1}|\tau_{10}+1}=\frac{1}{1+\frac{|\beta^{1}|\tau_{10}}{|q|+1}}\geq\frac{1}{2}.$

Furthermore, on $B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$ we also have by
(5.66) and Cauchy-Schwarz

(5.78) $\displaystyle 1$ $\displaystyle\geq\mathcal{E}(x,y,z)$ (5.79)
$\displaystyle=\frac{H_{0}^{2}}{H^{2}}\bigg{\\{}\mathcal{E}(x_{0},y_{0},z_{0})+2\tau_{10}\delta_{0}^{2}(a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}+a_{\mathbf{k}}\beta_{a_{\mathbf{k}}}^{1}+a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1})+\tau_{10}^{2}\delta_{0}^{2}\mathcal{E}(\beta_{a_{\mathbf{j}}}^{1},\beta_{a_{\mathbf{k}}}^{1},\beta_{a_{\mathbf{l}}}^{1})\bigg{\\}}$
(5.80)
$\displaystyle\geq\frac{H_{0}^{2}}{H^{2}}\bigg{\\{}\xi_{0}-|q|^{-1}2\Big{(}2+\tfrac{2C+2\zeta_{0}}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\Big{)}|\beta^{1}|\bigg{\\}}\geq\frac{\xi_{0}}{16}=:\xi_{1},\,\,\,\text{
and}$ (5.81) $\displaystyle|y|$
$\displaystyle=\frac{H_{0}}{H}|y_{0}+\tau_{10}\beta_{a_{\mathbf{k}}}^{1}/H_{0}|\geq\tfrac{1}{2}\sqrt{\xi_{0}}-\frac{\tau_{10}|\beta^{1}|}{|q|+1}\geq\frac{\sqrt{\xi_{0}}}{4}$

where we used the choice (5.76) and (5.77). Using the same argument as in
(5.68), we also have on $B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$

(5.82) $\displaystyle
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{j}|^{2}}-\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}y^{2}\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{j}|^{2}}-\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}\xi_{1}.$

Let

$\displaystyle\zeta_{1}:=\min\bigg{\\{}\xi_{0},\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}\xi_{1},\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\xi_{1}\bigg{\\}}.$

Then, under the hypothesis (5.73), we have on the event
$B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$

$\displaystyle\xi_{1}\leq\mathcal{E}(x,y,z)\leq 1\qquad\text{ and
}\qquad\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{1}\delta^{2}\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{j}|^{2}}-\zeta_{1}.$

In particular, Assumption 5.37 (A1) is satisfied for
$q^{\prime}=\varphi_{\tau_{10}}^{1}(q)$ on
$B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$. Applying Theorem 5.42, there exist
constants $\eta_{2}\in(0,1/4),c_{2}>0,\delta_{2}\in(0,1/4)$ depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h$ such that $R\geq 1/\delta_{2}$

$\displaystyle\mathbf{P}\big{(}\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{2}}\big{)}\geq\frac{c_{2}}{|\log H(q)|}.$

Hence using independence

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{\eta_{2}}))$
$\displaystyle\geq\mathbf{P}\big{(}\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{2}},u_{10}=1,u_{20}=a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},u_{30}=\text{damp}\big{)}$
$\displaystyle\geq\frac{c_{2}}{|\log H(q)|}\frac{1}{|S|(|S|-1)(|S|-2)}.$

On the other hand, suppose now that

(5.83)
$\displaystyle\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}-\zeta_{0}\delta_{0}^{2}\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})\leq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}+\zeta_{0}\delta_{0}^{2}\qquad\text{
and }$ $\displaystyle\qquad\qquad\qquad
a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\leq-C.$

Picking $C=\zeta_{0}+\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})$,
we consider the event

(5.84) $\displaystyle
B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}:=\bigg{\\{}1\leq\tau_{10}\leq
2\bigg{\\}}.$

Observe that

(5.85) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)$
$\displaystyle=\frac{H_{0}^{2}}{H^{2}}\bigg{\\{}E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x_{0},y_{0},z_{0})+2\delta_{0}^{2}\tau_{10}\bigg{(}\frac{a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}}{|\mathbf{j}|^{2}}+\frac{a_{\mathbf{k}}\beta_{a_{\mathbf{k}}}^{1}}{|\mathbf{k}|^{2}}+\frac{a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}}{|\mathbf{l}|^{2}}\bigg{)}$
$\displaystyle\qquad\qquad+\delta_{0}^{2}\tau_{10}^{2}E_{\mathbf{j}\mathbf{k}\mathbf{l}}(\beta_{a_{\mathbf{j}}}^{1},\beta_{a_{\mathbf{k}}}^{1},\beta_{a_{\mathbf{l}}}^{1})\bigg{\\}}$
$\displaystyle\leq\frac{H_{0}^{2}}{H^{2}}\bigg{\\{}\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{k}|^{2}}+\zeta_{0}\delta_{0}^{2}+2\delta_{0}^{2}\tau_{10}\bigg{(}\frac{a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}}{|\mathbf{j}|^{2}}+\frac{a_{\mathbf{k}}\beta_{a_{\mathbf{k}}}^{1}}{|\mathbf{k}|^{2}}+\frac{a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}}{|\mathbf{l}|^{2}}\bigg{)}$
$\displaystyle\qquad\qquad+\delta_{0}^{2}\tau_{10}^{2}E_{\mathbf{j}\mathbf{k}\mathbf{l}}(\beta_{a_{\mathbf{j}}}^{1},\beta_{a_{\mathbf{k}}}^{1},\beta_{a_{\mathbf{l}}}^{1})\bigg{\\}}.$

Thus on $B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$ we have

(5.86) $\displaystyle E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)$
$\displaystyle\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{0}\delta^{2}+2\delta^{2}\tau_{10}\bigg{\\{}a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\bigg{\\}}$
$\displaystyle\qquad\qquad+\delta^{2}\tau^{2}_{10}\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})$
$\displaystyle\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{0}\delta^{2}-2C\delta^{2}\tau_{10}+\delta^{2}\tau^{2}_{10}\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}-\zeta_{0}\delta^{2}$

and

(5.87)
$\displaystyle\frac{H_{0}}{H}=\frac{|q|+1}{|q+\beta^{1}\tau_{10}|+1}\geq\frac{|q|+1}{|q|+|\beta^{1}|\tau_{10}+1}=\frac{1}{1+\frac{|\beta^{1}|\tau_{10}}{|q|+1}}\geq\frac{1}{2},$

where we used the choice of $R$ in (5.76). Moreover, on
$B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$ we also have (5.78)-(5.80) by the
same arguments used above. Lastly, we note that on
$B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$ we have by the same arguments in
(5.81) and (5.82)

$\displaystyle
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\geq\frac{\mathcal{E}(x,y,z)}{|\mathbf{l}|^{2}}+\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}y^{2}\geq\frac{\mathcal{E}(x_{0},y_{0},z_{0})}{|\mathbf{l}|^{2}}+\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\xi_{1}.$

In particular, we have shown that on the event
$B^{2}_{\mathbf{j}\mathbf{k}\mathbf{l}}$

$\displaystyle\xi_{1}\leq\mathcal{E}(x,y,z)\leq 1\qquad\text{ and
}\qquad\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{1}\leq\mathcal{E}_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{j}|^{2}}-\zeta_{1}\delta^{2}.$

In particular, Assumption 5.37 (A2) is satisfied for
$q^{\prime}=\varphi_{\tau_{10}}^{1}(q)$ on
$B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$. Applying Theorem 5.42 and adjusting
the constants $\eta_{2},c_{2},\delta_{2}$ accordingly, it follows that if
$R\geq 1/\delta_{2}$ we have

$\displaystyle\mathbf{P}(\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{2}})\geq\frac{c_{2}}{|\log H(q)|}.$

Hence using independence

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{\eta_{2}}))$
$\displaystyle\geq\mathbf{P}\Big{(}\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{2}},u_{10}=1,u_{20}=a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},u_{30}=\text{damp}\Big{)}$
$\displaystyle\geq\frac{c_{2}}{|\log H(q)|}\frac{1}{|S|(|S|-1)(|S|-2)}.$

This finishes the proof.

∎

###### Proof of Theorem 5.24 under Assumption 5.21 (DF2).

Without loss of generality, throughout this argument we suppose that
$\\{(1,0),(N,N)\\}\subset\mathscr{D}$ as the other case is similar. Suppose
$q=(a,b)\in H_{>R+1}$ so that $|q|\geq R$ for some $R>3$ sufficiently large to
be determined below and suppose without loss of generality that
$a_{\mathbf{k}}$, $\mathbf{k}\in\mathcal{Z}^{2}$, has

$\displaystyle|a_{\mathbf{k}}|\geq\frac{|q|}{\sqrt{d}}.$

The case when $|b_{\mathbf{k}}|\geq|q|/\sqrt{d}$ proceeds in a nearly
identical fashion. Below, we only provide the details in cases of initial data
whose arguments are different than those considered under Assumption 5.21
(DF1).

_Case 1_. Suppose that $\mathbf{j}=(0,1)$,
$\mathbf{l}=\mathbf{j}+\mathbf{k}\in\mathcal{Z}^{2}$ and

$\displaystyle
q=(a,b)\notin\bigg{\\{}\frac{\mathcal{E}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})}{|\mathbf{k}|^{2}}+\zeta_{0}(\mathbf{j},\mathbf{k})\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})\leq\frac{\mathcal{E}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})}{|\mathbf{k}|^{2}}-\zeta_{0}(\mathbf{j},\mathbf{k})\bigg{\\}}.$

Using the arguments in the proof of Theorem 5.24 (DF1) _Case 1_ , it follows
that there exists constants
$\eta_{1}=\eta_{1}(|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h)>0$,
$c_{1}=c_{1}(|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h)>0$,
$\delta_{1}(|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h)>0$ such that $R\geq
1/\delta_{1}$ implies

(5.88)
$\displaystyle\mathbf{P}(|\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)\cdot
e_{a_{\mathbf{j}}}|\geq\eta_{1}|H(q)|)\geq\frac{c_{1}}{|\log H(q)|}.$

Note that the difference here under (DF2) versus (DF1) is that $\mathbf{j}$ is
not necessarily damped, so
$\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)$ may not
belong to $D_{\eta_{1}}$. However, letting $\mathbf{l}^{\prime}:=(1,1)$ and
noting that $\mathbf{l}^{\prime}\in\mathscr{F}$, we have that
$\beta_{a_{\mathbf{l}^{\prime}}}^{\ell}\neq 0$ for some
$\ell=1,2,\ldots,\mathfrak{m}$. Suppose without loss of generality that
$\beta_{a_{\mathbf{l}^{\prime}}}^{1}>0$ and define

$\displaystyle
q^{\prime}=(a^{\prime},b^{\prime})=\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)\quad\text{and}\quad
q^{\prime\prime}=(a^{\prime\prime},b^{\prime\prime})=\varphi_{\tau_{20}}^{1}\circ
q^{\prime}.$

If $a_{\mathbf{l}^{\prime}}\geq-1$, then on the event

$\displaystyle\bigg{\\{}\frac{2}{\beta_{a_{\mathbf{l}^{\prime}}}^{1}}\leq\tau_{20}\leq\frac{4}{\beta_{a_{\mathbf{l}^{\prime}}}^{1}}\bigg{\\}}$

and ensuring $R>0$ is large enough so that

$\displaystyle
R\geq\frac{8}{\eta_{1}}\frac{|\beta^{1}|}{\beta^{1}_{a_{\mathbf{l}^{\prime}}}}+1$

we have

$\displaystyle
a_{\mathbf{l}^{\prime}}^{\prime\prime}=a_{\mathbf{l}^{\prime}}+\beta^{1}_{a_{\mathbf{l}^{\prime}}}\tau_{20}\geq-1+2=1,\quad\frac{H(q^{\prime\prime})}{H(q^{\prime})}\geq\frac{|q^{\prime}|+1-|\beta^{1}|\tau_{20}}{|q^{\prime}|+1}\geq
1-\frac{|\beta^{1}|\tau_{20}}{|q|}\geq 1/2,$
$\displaystyle\frac{H(q^{\prime})}{H(q^{\prime\prime})}\geq\frac{1}{1+|\beta^{1}|\tau_{20}/H(q)}\geq\frac{1}{2},\quad|a_{\mathbf{j}}^{\prime\prime}|=|a_{\mathbf{j}}^{\prime}+\beta^{1}_{a_{\mathbf{j}}}\tau_{20}|\geq\eta_{1}H(q)-|\beta^{1}|\tau_{20}\geq\frac{\eta_{1}}{4}H(q^{\prime\prime}).$

Thus letting $\mathbf{k}^{\prime}=(1,0)$ and applying Lemma 5.30, we find that
there exists $\eta_{3},c_{3}>0$ small enough such that for all $R>0$ large
enough

(5.89)
$\displaystyle\mathbf{P}\Big{(}\varphi_{\tau_{30}}^{a_{\mathbf{j}}a_{\mathbf{k}^{\prime}}a_{\mathbf{l}^{\prime}}}\circ\varphi_{\tau_{20}}^{1}\circ\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)\in
D_{\eta_{3}}\Big{)}\geq\frac{c_{3}}{|\log(H(q))|}.$

In particular,

$\displaystyle\mathbf{P}_{q}\Big{(}A_{0}^{\text{damp}}(D_{\eta_{3}})\Big{)}$
$\displaystyle\geq\mathbf{P}\Big{(}\varphi_{\tau_{30}}^{a_{\mathbf{j}}a_{\mathbf{k}^{\prime}}a_{\mathbf{l}^{\prime}}}\circ\varphi_{\tau_{20}}^{1}\circ\varphi_{\tau_{10}}^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}(q)\in
D_{\eta_{3}},u_{10}=a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},u_{20}=1,u_{30}=a_{\mathbf{j}}a_{\mathbf{k}^{\prime}}a_{\mathbf{l}^{\prime}}\Big{)}$
$\displaystyle\geq\frac{c_{3}}{|\log H(q)|}\frac{1}{|S|(|S|-1)(|S|-2)}.$

One can apply a similar argument in the case when
$a_{\mathbf{l}^{\prime}}\leq-1$ to obtain the same conclusion, provided we
adjust $\eta_{3},c_{3}$ and $R$ accordingly.

_Case 2_ : Suppose that $\mathbf{j}=(0,1)$,
$\mathbf{l}=\mathbf{j}+\mathbf{k}\in\mathcal{Z}^{2}$ and

$\displaystyle
q=(a,b)\in\bigg{\\{}\frac{\mathcal{E}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})}{|\mathbf{k}|^{2}}+\zeta_{0}(\mathbf{j},\mathbf{k})\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})\leq\frac{\mathcal{E}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})}{|\mathbf{k}|^{2}}-\zeta_{0}(\mathbf{j},\mathbf{k})\bigg{\\}}.$

In this case, we slightly modify the arguments used in the proof of Theorem
5.24 (DF1) _Case 2_. Since $\mathbf{j}=(0,1)\in\mathscr{F}$, we may again
suppose without loss of generality that $\beta^{1}$ has
$\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})>0$. Let
$\mathbf{m}=(1,1)$ and recall that $\mathbf{m}\in\mathscr{F}$. We suppose
without loss of generality that $\beta^{1}_{a_{\mathbf{m}}}\neq 0$.

Let

$\displaystyle
q^{\prime}=(a^{\prime},b^{\prime}):=\varphi_{\tau_{10}}^{1}(q)\quad\text{ and
}\quad
q^{\prime\prime}=(a^{\prime\prime},b^{\prime\prime})=\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q),$

In this proof, to connect with the notation used in the analysis of (5.27)
above, we again offer a convenient abuse of notation and set
$H:=H(q^{\prime})$, $\delta=1/H$, $x=a_{\mathbf{j}}^{\prime}/H$,
$y=a_{\mathbf{k}}^{\prime}/H$, $z=a_{\mathbf{l}}^{\prime}/H$.

First, suppose furthermore that

(5.90) $\displaystyle
a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\geq-(\zeta_{0}+\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})).$

Observe that on the event

$\displaystyle
B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}:=\left\\{1+\frac{4\zeta_{0}+2\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\leq\tau_{10}\leq
2+\frac{4\zeta_{0}+2\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\right\\},$

for $R>0$ large enough following the calculations starting at (5.75), we have

$\displaystyle\xi_{1}\leq\mathcal{E}(x,y,z)\leq 1\qquad\text{ and
}\qquad\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{1}\delta^{2}\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{j}|^{2}}-\zeta_{1}.$

On the other hand

$\displaystyle|a_{\mathbf{m}}^{\prime}|=|a_{\mathbf{m}}+\beta^{1}_{a_{\mathbf{m}}}\tau_{10}|\leq\frac{|\beta^{1}_{a_{\mathbf{m}}}|}{4}\quad\text{
if and only if
}\quad-\frac{1}{4}-\frac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}}\leq\tau_{10}\leq\frac{1}{4}-\frac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}}.$

In particular, because $|(-1/4+\alpha,\alpha+1/4)|=1/2$ for any
$\alpha\in\mathbf{R}$ where $|\cdot|$ denotes Lebesgue measure, the event

$\displaystyle\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}:=B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}\cap\Big{\\{}\tau_{10}\notin[-\tfrac{1}{4}-\tfrac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}},\tfrac{1}{4}-\tfrac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}}]\Big{\\}}$

satisfies

$\displaystyle\mathbf{P}(\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1})\geq\mathbf{P}\bigg{\\{}\frac{3}{2}+\frac{4\zeta_{0}+2\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\leq\tau_{10}\leq
2+\frac{4\zeta_{0}+2\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}{\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})}\bigg{\\}}>0.$

Hence Assumption 5.37 (A1) is satisfied for
$q^{\prime}=\varphi_{\tau_{10}}^{1}(q)$ on
$\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$ provided (5.90) is also
satisfied. Applying Theorem 5.42, there exist constants
$\eta_{4}\in(0,1/4),c_{4}>0,\delta_{4}\in(0,1)$ depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h$ such that $R\geq 1/\delta_{4}$
and $q$ satisfying the above hypotheses implies

$\displaystyle\mathbf{P}\Big{(}|\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\cdot
a_{\mathbf{j}}|\geq\eta_{4}H(q)\Big{)}\geq\frac{c_{4}}{|\log H(q)|}.$

Additionally, we have that
$|a_{\mathbf{m}}^{\prime}|\geq|\beta^{1}_{a_{\mathbf{m}}}|/4>0$ on
$\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}$. Letting $\mathbf{n}=(1,0)$
and applying Lemma 5.30, there exists $\eta_{5},\delta_{5},c_{5}>0$ small
enough so that $R\geq 1/\delta_{5}$ and $q$ satisfying the hypotheses above
has

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{\eta_{5}}))$
$\displaystyle\geq\mathbf{P}\Big{(}\varphi_{\tau_{30}}^{a_{\mathbf{n}}a_{\mathbf{j}}a_{\mathbf{m}}}\circ\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{5}},u_{10}=1,u_{20}=a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},u_{30}=a_{\mathbf{n}}a_{\mathbf{j}}a_{\mathbf{m}}\Big{)}$
$\displaystyle\geq\frac{c_{5}}{|\log H(q)|}\frac{1}{|S|(|S|-1)(|S|-2)}.$

On the other hand, suppose that

(5.91)
$\displaystyle\frac{\mathcal{E}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})}{|\mathbf{k}|^{2}}-\zeta_{0}\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})\leq\frac{\mathcal{E}(a_{\mathbf{j}},a_{\mathbf{k}},a_{\mathbf{l}})}{|\mathbf{k}|^{2}}+\zeta_{0}\qquad\text{
and }$ $\displaystyle\qquad\qquad\qquad
a_{\mathbf{j}}\beta_{a_{\mathbf{j}}}^{1}\bigg{(}\frac{1}{|\mathbf{j}|^{2}}-\frac{1}{|\mathbf{k}|^{2}}\bigg{)}-a_{\mathbf{l}}\beta_{a_{\mathbf{l}}}^{1}\bigg{(}\frac{1}{|\mathbf{k}|^{2}}-\frac{1}{|\mathbf{l}|^{2}}\bigg{)}\leq-(\zeta_{0}+\Delta_{\mathbf{j}\mathbf{k}\mathbf{l}}^{1}(\beta^{1})).$

Recalling the event

(5.92) $\displaystyle
B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}=\bigg{\\{}1\leq\tau_{10}\leq
2\bigg{\\}},$

observe that on $B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$, following the
calculations starting at (5.85) we have

$\displaystyle\xi_{1}\leq\mathcal{E}(x,y,z)\leq 1\qquad\text{ and
}\qquad\frac{\mathcal{E}(x,y,z)}{|\mathbf{k}|^{2}}+\zeta_{1}\leq
E_{\mathbf{j}\mathbf{k}\mathbf{l}}(x,y,z)\leq\frac{\mathcal{E}(x,y,z)}{|\mathbf{j}|^{2}}-\zeta_{1}\delta^{2}.$

On the other hand

$\displaystyle|a_{\mathbf{m}}^{\prime}|=|a_{\mathbf{m}}+\beta^{1}_{a_{\mathbf{m}}}\tau_{10}|\leq\frac{|\beta^{1}_{a_{\mathbf{m}}}|}{4}\quad\text{
if and only if
}\quad-\frac{1}{4}-\frac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}}\leq\tau_{10}\leq\frac{1}{4}-\frac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}}.$

In particular, because $|(-1/4+\alpha,\alpha+1/4)|=1/2$ for any
$\alpha\in\mathbf{R}$, the event

$\displaystyle\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}:=B_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}\cap\Big{\\{}\tau_{10}\notin[-\tfrac{1}{4}-\tfrac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}},\tfrac{1}{4}-\tfrac{a_{\mathbf{m}}}{\beta^{1}_{a_{\mathbf{m}}}}]\Big{\\}}$

satisfies

$\displaystyle\mathbf{P}(\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2})\geq\mathbf{P}\bigg{\\{}\frac{3}{2}\leq\tau_{10}\leq
2\bigg{\\}}>0.$

By the above, we note that Assumption 5.37 (A2) is satisfied for
$q^{\prime}=\varphi_{\tau_{10}}^{1}(q)$ on
$\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$ provided (5.91) is also
satisfied. Applying Theorem 5.42, there exist constants
$\eta_{6}\in(0,1/4),c_{6}>0,\delta_{6}\in(0,1)$ depending only on
$|\mathbf{j}|,|\mathbf{k}|,|\mathbf{l}|,d,h$ such that $R\geq 1/\delta_{6}$
implies

$\displaystyle\mathbf{P}\Big{(}|\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\cdot
a_{\mathbf{j}}|\geq\eta_{6}H(q)\Big{)}\geq\frac{c_{6}}{|\log H(q)|}.$

Additionally, we have that
$|a_{\mathbf{m}}^{\prime}|\geq|\beta^{1}_{a_{\mathbf{m}}}|/4>0$ on
$\tilde{B}_{\mathbf{j}\mathbf{k}\mathbf{l}}^{2}$. Letting $\mathbf{n}=(1,0)$
and applying Lemma 5.30, there exists $\eta_{7},\delta_{7},c_{7}>0$ small
enough so that $R\geq 1/\delta_{7}$ and $q$ satisfying the above has

$\displaystyle\mathbf{P}\Big{(}\varphi_{\tau_{30}}^{a_{\mathbf{n}}a_{\mathbf{j}}a_{\mathbf{m}}}\circ\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{7}}\Big{)}\geq\frac{c_{7}}{|\log H(q)|}$

so that

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{\eta_{7}}))$
$\displaystyle\geq\mathbf{P}\Big{(}\varphi_{\tau_{30}}^{a_{\mathbf{n}}a_{\mathbf{j}}a_{\mathbf{m}}}\circ\varphi^{a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}}}_{\tau_{20}}\circ\varphi_{\tau_{10}}^{1}(q)\in
D_{\eta_{7}},u_{10}=1,u_{20}=a_{\mathbf{j}}a_{\mathbf{k}}a_{\mathbf{l}},u_{30}=a_{\mathbf{n}}a_{\mathbf{j}}a_{\mathbf{m}}\Big{)}$
$\displaystyle\geq\frac{c_{7}}{|\log H(q)|}\frac{1}{|S|(|S|-1)(|S|-2)}.$

Putting the above arguments together, we have shown that we can choose $R>3$
large enough, $c_{*},\eta_{*}>0$ small enough so that for $q\in H_{R+1}$ we
have

$\displaystyle\mathbf{P}_{q}(A_{0}^{\text{damp}}(D_{\eta_{*}}))\geq\frac{c_{*}}{\log(H(q))}.$

This concludes the proof.

∎

## Appendix: Derivation of the Galerkin splittings from the Euler equation

In this section, we derive the splitting studied in Section 5 starting from
the two-dimensional forced and damped Euler equation on the torus
$\mathbf{T}^{2}=[0,2\pi]^{2}$. Most of the derivation is done in [3], but we
provide the details for completeness.

Recall that in vorticity formulation, the damped and forced two-dimensional
Euler equation on $\mathbf{T}^{2}$ reads

(5.93)
$\displaystyle\begin{cases}\partial_{t}q+(\mathcal{K}q\cdot\nabla)q+\Lambda
q=\beta,\\\ \text{div}(q)=0\end{cases}$

where $q$ is a scalar quantity called the _vorticity_ ,
$\mathcal{K}=\nabla^{\perp}(-\Delta)^{-1}$ is the _Biot-Savart_ operator, and
$\nabla^{\perp}:=(\partial_{2},-\partial_{1})$. Although we will describe
$\Lambda,\beta$ more precisely below, the linear operator $\Lambda$ should be
thought of as a partial Laplacian while $\beta$ is a background forcing term
acting on select frequencies.

Assuming there is no mean flow; that is,

(5.94) $\displaystyle\int_{\mathbf{T}^{2}}q(t,x)\,dx=0\,\,\,\text{ for all
}t\geq 0,$

we express the solution $q$ of (5.93) in Fourier space as

(5.95) $\displaystyle q(t,x)=\sum_{\mathbf{j}\in\mathbf{Z}^{2}_{\neq
0}}q_{\mathbf{j}}(t)e_{\mathbf{j}}(x)$

where $\\{e_{\mathbf{j}}\\}_{\mathbf{j}\in\mathbf{Z}_{\neq 0}^{2}}$ is the
orthonormal family on $L^{2}(\mathbf{T}^{2};\mathbf{R})$ given by

$e_{\mathbf{j}}(x):=\frac{1}{2\pi}\exp(ix\cdot\mathbf{j}).$

Let $\mathscr{D}\subset\mathbf{Z}_{\neq 0}^{2}$. We suppose that

(5.96) $\displaystyle\Lambda
q:=\sum_{\mathbf{j}\in\mathscr{D}}\lambda_{\mathbf{j}}q_{\mathbf{j}}e_{\mathbf{j}}\quad\text{
and }\quad\beta=\sum_{\mathbf{j}\in\mathbf{Z}_{\neq
0}^{2}}(\beta_{a_{\mathbf{j}}}+i\beta_{b_{\mathbf{j}}})e_{\mathbf{j}}$

where $\lambda_{\mathbf{j}}$, $\mathbf{j}\in\mathbf{Z}_{\neq 0}^{2}$ and
$\beta_{a_{\mathbf{j}}},\beta_{b_{\mathbf{j}}}\in\mathbf{R}$.

Plugging this information into (5.93) we arrive at the following equation

(5.97)
$\displaystyle\dot{q}=-\sum_{\mathbf{j}+\mathbf{k}+\mathbf{l}=0}\theta_{\mathbf{k}\mathbf{l}}\bar{q}_{\mathbf{k}}\bar{q}_{\mathbf{l}}e_{\mathbf{j}}-\sum_{\mathbf{j}\in\mathscr{D}}\lambda_{\mathbf{j}}q_{\mathbf{j}}e_{\mathbf{j}}+\sum_{\mathbf{j}\in\mathbf{Z}_{\neq
0}^{2}}(\beta_{a_{\mathbf{j}}}+i\beta_{b_{\mathbf{j}}})e_{\mathbf{j}}$

where, in the sum above, $\mathbf{j},\mathbf{k},\mathbf{l}\in\mathbf{Z}_{\neq
0}^{2}$ and the coefficients $\theta_{\mathbf{k}\mathbf{l}}$ in (5.97) are as
in (5.3).

Letting $q_{\mathbf{j}}=a_{\mathbf{j}}+ib_{\mathbf{j}}$,