# A study of the Kuramoto model for synchronization phenomena based on
degenerate Kolmogorov-Fokker-Planck equations

G. Pecorella S. Polidoro C. Vernia  Dipartimento di Scienze Fisiche,
Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via Campi
213/b, 41125 Modena (Italy). E-mail<EMAIL_ADDRESS>di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio
Emilia, Via Campi 213/b, 41125 Modena (Italy). E-mail:
<EMAIL_ADDRESS>di Scienze Fisiche, Informatiche e
Matematiche, Università di Modena e Reggio Emilia, Via Campi 213/b, 41125
Modena (Italy). E-mail<EMAIL_ADDRESS>
###### Abstract

We study a nonlinear partial differential equation that arises when
introducing inertial effects in the Kuramoto model. Based on the known theory
of degenerate Kolmogorov operators, we prove existence, uniqueness and a
priori estimates of the solution to the relevant Cauchy problem. Moreover, a
stable numerical operator, which is consistent with the degenerate Kolmogorov
operator, is introduced in order to produce numerical solutions. Finally,
numerical experiments show how the synchronization phenomena depend on the
parameters of the Kuramoto model with inertia.

###### Contents

  1. 1 Introduction
  2. 2 Fundamental Solution
  3. 3 Proof of Theorem 1.3.
  4. 4 Finite difference method

##  1 Introduction

One of the most fascinating phenomena in nature is the tendency to
synchronization. It pervades nature at every scale from the nucleus to the
cosmos, and even our heart exhibits this phenomenon. A cluster of about
$10,000$ cells, called the sinoatrial node, generates the electrical rhythm
that commands the rest of the heart to beat, and it must do so reliably,
minute after minute, for about three billion beats in a lifetime. These cells
are a collection of oscillators, i.e. entities that cycle automatically in
more or less regular time intervals, and need to coordinate their rhythm in
order to achieve some kind of synchronization.

Huygens was one of the pioneers in the study of synchronization. In February
of 1665 the Dutch physicist was confined to his bedroom for several days, and
observed a curious phenomenon: the two pendulum clocks in the room, separated
by two feet, kept oscillating together without any variation; moreover, mixing
up the swings of the pendulums the synchronization returned after some time.
Intrigued by this phenomenon, he carried out several experiments, finding out
that the synchronization could take place only if they could communicate in
some way.

In the following centuries many mathematicians studied the problem, coming up
with different models: some of these describe specific problems (see [18] for
the description of sinoatrial node synchronization), while others have been
created with the aim of being general enough to describe the dynamics of
different systems, even relevant to very different scientific fields such as
biology, physics, and social sciences as well. One of the most famous model
that belongs to the latter group is the one proposed by Kuramoto, from which
many variations have been devised. We refer to the article [22] by Spigler for
an overview of this model and some of its derivatives. We next describe the
Kuramoto model and some of its modifications we are interested in, then we
discuss our contributions on this subject.

The Kuramoto model is a system of ordinary differential equations introduced
in the late decades of the last century in [12], that describes the collective
behaviour of a population that can be seen as a group of oscillators. Let’s
consider a population of $N$ oscillators, we denote as $\vartheta_{k}(t)$ the
phase related to the $k$-th oscillator, and as $\Omega_{k}$ its natural
frequency, which is an intrinsic parameter of the $k$-th oscillator. We
suppose that these natural frequencies are distributed according to an
unimodal distribution $g=g(\Omega)$ that is normalized and symmetric with
respect to its mean frequency $\overline{\Omega}$. The governing equation is
given by the following system of ordinary differential equations

$\dot{\vartheta}_{k}=\Omega_{k}+\frac{K}{N}\sum_{j=1}^{N}\sin(\vartheta_{j}-\vartheta_{k}),\qquad\;k=1,...,N,$
(1.1)

where $K$ is a real positive constant sizing the coupling strength between the
oscillators. Note that the coupling in the above system is nonlinear, thus the
ensuing phenomena is expected to be rather complicated. On the other hand, a
mean-field coupling, which is present in (1.1), significantly simplify the
evolution of the system. In order to point out this coupling effect, we
introduce the following complex number, called order parameter

$r_{N}(t)\mathrm{e}^{i\psi_{N}(t)}=\frac{1}{N}\sum_{j=1}^{N}\mathrm{e}^{i\vartheta_{j}(t)}.$
(1.2)

The phase $\psi_{N}(t)$ is the mean phase of the system. The modulus
$|r_{N}(t)|$ provides us with a measure of the synchronization of the system.
When $|r_{N}(t)|=0$ the system is fully incoherent, when $0<|r_{N}(t)|<1$ the
system is partly synchronized, and $|r_{N}(t)|=1$ matches the full coherence
state. Equation (1.1) can be rewritten using the order parameter

$\dot{\vartheta}_{k}=\Omega_{k}+Kr_{N}\sin(\psi_{N}-\vartheta_{k}),\qquad\;k=1,...,N,$
(1.3)

which is the equation of an overdamped pendulum with torque $\Omega_{k}$ and
restoring force proportional to $Kr_{N}$. This formulation simplifies the
analytical treatment.

Despite the simplicity of this model, Wiesenfeld, Colet and Strogatz show in
[26, 27] that (1.3) describes some important physical phenomena such as the
interaction of quasi-optical oscillators with a cavity and some Josephson
arrays, and it gives us a starting point to study synchronization in generic
oscillators. Moreover, Kuramoto model shows how synchronization in a
population of many coupled oscillators may occur, and the type of
synchronization: indeed it may occur frequency synchronization, where each
oscillator completes its cycle at the same time, phase synchronization, where
each oscillator is at the same point in the cycle, or even a more complex
scenario.

The Kuramoto model can be extended to a population of infinitely many
oscillators, as it was described by Strogatz in [23]. Let’s consider a
continuum of oscillators, and let $\varrho=\varrho(\vartheta,{\Omega},t)$ be a
density function describing the fraction of oscillators at phase $\vartheta$
with natural frequency ${\Omega}$ at time $t$. Then $\varrho$ is a nonnegative
function $2\pi$ periodic in $\vartheta$ and satisfies

$\int_{0}^{2\pi}\varrho(\vartheta,{\Omega},t)d\vartheta=1,\qquad\forall\;({\Omega},t)\in\mathbb{R}\times[0,+\infty[.$
(1.4)

The evolution of $\varrho$ is described by the usual continuity equation

$\frac{\partial\varrho}{\partial
t}=-\frac{\partial}{\partial\vartheta}(\varrho v),$ (1.5)

which expresses conservation of oscillators of frequency ${\Omega}$. Here
$v=v(\vartheta,{\Omega},t)$ is the instantaneous velocity of an oscillator at
position $\vartheta$, given that it has natural frequency ${\Omega}$, and is
interpreted in the Eulerian sense. From (1.3) we see that that velocity is

$v(\vartheta,\Omega,t)=\Omega+Kr(t)\sin(\psi(t)-\vartheta),$ (1.6)

where $r(t)$ and $\psi(t)$ follow from (1.2) by letting $N\to+\infty$

$r(t)\mathrm{e}^{\mathrm{i}\psi(t)}=\int_{0}^{2\pi}\int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}\vartheta^{\prime}}\varrho(\vartheta^{\prime},\Omega^{\prime},t)g(\Omega^{\prime})d\Omega^{\prime}d\vartheta^{\prime}.$
(1.7)

Combining (1.5) and(1.7) we obtain the following nonlinear partial integro-
differential equation for the density $\varrho$

$\frac{\partial\varrho}{\partial
t}(\vartheta,\Omega,t)=-\frac{\partial}{\partial\vartheta}\left[\varrho(\vartheta,\Omega,t)\left(\Omega+K\int_{0}^{2\pi}\int_{\mathbb{R}}\sin(\vartheta^{\prime}-\vartheta)\varrho(\vartheta^{\prime},\omega^{\prime},t)g(\Omega^{\prime})d\Omega^{\prime}d\vartheta^{\prime}\right)\right].$
(1.8)

Sakaguchi in [21] modified the deterministic equations (1.3) and (1.8) by
adding noise to describe stochastic phenomena, in order to consider rapid
stochastic fluctuations in the natural frequencies. The governing equations
(1.3) for $N$ oscillators take now the form of a system of Langevin equations

$\dot{\vartheta}_{k}=\Omega_{k}+\xi_{k}+\frac{K}{N}\sum_{j=1}^{N}\sin(\vartheta_{j}-\vartheta_{k}),\qquad\;k=1,...,N,$
(1.9)

where $\xi(t)=D\,W_{t}$ being $(W_{t})_{t\geq 0}$ a $N$-dimensional Wiener
process, with $D>0$.

Then, Sakaguchi argued intuitively that since (1.9) is a system of Langevin
equations with mean-field coupling, as $N\rightarrow\infty$ the density
$\varrho(\vartheta,\omega,t)$ should satisfy the following Fokker–Planck
equation

$\frac{\partial\varrho}{\partial
t}=D\frac{\partial^{2}\varrho}{\partial\vartheta^{2}}-\frac{\partial}{\partial\vartheta}\left[\varrho\left(\Omega+K\int_{0}^{2\pi}\int_{\mathbb{R}}\sin(\vartheta^{\prime}-\vartheta)\varrho(\vartheta^{\prime},\Omega^{\prime},t)g(\Omega^{\prime})d\Omega^{\prime}d\vartheta^{\prime}\right)\right].$
(1.10)

Note that (1.10) is a nonlinear parabolic partial integro-differential
equation, that reduces to (1.8) when $D=0$. We finally recall that linear
stability of the incoherent state, concerning both the deterministic and the
stochastic equations (1.8) and (1.10), respectively, have been studied by
Strogatz and Mirollo in [24]. The Cauchy problem relevant to (1.10) has been
studied in recent years by Lavrentiev, Spigler and Tani in [14], [15] and
[16].

Based on some ideas developed by Ermentrout in his studies about fireflies
synchronization [9], Tanaka, Lichtenberg and Oishi in [25] elaborated the
following second order variation of the Kuramoto model

$m\ddot{\vartheta}_{k}+\dot{\vartheta}_{k}=\Omega_{k}+\frac{K}{N}\sum_{j=1}^{N}\sin(\vartheta_{j}-\vartheta_{k}),\qquad
k=1,...,N,$ (1.11)

with $m>0$. We point out that the term $m\ddot{\vartheta}_{k}$ introduces an
inertial effect in the model, in order to better explain some synchronization
phenomena. Unlike the original Kuramoto model, the Ermentrout’s second order
model has the property to allow near-perfect phase synchronization, where the
phase shift between synchronized oscillators is inversely proportional to the
mass. The equation (1.11) can be rewritten in terms of the order parameter
(1.2) as follows

$m\ddot{\vartheta}_{k}+\dot{\vartheta}_{k}=\Omega_{k}+Kr_{N}\sin(\psi_{N}-\vartheta_{k}),\qquad
i=1,...,N,$ (1.12)

which is the governing equation for a single damped driven pendulum with
torque $\Omega_{k}$.

Starting from the contributions due to Tanaka, Lichtenberg and Oishi [25],
Acebroan and Spigler introduced a noise term $\xi_{k}=\xi_{k}(t)$ in (1.11),
so that they considered in [1] a system of second order Langevin equations
that consider an inertial term

$m\ddot{\vartheta}_{k}+\dot{\vartheta}_{k}=\Omega_{k}+Kr_{N}\sin(\psi_{N}-\vartheta_{k})+\xi_{k},\qquad
k=1,...,N.$ (1.13)

If we finally set $\dot{\vartheta}_{k}=\omega_{k}$, and we let $N\to\infty$ as
we did in (1.10), we find the following nonlinear degenerate parabolic
integro-differential equation

$\frac{D}{m^{2}}\frac{\partial^{2}\varrho}{\partial\omega^{2}}+\frac{1}{m}\frac{\partial}{\partial\omega}[(\omega-\Omega-
K_{\varrho}(\vartheta,t))\varrho]-\omega\frac{\partial\varrho}{\partial\vartheta}-\frac{\partial\varrho}{\partial
t}=0,$ (1.14)

where

$K_{\varrho}(\vartheta,t)=K\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{2\pi}g(\Omega^{\prime})\sin(\vartheta^{\prime}-\vartheta)\varrho(\omega^{\prime},\vartheta^{\prime},\Omega^{\prime},t)d\vartheta^{\prime}d\omega^{\prime}d\Omega^{\prime}.$
(1.15)

In order to simplify the notation, in the sequel we set $D=1$ and $m=1$. For a
given $\varrho_{0}=\varrho_{0}(\omega,\vartheta,\Omega)$ we consider the
following Cauchy problem

$\begin{cases}\frac{\partial^{2}\varrho}{\partial\omega^{2}}+\frac{\partial}{\partial\omega}[(\omega-\Omega-
K_{\varrho}(\vartheta,t))\varrho]-\omega\frac{\partial\varrho}{\partial\vartheta}-\frac{\partial\varrho}{\partial
t}=0,\qquad&({\omega},\vartheta,{\Omega},t)\in\mathbb{R}^{3}\times[0,T],\\\
\varrho(\omega,\vartheta,{\Omega},0)=\varrho_{0}(\omega,\vartheta,\Omega),&(\omega,\vartheta,\Omega)\in\mathbb{R}^{3}.\end{cases}$
(1.16)

The Cauchy problem (1.16) has been addressed by Akhmetov, Lavrentiev and
Spigler in [2], [3], [4] and [13] by using a parabolic regularization of the
governing equation.

The main result proved in [4] is the existence and the uniqueness of a
classical solution $\varrho$ to (1.16). Since
$(\omega,\vartheta)\mapsto\varrho(\omega,\vartheta,\Omega,t)$ denotes the
density of a probability measure, the authors of the afore-mentioned
references focus on positive solutions with the property

$\int_{]0,2\pi[\times\mathbb{R}}\varrho(\omega,\vartheta,\Omega,t)d\vartheta\,d\omega=1,\qquad
t\geq 0,\ \Omega\in\mathbb{R}.$

Moreover, the solutions considered there have the following _exponential decay
property_

###### Definition 1.1

We say that a function $f:\mathbb{R}^{4}\rightarrow\mathbb{R}$ has the
_exponential decay property_ if there exist two positive constants $M,C$ such
that:

$(\text{\bf E})\hskip 105.2751pt|f({\omega},\vartheta,\Omega,t)|\leq
C\mathrm{e}^{-M\omega^{2}},\qquad\forall({\omega},\vartheta,\Omega,t)\in\mathbb{R}^{4}.\hskip
105.2751pt$ (1.17)

The main result proved in [4] is the existence of a unique positive classical
solution $\varrho$ to the Cauchy problem (1.16), with the properties listed
above, provided that $\varrho_{0}$ and some of its derivatives have the
exponential decay property.

In this work we address the Cauchy problem (1.16) by using the theory of
subelliptic equations in Lie groups as described in the survey article [5]. In
particular, the first order differential operator
$Y={\omega}\partial_{\omega}-{\omega}\partial_{\vartheta}-\partial_{t}$ is
considered as a _Lie derivative_ , that is

###### Definition 1.2

Consider the integral curve $\exp(sY)(z)$ of $Y$ defined for
$z=({\omega},\vartheta,{\Omega},t)\in\mathbb{R}^{4}$ as the unique solution to

$\begin{cases}\frac{d}{ds}\exp(sY)(z)=Y\exp(sY)(z),\\\
\exp(sY)(z)|_{s=0}=(z).\end{cases}$ (1.18)

A function $\varrho$ is Lie differentiable with respect to $Y$ at the point
$z=({\omega},\vartheta,{\Omega},t)$ if the following limit

$Y\varrho(z)=\frac{d}{ds}\varrho\left(\exp(sY)(z)\right)\big{|}_{s=0}$ (1.19)

exists and is finite.

This definition clarifies the meaning of _classical solution_ to (1.14). A
function $\varrho$ is a classical solution if it is continuous, its
derivatives $\partial_{{\omega}}\varrho$, and
$\partial^{2}_{{\omega}{\omega}}\varrho$ are defined as continuous functions,
the Lie derivative $Y\varrho$ is continuous and the equation (1.14) is
satisfied at every point.

The main advantage in our approach, with respect to [4], is that we don’t
require unnecessary regularity on the data $\varrho_{0}$. Moreover, our
hypothesis (1.21) below does not require the compactness of the support of $g$
assumed in[4]. Finally, a numerical approximation of the Lie derivative
$Y\varrho$ allows us to define a _stable_ numerical method for the Cauchy
problem (1.16). Our first result is the following

###### Theorem 1.3

Let $g$ be a non-negative function such that

$\int_{\mathbb{R}}g({\Omega})d{\Omega}=1,$ (1.20)

and assume that exists ${\beta}>2$ such that

$\int_{\mathbb{R}}g({\Omega})\mathrm{e}^{|{\Omega}|^{\beta}}d{\Omega}<+\infty.$
(1.21)

Let $\varrho_{0}\in C(\mathbb{R}^{3})$ be a strictly positive function, $2\pi$
periodic in $\vartheta$ such that

  1. 1.

$\varrho_{0}$ verifies the exponential decay property (E) as stated in
Definition 1.1;

  2. 2.

for every $\Omega\in\mathbb{R}$ we have

$\int_{]0,2\pi[\times\mathbb{R}}\varrho_{0}({\omega},\vartheta,\Omega)d\vartheta
d\omega=1.$ (1.22)

Then there exists a strictly positive classical solution $\varrho$ to the
Cauchy problem (1.16), defined for
$({\omega},\vartheta,\Omega,t)\in\mathbb{R}^{3}\times[0,+\infty[$ such that

$\int_{]0,2\pi[\times\mathbb{R}}\varrho({\omega},\vartheta,\Omega,t)d\vartheta
d\omega=1,\qquad\text{for every}\quad t\geq 0,\ \Omega\in\mathbb{R}.$ (1.23)

Moreover $\varrho$ is $2\pi$ periodic in $\vartheta$, continuously depends on
$\Omega$ and for every $T>0$ and ${\Omega}\in\mathbb{R}$ there exist positive
constants $C_{{\Omega},T}$, $\overline{M}$ such that

  1. 1.

the function
$({\omega},\vartheta,\Omega,t)\mapsto\varrho(\omega,\vartheta,\Omega,t)$ has
the exponential decay property (E), with constants $\overline{M}$ and
$C_{{\Omega},T}$;

  2. 2.

the function
$({\omega},\vartheta,\Omega,t)\mapsto\partial_{\omega}\varrho(\omega,\vartheta,\Omega,t)$
has the exponential decay property (E), with constants $\overline{M}$ and
$C_{{\Omega},T}/\sqrt{t}$;

  3. 3.

the function
$({\omega},\vartheta,\Omega,t)\mapsto\partial^{2}_{\omega}\varrho(\omega,\vartheta,\Omega,t)$
has the exponential decay property (E), with constants $\overline{M}$ and
$C_{{\Omega},T}/t$. The same assertion holds for the Lie derivative
$Y\varrho=(\omega\partial_{\omega}-\omega\partial_{\vartheta}-\partial_{t})\varrho$.

Furthermore, $\varrho$ is the unique positive solution satisfying the
properties listed above whenever $g$ has compact support.

Following [4], we look for a solution to the nonlinear Cauchy problem (1.16)
as the fixed point of a map that associates to a given function $\varrho$ the
solution to a suitable linearized problem, which in our case is relevant to a
_degenerate Kolmogorov equation_. With this aim, we introduce the following
Cauchy problem

$\begin{dcases}\partial^{2}_{\omega\omega}u+\omega\partial_{\omega}u-\omega\partial_{\vartheta}u+\Phi_{\widetilde{\varrho}}(\vartheta,\Omega,t)\partial_{\omega}u+u-\partial_{t}u=f,\;\;\;\;&({\omega},\vartheta,{\Omega},t)\in\mathbb{R}^{3}\times]T_{0},T_{1}[,\\\
u({\omega},\vartheta,{\Omega},T_{0})=u_{0}(\omega,\vartheta,\Omega),\;\;&(\omega,\vartheta,\Omega)\in\mathbb{R}^{3},\end{dcases}$
(1.24)

where

$\Phi_{\widetilde{\varrho}}(\vartheta,\Omega,t)=-\Omega-K\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{2\pi}g(\Omega^{\prime})\sin(\vartheta^{\prime}-\vartheta)\widetilde{\varrho}({\omega}^{\prime},\vartheta^{\prime},\Omega^{\prime},t)d\vartheta^{\prime}d\omega^{\prime}d\Omega^{\prime}.$
(1.25)

If $\widetilde{\varrho}$ ia s given continuous function satisfying the
exponential decay property (see Definition 1.1), then there exists a
_fundamental solution_ ${\Gamma}_{\Omega,\widetilde{\varrho}}$ useful to
represent the solution $u$ to (1.24) as follows

$\displaystyle u({\omega},\vartheta,\Omega,t)=$
$\displaystyle\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;\xi,\eta,T_{0})u_{0}(\xi,\eta,\Omega)d\eta
d\xi-$ (1.26) $\displaystyle-$
$\displaystyle\int_{T_{0}}^{t}\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;\xi,\eta,\tau)f(\xi,\eta,\Omega,\tau)d\eta
d\xi d\tau.$

when $f,u_{0}$ verify appropriate assumptions. In Section 2 we study the
existence of ${\Gamma}_{\Omega,\widetilde{\varrho}}$ and some bounds of
${\Gamma}_{\Omega,\widetilde{\varrho}}$ and its derivatives, that provide us
with the proof of the existence of the unique solution to (1.16).

This paper is organized as follows: in Section 2 we recall some results about
the fundamental solution and we give some bounds useful in the proof of
Theorem 1.3, Section 3 is devoted to the proof of Theorem 1.3. In Section 4 we
define a stable and consistent finite difference scheme that follows the
structure of (1.14).

##  2 Fundamental Solution

In this Section we recall some known facts about the partial differential
operator

$\mathscr{L}_{\Omega,\widetilde{\varrho}}=\partial^{2}_{\omega\omega}+\omega\partial_{\omega}-\omega\partial_{\vartheta}+\Phi_{\widetilde{\varrho}}(\vartheta,{\Omega},t)\partial_{\omega}+1-\partial_{t},$
(2.1)

appearing in the Cauchy problem (1.24), where the function
$\Phi_{\widetilde{\varrho}}$ has been introduced in (1.25)

$\Phi_{\widetilde{\varrho}}(\vartheta,{\Omega},t)=-\Omega-K\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{2\pi}g(\Omega^{\prime})\sin(\vartheta^{\prime}-\vartheta)\widetilde{\varrho}({\omega}^{\prime},\vartheta^{\prime},\Omega^{\prime},t)d\vartheta^{\prime}d\omega^{\prime}d\Omega^{\prime}.$
(2.2)

Note that no derivatives with respect to ${\Omega}$ appear in
$\mathscr{L}_{\Omega,\widetilde{\varrho}}$, then, from now on we fix
${\Omega}\in\mathbb{R}$ and we consider it as a parameter.

Let us now recall some known facts about the degenerate differential operator
$\mathscr{L}_{\Omega,\widetilde{\varrho}}$ that will be used in the proof of
Theorem 1.3. For a reason that will be clear in the following, we also
introduce a parameter ${\varepsilon}>0$, and we set

$\mathscr{L}^{\varepsilon}:=(1+{\varepsilon})\partial^{2}_{{\omega}{\omega}}+\omega\partial_{\omega}-\omega\partial_{\vartheta}-\partial_{t}.$
(2.3)

This strongly degenerate operator has a fundamental solution
${\Gamma}^{\varepsilon}={\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\omega}_{0},\vartheta_{0},t_{0})$
which is smooth with respect to the variable $({\omega},\vartheta,t)$
belonging to the set
$\mathbb{R}^{3}\setminus\\{({\omega}_{0},\vartheta_{0},t_{0})\\}$. We
introduce some notation in order to write the explicit expression of
${\Gamma}^{\varepsilon}$. We first write $\mathscr{L}^{\varepsilon}$ in the
form

$\mathscr{L}^{\varepsilon}:=\sum_{i,j=1}^{2}a_{ij}\partial^{2}_{x_{i}x_{j}}+\sum_{i,j=1}^{2}b_{ij}x_{j}\partial_{x_{i}}-\partial_{t},$
(2.4)

where $x=(x_{1},x_{2})=({\omega},\vartheta)$, and

$A_{\varepsilon}:=\left(a_{ij}\right)_{i,j=1,2}=\begin{pmatrix}1+{\varepsilon}&0\\\
0&0\end{pmatrix},\quad B:=\left(b_{ij}\right)_{i,j=1,2}=\begin{pmatrix}1&0\\\
-1&0\end{pmatrix}.$ (2.5)

Following Hörmander (see p. 148 in [10]), we set, for every $t\in\mathbb{R}$,

$E(t):=\exp\big{(}-tB\big{)},\qquad
C_{\varepsilon}(t):=\int_{0}^{t}E(s)A_{\varepsilon}E^{T}(s)\,ds.$ (2.6)

A plain computation shows that

$E(t)=\begin{pmatrix}\mathrm{e^{-t}}\;\;\;&0\\\
1-\mathrm{e^{-t}}&1\end{pmatrix},\qquad
C_{\varepsilon}(s)=\frac{1+{\varepsilon}}{2}\begin{pmatrix}1-\mathrm{e}^{-2t}&1-2\mathrm{e}^{-t}+\mathrm{e}^{-2t}\\\
1-2\mathrm{e}^{-t}+\mathrm{e}^{-2t}&2t-3+4\mathrm{e}^{-t}-\mathrm{e}^{-2t}\end{pmatrix}.$
(2.7)

Note that the matrix $C_{\varepsilon}(t)$ is symmetric and strictly positive
for every $t>0$, then it is invertible and the fundamental solution
${\Gamma}^{\varepsilon}$ of $\mathscr{L}^{\varepsilon}u=0$ is defined for
every $(x,t)=({\omega},\vartheta,t)\in\mathbb{R}^{2}\times]0,+\infty[$ as
follows

${\Gamma}^{\varepsilon}(x,t)=\frac{1}{4\pi\sqrt{\text{det}\,C_{\varepsilon}(t)}}\exp\left(-\tfrac{1}{4}\langle
C_{\varepsilon}^{-1}(t)x,x\rangle-t\right).$ (2.8)

As customary in the setting of parabolic equation, we set
${\Gamma}^{\varepsilon}(x,t)=0$ whenever $t\leq 0$. Finally, the fundamental
solution with singularity at any point $(x_{0},t_{0})\in\mathbb{R}^{3}$ is
defined as

${\Gamma}^{\varepsilon}(x,t;x_{0},t_{0}):={\Gamma}^{\varepsilon}(x-E(t-t_{0})x_{0},t-t_{0}).$
(2.9)

The following properties of ${\Gamma}^{\varepsilon}$ will be useful in the
sequel

$\int_{\mathbb{R}^{2}}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\xi},{\eta},{\tau})d{\eta}d{\xi}=1,\qquad\forall({\omega},\vartheta)\in\mathbb{R}^{2},\tau<t.$
(2.10)

Moreover, a direct computation shows that the following identity holds true

$\int_{\mathbb{R}}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\xi},{\eta},{\tau})d{\eta}=\widetilde{\Gamma}^{\varepsilon}({\omega},t;{\xi},{\tau}),\qquad\forall({\omega},\vartheta,t,{\xi},\tau)\in\mathbb{R}^{5},\quad\text{with}\
\tau<t.$ (2.11)

Here

$\widetilde{\Gamma}^{\varepsilon}({\omega},t;{\xi},{\tau})=\frac{1}{\sqrt{(1+{\varepsilon})\pi(1-e^{-2(t-\tau)})}}\exp\left(-\tfrac{|{\omega}-e^{-(t-\tau)}\xi|^{2}}{(1+{\varepsilon})(1-e^{-2(t-\tau)})}-(t-\tau)\right),$
(2.12)

is the fundamental solution to the operator
$(1+{\varepsilon})\partial^{2}_{{\omega}{\omega}}+\omega\partial_{\omega}-\partial_{t}$.

Let’s turn our attention to the Cauchy problem (1.24)

$\begin{dcases}\mathscr{L}_{\Omega,\widetilde{\varrho}}\
u=f,\;\;\;\;&({\omega},\vartheta,t)\in\mathbb{R}^{2}\times]T_{0},T_{1}[,\\\
u({\omega},\vartheta,T_{0})=u_{0}({\omega},\vartheta),\;\;&({\omega},\vartheta)\in\mathbb{R}^{2}.\end{dcases}$
(2.13)

The existence of a fundamental solution
${\Gamma}_{\Omega,\widetilde{\varrho}}$ to the equation
$\mathscr{L}_{\Omega,\widetilde{\varrho}}\,u=0$ has been proved by Di
Francesco and Pascucci in [6] via the Levi’s parametrix method. In order to
state the main results of [6], which include some bounds for
${\Gamma}_{\Omega,\widetilde{\varrho}}$ that will be useful in the proof of
Theorem 1.3, we recall some further notation and known facts on degenerate
Kolmogorov operators.

The expression $(x-E(t-t_{0})x_{0},t-t_{0})$ appearing in (2.9) is related to
an invariance property of the differential operator
$\mathscr{L}^{\varepsilon}$, which is needed to state the Hölder continuity
assumption on the coefficient $\Phi_{\widetilde{\varrho}}$ appearing in the
operator $\mathscr{L}_{\Omega,\widetilde{\varrho}}$. We first recall that the
operator $\mathscr{L}^{\varepsilon}$ is invariant with respect to the change
of variable
$({\omega},\vartheta,t)\mapsto({\omega},\vartheta,t)\circ({\xi},{\eta},\tau)$,
where

$\displaystyle({\omega},\vartheta,t)\circ({\xi},{\eta},\tau):=$
$\displaystyle(({\xi},{\eta})+E(\tau)({\omega},\vartheta),t+\tau)$ (2.14)
$\displaystyle=$
$\displaystyle({\xi}+{\omega}\mathrm{e}^{-{\tau}},{\eta}+\vartheta+{\omega}(1-\mathrm{e}^{-{\tau}}),t+{\tau}),\qquad({\omega},\vartheta,t),({\xi},{\eta},\tau)\in\mathbb{R}^{3}.$

The set $\big{(}\mathbb{R}^{3},\circ\big{)}$ is a Lie group with identity
$(0,0,0)$, and the inverse of a point
$({\omega},\vartheta,t)\in\mathbb{R}^{3}$ is

$({\omega},\vartheta,t)^{-1}=(-E(-t)({\omega},\vartheta),-t)=(-{\omega}\mathrm{e}^{t},-\vartheta-{\omega}(1-\mathrm{e}^{t}),-t).$
(2.15)

Moreover, as usual in the regularity theory for subelliptic operators on Lie
groups, an _anisotropic_ norm is used to define a quasi-metric. In this case
we consider the norm

$\|({\omega},\vartheta,t)\|=|{\omega}|+|\vartheta|^{\frac{1}{3}}+|t|^{\frac{1}{2}},\qquad\forall({\omega},\vartheta,t)\in\mathbb{R}^{3},$
(2.16)

and we define the quasi-distance as follows: for every
$({\omega},\vartheta,t),({\xi},{\eta},{\tau})\in\mathbb{R}^{3}$, we define

$\displaystyle d(({\omega},\vartheta,t),({\xi},{\eta},{\tau}))$
$\displaystyle=\|({\xi},{\eta},{\tau})^{-1}\circ({\omega},\vartheta,t)\|$
(2.17)
$\displaystyle=|{\omega}-{\xi}\mathrm{e}^{{\tau}-t}|+|\vartheta-{\eta}+{\xi}(\mathrm{e}^{{\tau}-t}-1)|^{\frac{1}{3}}+|t-{\tau}|^{\frac{1}{2}}.$

We are now in position to give the following

######  Definition 2.1 (Hölder continuous function)

Let $\alpha\in]0,1]$, and let $U$ be an open subset of $\mathbb{R}^{3}$. We
say that a function $f:U\longrightarrow\mathbb{R}$ is Hölder continuous with
exponent $\alpha$ in $U$ (in short $f\in C^{\alpha}(U)$) if there exists a
positive constant $k$ such that

$|f({\omega},\vartheta,t)-f({\xi},{\eta},{\tau})|\leq
k\,d(({\omega},\vartheta,t),({\xi},{\eta},{\tau}))^{\alpha},\qquad\forall({\omega},\vartheta,t),({\xi},{\eta},{\tau})\in
U.$ (2.18)

We next recall Theorem 1.4 in [6], which provides us with an existence result
for a fundamental solution ${\Gamma}_{\Omega,\widetilde{\varrho}}$ to
$\mathscr{L}_{\Omega,\widetilde{\varrho}}u=0$.

###### Theorem 2.2

Let suppose that the coefficient
$\Phi_{\widetilde{\varrho}}({\omega},\vartheta,t)$ of
$\mathscr{L}_{{\Omega},\widetilde{\varrho}}$ is bounded and such that there
exists ${\alpha}\in]0,1]$ and a positive constant $k$ such that

$|\Phi_{\widetilde{\varrho}}(\vartheta,{\Omega},t)-\Phi_{\widetilde{\varrho}}(\widetilde{\vartheta},\Omega,t)|\leq
k\
d((\vartheta,{\Omega},t),(\widetilde{\vartheta},{\Omega},t))^{\alpha},\qquad\forall\vartheta,\widetilde{\vartheta}\in\mathbb{R},({\Omega},t)\in\mathbb{R}^{2}.$
(2.19)

Then there exists a fundamental solution
${\Gamma}_{\Omega,\widetilde{\varrho}}$ to
$\mathscr{L}_{\Omega,\widetilde{\varrho}}$ with the following properties:

  1. 1.

${\Gamma}_{\Omega,\widetilde{\varrho}}(\cdot;{\xi},{\eta},{\tau})\in\mathrm{L}_{{\text{\rm
loc}}}^{1}(\mathbb{R}^{3})\cap
C(\mathbb{R}^{3}\setminus\\{({\xi},{\eta},{\tau})\\})$ for every
$({\xi},{\eta},{\tau})\in\mathbb{R}^{3}$;

  2. 2.

${\Gamma}_{\Omega,\widetilde{\varrho}}(\cdot;{\xi},{\eta},{\tau})$ is a
classical solution to $\mathscr{L}_{\Omega,\widetilde{\varrho}}\ u=0$ in
$\mathbb{R}^{3}\setminus\\{({\xi},{\eta},{\tau})\\}$ for every
$({\xi},{\eta},{\tau})\in\mathbb{R}^{3}$;

  3. 3.

the following identity holds

$\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;{\xi},{\eta},{\tau})d{\eta}d{\xi}=\mathrm{e}^{-(t-\tau)},\qquad\forall({\omega},\vartheta)\in\mathbb{R}^{2},\tau<t.$
(2.20)

  4. 4.

for every ${\varepsilon}>0$ and $T>0$ there exists a constant
$\overline{C}=\overline{C}\big{(}{\varepsilon},T,\|\Phi_{\widetilde{\varrho}}\|_{L^{\infty}(\mathbb{R}^{3})}\big{)}$,
such that

$\displaystyle
0<\;{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;{\xi},{\eta},{\tau})$
$\displaystyle\leq\overline{C}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\xi},{\eta},{\tau}),$
(2.21)
$\displaystyle|\partial{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;{\xi},{\eta},{\tau})|$
$\displaystyle\leq\frac{\overline{C}}{\sqrt{(t-\tau)^{i}}}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\xi},{\eta},{\tau}),$

where ${\Gamma}^{\varepsilon}$ is the function defined in (2.9) $\partial$
stands for $\partial_{{\omega}}$ ($i=1$), $\partial^{2}_{{\omega}{\omega}}$ or
Lie derivative $Y$ ($i=2$), for every
$({\omega},\vartheta,t),({\xi},{\eta},{\tau})\in\mathbb{R}^{3}$ with
$0<t-\tau<T$.

Let’s consider the following Cauchy problem

$\begin{dcases}\mathscr{L}_{\Omega,\widetilde{\varrho}}u({\omega},\vartheta,t)=f({\omega},\vartheta,t),\qquad&({\omega},\vartheta,t)\in\mathbb{R}^{2}\times]T_{0},T_{1}[,\\\
u({\omega},\vartheta,T_{0})=u_{0}({\omega},\vartheta),\qquad&({\omega},\vartheta)\in\mathbb{R}^{2},\end{dcases}$
(2.22)

where $u_{0}\in C(\mathbb{R}^{2})$ is such that

$|u_{0}({\omega},\vartheta)|\leq
C_{1}\mathrm{e}^{C_{1}(\omega^{2}+\vartheta^{2})},\qquad\forall({\omega},\vartheta)\in\mathbb{R}^{2},$
(2.23)

and $f\in C(\mathbb{R}^{2}\times(T_{0},T_{1}))$ is such that

$|f({\omega},\vartheta,t)|\leq\frac{C_{1}\mathrm{e}^{C_{1}(\omega^{2}+\vartheta^{2})}}{\sqrt{t}},\qquad\forall({\omega},\vartheta,t)\in\mathbb{R}^{2}\times(T_{0},T_{1}),$
(2.24)

for some positive constant $C_{1}$, and for any compact subset $M$ of
$\mathbb{R}^{2}$ there exists a positive constant $C$, $\beta\in]0,1[$ and
$\delta>0$ such that

$|f({\omega},\vartheta,t)-f({\xi},{\eta},t)|\leq
C\frac{d(({\omega},\vartheta,t),({\xi},{\eta},t))^{\beta}}{t^{1-\delta}},\qquad\forall({\omega},\vartheta),({\xi},{\eta})\in
M,t\in]T_{0},T_{1}[.$ (2.25)

Then there exists $T\in]T_{0},T_{1}[$, only depending on growth constant
$C_{1}$, such that the function

$\displaystyle u({\omega},\vartheta,t)=$ (2.26)
$\displaystyle\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;{\xi},{\eta},T_{0})u_{0}({\xi},{\eta})d{\eta}d{\xi}-\int_{T_{0}}^{t}\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,\widetilde{\varrho}}({\omega},\vartheta,t;{\xi},{\eta},{\tau})f({\xi},{\eta},\tau)d{\eta}d{\xi}d\tau,$

is solution to the previous Cauchy problem in $]T_{0},T[$.

Moreover if $u$ is a solution to the Cauchy problem with null $f$ and $g$, and
verifies the following estimate

$|u({\omega},\vartheta,t)|\leq
C\mathrm{e}^{C({\omega}^{2}+\vartheta^{2})},\qquad\forall({\omega},\vartheta,t)\in\mathbb{R}^{2}\times]T_{0},T_{1}[,$
(2.27)

for some $C>0$, then $u\equiv 0$.

##  3 Proof of Theorem 1.3.

In this Section we prove existence, uniqueness and regularity of classical
solutions to the Cauchy problem (1.16) in $S_{T}=\mathbb{R}^{3}\times]0,T[$,
which we recall here for reader’s convenience

$\begin{dcases}\mathscr{L}_{{\Omega},{\varrho}}{\varrho}=\frac{\partial^{2}\varrho}{\partial\omega^{2}}+\frac{\partial}{\partial\omega}\left[(\omega+\Phi_{\varrho}(\vartheta,{\Omega},t))\varrho\right]-\omega\frac{\partial\varrho}{\partial\vartheta}-\frac{\partial\varrho}{\partial
t}=0,\qquad&\mathrm{in}\ S_{T},\\\
\varrho({\omega},\vartheta,\Omega,0)=\varrho_{0}({\omega},\vartheta,\Omega),\qquad&\mathrm{in}\
\mathbb{R}^{3},\end{dcases}$ (3.1)

where

$\Phi_{\varrho}(\vartheta,{\Omega},t)=-{\Omega}-K\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{2\pi}g(\Omega^{\prime})\sin(\vartheta^{\prime}-\vartheta)\varrho({\omega}^{\prime},\vartheta^{\prime},\Omega^{\prime},t)d\vartheta^{\prime}d\omega^{\prime}d\Omega^{\prime}.$
(3.2)

As said in the Introduction, we find a solution to the nonlinear problem
(1.16) as the fixed point of a map that associates to a given function
$\varrho$ the solution to the linearized problem (1.24), which is obtained by
using the fundamental solution ${\Gamma}_{\Omega,\widetilde{\varrho}}$. We
list the assumptions about the initial datum $\varrho_{0}$ we will adopt in
this iterative argument.

  1. 1.

$\varrho_{0}\in C(\mathbb{R}^{3})$;

  2. 2.

$\varrho_{0}$ is strictly positive and verifies (E) in Definition 1.1;

  3. 3.

$\varrho_{0}$ is $2\pi$ periodic with respect to $\vartheta$;

  4. 4.

for every $\Omega\in\mathbb{R}$ we have

$\int_{]0,2\pi[\times\mathbb{R}}\varrho_{0}({\omega},\vartheta,\Omega)d\vartheta
d\omega=1.$ (3.3)

Moreover we recall that the natural frequency distributions of oscillators
$g({\Omega})$ is a probability density, satisfying (1.21). We first prove a
preliminary result. Here and in the sequel, we set
$S_{T}^{*}:=\mathbb{R}^{2}\times]0,T[$, and we fix ${\Omega}\in\mathbb{R}$.
Moreover, we consider the function ${\Gamma}^{\varepsilon}$, defined in (2.9),
for a fixed ${\varepsilon}>0$, as we will rely on (2.21).

###### Lemma 3.1

Consider for $T>0$ the Cauchy problem (1.24) in $S_{T}$ with $f=0$. If
$\widetilde{\varrho}$ is continuous and verifies property (E), then the
function $u$ defined as

$u({\omega},\vartheta,\Omega,t)=\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,{{\widetilde{\varrho}}}}({\omega},\vartheta,t;\xi,\eta,0)\varrho_{0}(\xi,\eta,\Omega)d\eta
d\xi,$ (3.4)

is a classical solution $u$ to (1.24) with $T_{0}=0,T_{1}=T$, and
$u_{0}=\varrho_{0}$ . Moreover

  1. 1.

$u$ verifies property (E) with constants $\overline{M},C_{\Omega,T}$. In
addition we have

$|\partial
u({\omega},\vartheta,\Omega,t)|\leq\frac{C_{\Omega,T}}{\sqrt{t^{i}}}\mathrm{e}^{-\overline{M}\omega^{2}},\quad\forall(\omega,\vartheta,t)\in
S_{T}^{*},$ (3.5)

where $\partial$ stands for $\partial_{\omega}$ (and (3.5) holds with $i=1$),
$\partial^{2}_{\omega\omega}$ or Lie derivative $Y$ (and (3.5) holds with
$i=2$);

  2. 2.

$u$ is $2\pi$ periodic with respect to $\vartheta$;

  3. 3.

for every $\Omega\in\mathbb{R}$ and for every $t\in[0,T[$ we have

$\int_{]0,2\pi[\times\mathbb{R}}u({\omega},\vartheta,\Omega,t)d\vartheta
d\omega=1.$ (3.6)

  4. 4.

$u({\omega},\vartheta,\Omega,t)>0$ for every $t\geq 0$.

Proof. As a direct consequence of the property (E), we can easily see that the
function $\Phi_{\widetilde{\varrho}}$ defined in (1.25) is bounded. Moreover,
if we consider $\Phi_{\widetilde{\varrho}}$ as a function of the variable
$(\omega,\vartheta,{\Omega},t)$ by letting
$\Phi_{\widetilde{\varrho}}(\omega,\vartheta,{\Omega},t):=\Phi_{\widetilde{\varrho}}(\vartheta,{\Omega},t)$,
there exists a positive constant $L$ such that

$|\Phi_{\widetilde{\varrho}}(\omega_{1},\vartheta_{1},{\Omega},t)-\Phi_{\widetilde{\varrho}}(\omega_{2},\vartheta_{2},\Omega,t)|\leq
L\;\mathrm{d}((\omega_{1},\vartheta_{1},{\Omega},t),(\omega_{2},\vartheta_{2},{\Omega},t)),$
(3.7)

for every
$(\omega_{1},\vartheta_{1},{\Omega},t),(\omega_{2},\vartheta_{2},{\Omega},t)\in
S_{T}$. Therefore the function $u$ defined in (3.4) is a classical solution
$u$ to (1.24), by Theorem 2.2. We next prove that $u$ has the properties
listed in the statement of the Lemma.

In order to prove that $u$ and all the derivatives $\partial u$ listed in the
point _(.1)_ verify the property (E), we first observe that the identity (3.4)
can be differentiated under the integral sign, _i.e._

$\partial
u({\omega},\vartheta,\Omega,t)=\int_{\mathbb{R}^{2}}\partial{\Gamma}_{\Omega,{{\widetilde{\varrho}}}}({\omega},\vartheta,t;\xi,\eta,0)\varrho_{0}(\xi,\eta,\Omega)d\eta
d\xi.$ (3.8)

Then by point _(.4)_ of Theorem 2.2 we have

$|\partial
u({\omega},\vartheta,\Omega,t)|\leq\frac{\overline{C}_{{\Omega},T}}{\sqrt{t^{i}}}\int_{\mathbb{R}^{2}}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;\xi,\eta,0)\varrho_{0}(\xi,\eta,\Omega)d\eta
d\xi,$ (3.9)

for some $\overline{C}_{{\Omega},T}>0$ that depend on
${\varepsilon},T,\|\Phi_{\widetilde{\varrho}\|_{L^{\infty}(\mathbb{R}^{3})}}$,
and therefore on ${\Omega}$.

Property (E) follows from (2.21) and from the exponential decay of the initial
datum. Consider a given $\omega\in\mathbb{R}$, by (2.10) we have

$|\partial
u({\omega},\vartheta,\Omega,t)|\leq\frac{\overline{C}_{{\Omega},T}C}{\sqrt{t^{i}}}\int_{\mathbb{R}^{2}}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\xi},{\eta},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}d{\eta}.$
(3.10)

If $|\omega|>1$, we first recall the equation (2.12) where
$\widetilde{{\Gamma}}^{\varepsilon}$ has been defined, then we split the above
integral in two parts

$\begin{split}\int_{\mathbb{R}^{2}}&{\Gamma}^{\varepsilon}({\omega},\vartheta,t;{\xi},{\eta},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}d{\eta}=\int_{-\infty}^{+\infty}\widetilde{{\Gamma}}^{\varepsilon}({\omega},t;{\xi},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}\\\
&\qquad=\int_{\big{\\{}|{\xi}|\geq\frac{|{\omega}|}{2}\big{\\}}}\widetilde{{\Gamma}}^{\varepsilon}({\omega},t;{\xi},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}+\int_{\big{\\{}|{\xi}|<\frac{|{\omega}|}{2}\big{\\}}}\widetilde{{\Gamma}}^{\varepsilon}({\omega},t;{\xi},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}.\end{split}$
(3.11)

Concerning the first integral, we immediately have

$\int_{\big{\\{}|{\xi}|\geq\frac{|{\omega}|}{2}\big{\\}}}\widetilde{{\Gamma}}^{\varepsilon}({\omega},t;{\xi},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}\leq\mathrm{e}^{-M\frac{{\omega}^{2}}{4}}\int_{-\infty}^{+\infty}\widetilde{{\Gamma}}^{\varepsilon}({\omega},t;{\xi},0)d{\xi}=\mathrm{e}^{-M\frac{{\omega}^{2}}{4}}.$
(3.12)

We use the change of variable
$x=\frac{\mathrm{e}^{-t}{\xi}-{\omega}}{\sqrt{(1+{\varepsilon})(1-e^{-2t})}}$
in the second integral. Note that, if $|{\xi}|<\frac{|{\omega}|}{2}$, then
$|x|>\frac{|{\omega}|}{2\sqrt{(1+{\varepsilon})(1-e^{-2t})}}$, so that

$\begin{split}\int_{\big{\\{}|{\xi}|<\frac{|{\omega}|}{2}\big{\\}}}&\widetilde{{\Gamma}}^{\varepsilon}({\omega},t;{\xi},0)\mathrm{e}^{-M{\xi}^{2}}d{\xi}\leq\\\
\leq&\int_{\big{\\{}|x|>\frac{|{\omega}|}{2\sqrt{(1+{\varepsilon})(1-e^{-2t})}}\big{\\}}}\frac{\mathrm{e}^{-x^{2}}}{\sqrt{\pi}}\mathrm{e}^{-M\mathrm{e}^{2t}\left(x\sqrt{M(1+{\varepsilon})(1-e^{-2t})}+{\omega}\right)^{2}}dx\leq\\\
\leq&\tfrac{\mathrm{e}^{-\frac{{\omega}^{2}}{4(1+{\varepsilon})(1-e^{-2t})}}}{M\sqrt{(1+{\varepsilon})(1-e^{-2t})}}<\tfrac{\mathrm{e}^{-\frac{1}{8(1+{\varepsilon})(1-e^{-2t})}}}{M\sqrt{(1+{\varepsilon})(1-e^{-2t})}}\cdot\mathrm{e}^{-\frac{{\omega}^{2}}{8(1+{\varepsilon})}}\leq\widetilde{C}_{{\varepsilon},T}\cdot\mathrm{e}^{-\frac{{\omega}^{2}}{8(1+{\varepsilon})}},\end{split}$
(3.13)

where we have used the assumption that $|\omega|>1$ and the fact that the
function appearing in the last line of the above display is bounded in its
domain $\mathbb{R}^{+}$. Then (3.5) in the case $|\omega|>1$ follows from the
above inequality and (3.12), if we set

$\overline{M}=\min\big{\\{}\tfrac{M}{4},\tfrac{1}{8(1+{\varepsilon})}\big{\\}}.$
(3.14)

If $|\omega|\leq 1$ by (3.10) we have

$|\partial
u({\omega},\vartheta,\Omega,t)|\leq\frac{\overline{C}_{{\Omega},T}\,C\,\mathrm{e}^{\overline{M}}}{\sqrt{t^{i}}}\mathrm{e}^{-\overline{M}{\omega}^{2}},$
(3.15)

then the thesis follows if we set

$C_{{\Omega},T}:=\overline{C}_{{\Omega},T}\,C\max\\{2,2\,\widetilde{C}_{{\varepsilon},T},\mathrm{e}^{\overline{M}}\\}.$
(3.16)

The proof of the periodicity stated in _(.2)_ follows from a standard
uniqueness argument. We omit the details.

In order to prove the assertion _(.3)_ of the Lemma, we first notice that the
continuity of the functions $\partial_{\omega}u$ and
$Yu={\omega}\partial_{\omega}u-{\omega}\partial_{\vartheta}u-\partial_{t}u$
implies the continuity of the Lie derivative
$\widetilde{Y}u:=Yu-{\omega}\partial_{\omega}u=-{\omega}\partial_{\vartheta}u-\partial_{t}u$.
Let us define for every $R>0$ and $0<t_{0}<t_{1}$ the sets

$\displaystyle
C_{R}=\left\\{({\omega},\vartheta,t)\in\mathbb{R}^{3}:-R\leq{\omega}\leq
R,\;0\leq\vartheta\leq 2\pi,\;t_{0}\leq t\leq t_{1}\right\\},$ (3.17)
$\displaystyle
B_{R}=\left\\{({\omega},\vartheta)\in\mathbb{R}^{2}:-R\leq{\omega}\leq
R,\;0\leq\vartheta\leq 2\pi\right\\}.$

If we integrate in $C_{R}$ the equation in (1.24) we obtain

$\int_{C_{R}}-\widetilde{Y}ud\omega d\vartheta
dt=\int_{C_{R}}\partial_{\omega}(\partial_{\omega}u+(\omega+\Phi_{\widetilde{\varrho}}(\vartheta,\Omega,t))u)d\omega
d\vartheta dt.$ (3.18)

Then by the divergence theorem we have

$\displaystyle\int_{B_{R}}u({\omega},\vartheta,{\Omega},t_{1})d{\omega}d\vartheta-\int_{B_{R}}u({\omega},\vartheta,{\Omega},t_{0})d{\omega}d\vartheta+\int_{t_{0}}^{t_{1}}\int_{-R}^{R}{\omega}u({\omega},0,{\Omega},t)d{\omega}dt-$
(3.19)
$\displaystyle-\int_{t_{0}}^{t_{1}}\int_{-R}^{R}{\omega}u({\omega},2\pi,{\Omega},t)d{\omega}dt=\int_{C_{R}}\partial_{\omega}(\partial_{\omega}u+(\omega+\Phi_{\widetilde{\varrho}}(\vartheta,\Omega,t))u)d\omega
d\vartheta dt.$

The last two integrals in the left-hand side sum to zero by periodicity,
moreover by the exponential decay in $\omega$ we have

$\displaystyle\lim_{R\to\infty}\int_{C_{R}}\partial_{\omega}(\partial_{\omega}u+(\omega+\Phi_{\widetilde{\varrho}}(\vartheta,\Omega,t))u)d\omega
d\vartheta dt=0.$ (3.20)

Then for every $0\leq t_{0}<t_{1}$ we have

$\int_{]0,2\pi[\times\mathbb{R}}u({\omega},\vartheta,{\Omega},t_{0})d{\omega}d\vartheta=\int_{]0,2\pi[\times\mathbb{R}}u({\omega},\vartheta,{\Omega},t_{1})d{\omega}d\vartheta.$
(3.21)

The thesis then follows from the property of $\varrho_{0}$ by choosing
$t_{0}=0$.

The proof of the point _(.4)_ is a consequence of the representation formula
(3.4), since both ${\Gamma}_{\Omega,{\widetilde{\varrho}}}$ and $\varrho_{0}$
are positive functions. $\square$

We are now ready to prove an existence result for the Cauchy problem (1.16).

###### Proposition 3.2

For every $\Omega,T\in\mathbb{R}$, with $T>0$ and $\varrho_{0}$ satisfying the
property (E), there exists a classical solution
$\varrho=\varrho({\omega},\vartheta,\Omega,t)$ to (1.16) that verifies
property (E) in the set $S_{T}^{*}$. Moreover $\varrho$ is unique and can be
represented by the fundamental solution ${\Gamma}_{\Omega,{{\varrho}}}$ of
$\mathscr{L}_{\Omega,{{\varrho}}}$ as follows

$\varrho({\omega},\vartheta,\Omega,t)=\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,{{\varrho}}}({\omega},\vartheta,t;\xi,\eta,0)\varrho_{0}(\xi,\eta,\Omega)d\eta
d\xi.$ (3.22)

Proof. We define a sequence of function
$\big{\\{}\varrho_{n}\big{\\}}_{n\in\mathbb{N}\cup\\{0\\}}$ as follows. We set
$\varrho_{0}({\omega},\vartheta,\Omega,t)=\varrho_{0}({\omega},\vartheta,\Omega)$
for every $({\omega},\vartheta,t)\in S_{T}^{*}$. Then, for every
$n\in\mathbb{N}$, we let

$\Phi_{\varrho_{n-1}}(\vartheta,{\Omega},t)=-{\Omega}-K\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{0}^{2\pi}g(\Omega^{\prime})\sin(\vartheta^{\prime}-\vartheta)\varrho_{n-1}({\omega}^{\prime},\vartheta^{\prime},\Omega^{\prime},t)d\vartheta^{\prime}d\omega^{\prime}d\Omega^{\prime},$
(3.23)

and we define $\varrho_{n}$ as the solution to

$\begin{dcases}\mathscr{L}_{{\Omega},{\varrho}_{n-1}}{\varrho}_{n}=\partial^{2}_{\omega\omega}\varrho_{n}+\partial_{\omega}[(\omega+\Phi_{\varrho_{n-1}}(\vartheta,{\Omega},t))\varrho_{n}]-\omega\partial_{\vartheta}\varrho_{n}-\partial_{t}\varrho_{n}=0,\\\
\varrho_{n}({\omega},\vartheta,\Omega,0)=\varrho_{0}({\omega},\vartheta,\Omega).\end{dcases}$
(3.24)

The solution to the non-linear Cauchy problem will be defined as the limit of
the sequence $\big{\\{}\varrho_{n}\big{\\}}_{n\in\mathbb{N}\cup\\{0\\}}$. We
rely on a compactness argument to prove the convergence of a subsequence of
$\big{\\{}\varrho_{n}\big{\\}}_{n\in\mathbb{N}\cup\\{0\\}}$. However, due the
presence of the term $\varrho_{n-1}$ in the coefficient $\Phi_{\varrho_{n-1}}$
in (3.24), we need to show that the whole sequence converges.

We now prove that the sequence
$\big{\\{}\varrho_{n}\big{\\}}_{n\in\mathbb{N}\cup\\{0\\}}$ has the property
(E), with constants that does not depend on $n$. As $\varrho_{0}\in
C(\mathbb{R}^{3})$ and verifies property (E), we can apply Lemma 3.1,
therefore for every $n\in\mathbb{N}$ there exists a classical solution
${\varrho}_{n}$ in $\mathbb{R}^{2}\times[0,T[$ such that:

  * •

$\varrho_{n}$ is $2\pi$ periodic in $\vartheta$;

  * •

$\varrho_{n}$ is strictly positive for every $t\geq 0$;

  * •

$\varrho_{n}$ is normalized

$\int_{]0,2\pi[\times\mathbb{R}}\varrho_{n}({\omega},\vartheta,\Omega,t)d\vartheta
d\omega=1,\qquad\forall\;t\in[0,T[.$ (3.25)

We recall that ${\Gamma}_{\Omega,{\varrho}_{n-1}}$ denotes the fundamental
solution of the operator $\mathscr{L}_{{\Omega},{\varrho}_{n-1}}$ appearing in
(3.24). The assertion _(.4)_ of Theorem 2.2 yields

$\displaystyle{\Gamma}_{\Omega,{\varrho}_{n-1}}({\omega},\vartheta,t;\xi,\eta,0)$
$\displaystyle\leq\overline{C}_{{\Omega},T}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;\xi,\eta,0),$
(3.26)
$\displaystyle|\partial{\Gamma}_{\Omega,{\varrho}_{n-1}}({\omega},\vartheta,t;\xi,\eta,0)|$
$\displaystyle\leq\frac{\overline{C}_{{\Omega},T}}{\sqrt{t^{i}}}{\Gamma}^{\varepsilon}({\omega},\vartheta,t;\xi,\eta,0),$

where the constant $\overline{C}_{{\Omega},T}>0$ only depends on
${\varepsilon},T$, and on
$\|\Phi_{{\varrho_{n-1}}}\|_{L^{\infty}(S_{T}^{*})}$. Moreover, also using
(3.7) and (3.25), we see that

$\begin{split}\|\Phi_{{\varrho_{n-1}}}\|_{L^{\infty}(S_{T}^{*})}&\leq|\Omega|+K,\\\
|\Phi_{{\varrho_{n-1}}}({\omega}_{1},\vartheta_{1},\Omega,t)-\Phi_{{\varrho_{n-1}}}({\omega}_{2},\vartheta_{2},\Omega,t)|&\leq
K\,\mathrm{d}(({\omega}_{1},\vartheta_{1},\Omega,t),({\omega}_{2},\vartheta_{2},\Omega,t)),\end{split}$
(3.27)

for every
$({\omega}_{1},\vartheta_{1},\Omega,t),({\omega}_{2},\vartheta_{2},\Omega,t)\in
S_{T}$, and for every $n\in\mathbb{N}$. Note that, in particular, the constant
$\overline{C}_{{\Omega},T}$ in (3.26) does not depend on $n$. In the sequel,
we will use the fact that, from the boundedness and the Lipschitz continuity
of the function $\Phi_{{\varrho_{n-1}}}$, the $\beta$-Hölder continuity of
$\Phi_{{\varrho_{n-1}}}$ directly follows for every $\beta\in]0,1[$.

As a consequence of Lemma 3.1, the sequence
$\\{\varrho_{n}\\}_{n\in\mathbb{N}}$ has an uniform exponential decay in
$\omega$, that is

$0<\varrho_{n}({\omega},\vartheta,t)\leq
C_{{\Omega},T}\,\mathrm{e}^{-\overline{M}\omega^{2}},\qquad\forall({\omega},\vartheta,t)\in\mathbb{R}^{2}\times]0,T[,\quad
n\in\mathbb{N}.$ (3.28)

Moreover, $\partial^{2}_{\omega}\varrho_{n}$, and $Y\varrho_{n}$ have an
uniform exponential decay in $\omega$ as well, with a constant
$C_{{\Omega},T}/{t}$. Thus, the sequence $\\{\varrho_{n}\\}_{n\in\mathbb{N}}$
is equi-bounded and equi-continuous on every compact subset of
$\mathbb{R}^{2}\times]0,T]$. Hence, there exists a subsequence
$\\{\varrho_{n_{k}}\\}_{k\in\mathbb{N}}$ and a function $\varrho\in
C(S^{*}_{T})$ such that

$\varrho_{n_{k}}\rightarrow\varrho,\quad\text{as}\quad{k\to+\infty},$ (3.29)

uniformly on every compact subset of $\mathbb{R}^{2}\times]0,T]$.

As said above, we need to show that the whole sequence
$\\{\varrho_{n}\\}_{n\in\mathbb{N}}$ converges to $\varrho$. We will prove a
stronger result, namely

$\|\varrho_{n}-\varrho\|_{L^{\infty}(S_{T}^{*})}\rightarrow
0,\quad\text{as}\quad{n\to+\infty},$ (3.30)

where ${S_{T}^{*}}=\mathbb{R}^{2}\times]0,T[$. Aiming at proving (3.30), we
preliminarily note that the constant $\overline{C}_{{\Omega},T}$ in (3.26)
depends on the quantities ${\Omega}$ and $T$ as follows

$\overline{C}_{{\Omega},T}=C_{\varepsilon}+\frac{\sqrt{T}}{2}\sum_{j=1}^{+\infty}(|{\Omega}|+K)^{j}(1+\sqrt{T})^{j}\frac{\sqrt{\pi}}{{\Gamma}_{E}(\frac{j}{2})},$
(3.31)

where $C_{\varepsilon}>0$ depends only on ${\varepsilon}$, which is fixed, and
${\Gamma}_{E}$ denotes the Euler Gamma function. We obtain the above
expression by repeating the computations made in [6] in the particular case of
the operator $\mathscr{L}$. Then, by recalling the definition (3.16) of
$C_{{\Omega},T}$, the hypothesis (1.21) yields

$G:=\int_{\mathbb{R}}g({\Omega}){C}_{{\Omega},T}\,d{\Omega}<+\infty.$ (3.32)

Let us now fix a positive $t_{0}$, which needs to be chosen small enough to
have

$q:=2K\pi^{\frac{3}{2}}\overline{C}_{{\Omega},T}G\sqrt{\tfrac{t_{0}}{A}}<1,$
(3.33)

where we have set

$A:=\frac{\overline{M}}{1+2(1+{\varepsilon})\overline{M}(\mathrm{e}^{2T}-1)}.$
(3.34)

Here $\overline{M}$ is the constants in the exponential decay. Then, we define
${\widehat{k}}=\lfloor T/t_{0}\rfloor+1$ and we set $T_{k}=kt_{0}$ for
$k=0,...,{\widehat{k}}-1$, and $T_{\widehat{k}}=T$. For every $n\in\mathbb{N}$
we consider the function

$v_{n}({\omega},\vartheta,\Omega,t):=\varrho_{n+1}({\omega},\vartheta,\Omega,t)-\varrho_{n}(\omega,\vartheta,\Omega,t).$
(3.35)

It is the unique bounded solution to the following Cauchy problem

$\begin{cases}\mathscr{L}_{{\Omega},\varrho_{n}}v_{n}=(\Phi_{\varrho_{n}}-\Phi_{\varrho_{n-1}})\partial_{\omega}\varrho_{n},\quad\mathrm{in}\;\mathbb{R}^{2}\times]T_{k},T_{k+1}[,\\\
v_{n}({\omega},\vartheta,\Omega,T_{k})=\varrho_{n+1}({\omega},\vartheta,\Omega,T_{k})-\varrho_{n}({\omega},\vartheta,\Omega,T_{k}).\end{cases}$
(3.36)

Now we represent the function $v_{n}$ by using the fundamental solution
${\Gamma}_{{\Omega},\varrho_{n}}$ of $\mathscr{L}_{{\Omega},\varrho_{n}}$. The
above PDE can be written as $\mathscr{L}_{{\Omega},\varrho_{n}}v_{n}=f_{n}$,
with

$f_{n}({\omega},\vartheta,\Omega,t):=(\Phi_{\varrho_{n}}(\vartheta,\Omega,t)-\Phi_{\varrho_{n-1}}(\vartheta,\Omega,t))\partial_{\omega}\varrho_{n}({\omega},\vartheta,\Omega,t).$
(3.37)

In order to apply Theorem 2.2, we claim that there exist three positive
constants $C_{1},\gamma$ and $\beta$, with $\beta,\gamma<1$, such that

$\begin{split}|f_{n}({\omega},\vartheta,\Omega,t)|&\leq\frac{C_{1}}{\sqrt{t}},\\\
|f_{n}({\omega}_{1},\vartheta_{1},\Omega,t)-f_{n}({\omega}_{2},\vartheta_{2},\Omega,t)|&\leq\frac{C_{1}}{t^{1-\gamma}}\;\mathrm{d}(({\omega}_{1},\vartheta_{1},\Omega,t),({\omega}_{2},\vartheta_{2},\Omega,t))^{\beta},\end{split}$
(3.38)

for every
$({\omega},\vartheta,\Omega,t)({\omega}_{1},\vartheta_{1},\Omega,t),({\omega}_{2},\vartheta_{2},\Omega,t)\in
S_{T}$, and for every $n\in\mathbb{N}$. The first inequality in (3.38)
directly follows from the first inequality in (3.27) and from the uniform
exponential decay of $\partial_{\omega}\varrho_{n}$. In order to prove the
second inequality, we claim that there exist two positive constants $C$ and
$\beta$, with $\beta<1$, such that

$|\partial_{\omega}\varrho_{n}({\omega}_{1},\vartheta_{1},\Omega,t)-\partial_{\omega}\varrho_{n}({\omega}_{2},\vartheta_{2},\Omega,t)|\leq\frac{C}{t^{\frac{1+\beta}{2}}}\;\mathrm{d}(({\omega}_{1},\vartheta_{1},\Omega,t),({\omega}_{2},\vartheta_{2},\Omega,t))^{\beta},$
(3.39)

for every
$({\omega}_{1},\vartheta_{1},\Omega,t),({\omega}_{2},\vartheta_{2},\Omega,t)\in
S_{T}$, and for every $n\in\mathbb{N}$. The conclusion of the proof of (3.27)
plainly follows from the exponential decay of $\partial_{\omega}\varrho_{n}$
combined with (3.27) and (3.39).

The estimate (3.39) is a direct consequence of the identity

$\partial\varrho_{n}({\omega},\vartheta,\Omega,t)=\int_{\mathbb{R}^{2}}\partial_{\omega}{\Gamma}_{\Omega,{{\varrho_{n}}}}({\omega},\vartheta,t;\xi,\eta,0)\varrho_{0}(\xi,\eta,\Omega)d\eta
d\xi.$ (3.40)

Then we use the following property the fundamental solution, proved in Theorem
3.2 of [7]: for every $\beta\in]0,1[$ there exists a positive constant
$c_{\beta}$ such that

$\displaystyle\big{|}\partial_{\omega}{\Gamma}_{{\Omega},\varrho_{n}}({\omega}_{1},\vartheta_{1},t;{\xi},\eta,0)-\partial_{\omega}{\Gamma}_{{\Omega},\varrho_{n}}$
$\displaystyle({\omega}_{2},\vartheta_{2},t;{\xi},\eta,0)\big{|}\leq$ (3.41)
$\displaystyle
c_{\beta}\frac{\mathrm{d}(({\omega}_{1},\vartheta_{1},t),(\omega_{2},\vartheta_{2},t))^{\beta}}{t^{\frac{1+\beta}{2}}}{\Gamma}^{\varepsilon}({\omega}_{1},\vartheta_{1},t;{\xi},\eta,0),$

for every
$t\in]0,T[,({\omega}_{1},\vartheta_{1}),({\omega}_{2},\vartheta_{2}),({\xi},{\eta})\in\mathbb{R}^{2}$,
being ${\Gamma}^{\varepsilon}$ the function defined in (2.3). Finally, by
using the exponential decay of the initial datum and (2.10) we find

$\displaystyle\big{|}\partial_{\omega}\varrho_{n}({\omega}_{1},\vartheta_{1},\Omega,t)-\partial_{\omega}\varrho_{n}({\omega}_{2},\vartheta_{2},\Omega,t)\big{|}\leq$
(3.42) $\displaystyle
c_{\beta}\frac{\mathrm{d}(({\omega}_{1},\vartheta_{1},t),(\omega_{2},\vartheta_{2},t))^{\beta}}{t^{\frac{1+\beta}{2}}}$
$\displaystyle\int_{\mathbb{R}^{2}}{\Gamma}^{\varepsilon}({\omega}_{1},\vartheta_{1},t;{\xi},\eta,0)\varrho_{0}({\xi},{\eta})d{\xi}d{\eta}\leq$
$\displaystyle\qquad\qquad\qquad\qquad\overline{c}_{\beta}\frac{\mathrm{d}(({\omega}_{1},\vartheta_{1},t),(\omega_{2},\vartheta_{2},t))^{\beta}}{t^{\frac{1+\beta}{2}}},$

for some $\overline{c}_{\beta}>0$. This concludes the proof of (3.38) with
$\beta$ arbitrarily chosen in $]0,1[$ and $\gamma=\frac{1-\beta}{2}$.

Once the bounds (3.38) have been proved, we represent $v_{n}$ as follows

$\displaystyle
v_{n}({\omega},\vartheta,\Omega,t)=\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,{{\varrho_{n}}}}$
$\displaystyle({\omega},\vartheta,t;\xi,\eta,T_{k})v_{n}({\xi},\eta,\Omega,T_{k})\,d\eta
d\xi$ (3.43)
$\displaystyle-\int_{T_{k}}^{t}\int_{\mathbb{R}^{2}}{\Gamma}_{\Omega,{{\varrho_{n}}}}({\omega},\vartheta,t;{\xi},\eta,\tau)f_{n}(\xi,\eta,{\Omega},\tau)\,d\eta
d\xi d\tau.$

In order to simplify the remaining part of the proof, we define recursively
the constants

$M_{1}:=\frac{\overline{M}}{1+\overline{M}(1+{\varepsilon})(\mathrm{e}^{2T_{1}}-1)},\quad
M_{k+1}:=\frac{M_{k}}{1+M_{k}(1+{\varepsilon})(\mathrm{e}^{2(T_{k+1}-T_{k})}-1)},\quad
k=1,...,{\widehat{k}}-1,$ (3.44)

and we note that, by a plain induction argument, we find

$M_{k}=\frac{\overline{M}}{1+\overline{M}(1+{\varepsilon})k(\mathrm{e}^{2t_{0}}-1)},\quad
M_{\hat{k}}=\frac{\overline{M}}{1+\overline{M}(1+{\varepsilon})((\hat{k}-1)(\mathrm{e}^{2t_{0}}-1)+(\mathrm{e}^{2(T-T_{\hat{k}-1})}-1))}.$
(3.45)

In particular, the inequalities below hold

$\overline{M}>M_{k}>M_{k+1}\geq A,\qquad k=1,...,{\widehat{k}}-1,$ (3.46)

since

$(\hat{k}-1)(\mathrm{e}^{2t_{0}}-1)+(\mathrm{e}^{2(T-T_{\hat{k}-1})}-1)\leq\hat{k}(\mathrm{e}^{2t_{0}}-1)\leq\left(\frac{T}{t_{0}}+1\right)(\mathrm{e}^{2t_{0}}-1)\leq
2(\mathrm{e}^{2T}-1).$ (3.47)

We now introduce the weighted norms

$\|\varrho_{n+1}-\varrho_{n}\|_{k}:=\sup_{S^{*}_{T}\cap]T_{k-1},T_{k}]}|\varrho_{n+1}({\omega},\vartheta,\Omega,t)-\varrho_{n}(\omega,\vartheta,\Omega,t)|\frac{\mathrm{e}^{M_{k}\omega^{2}}}{{C}_{{\Omega},T}},\qquad
k=1,...,{\widehat{k}}.$ (3.48)

From (3.46) and (3.28) it follows that

$\|\varrho_{n+1}-\varrho_{n}\|_{k}\leq 2,\qquad k=1,...,{\widehat{k}}.$ (3.49)

The term $\frac{\mathrm{e}^{M_{k}\omega^{2}}}{{C}_{{\Omega},T}}$ appearing in
the norm introduced in (3.48) is useful in order to deal with the term $f_{n}$
in (3.43). Indeed, later on we will rely on the following bound

$\displaystyle|\Phi_{\varrho_{n}}(\vartheta,t)-\Phi_{\varrho_{n-1}}(\vartheta,t)|\leq$
(3.50) $\displaystyle\leq
K\int_{\mathbb{R}}g(\Omega)\int_{\mathbb{R}}\int_{0}^{2\pi}|\sin(\vartheta-\vartheta^{\prime})(\varrho_{n}(\omega^{\prime},\vartheta^{\prime},\Omega,t)-\varrho_{n-1}(\omega^{\prime},\vartheta^{\prime},\Omega,t))|d\vartheta^{\prime}d\omega^{\prime}d\Omega\leq$
$\displaystyle\leq
K\|\varrho_{n}-\varrho_{n-1}\|_{k+1}\int_{\mathbb{R}}\int_{0}^{2\pi}\mathrm{e}^{-{M_{k+1}}\omega^{\prime
2}}d\vartheta^{\prime}d\omega^{\prime}\int_{\mathbb{R}}g(\Omega){C}_{{\Omega},T}d{\Omega}=\frac{2\pi^{\frac{3}{2}}KG}{\sqrt{M_{k+1}}}\|\varrho_{n}-\varrho_{n-1}\|_{k+1},$

for every $\vartheta\in\mathbb{R}$ and $t\in]T_{k},T_{k+1}[$, which will be
used to estimate $|f_{n}|$.

In order to prove (3.30) we claim that the following estimates

$\displaystyle\|\varrho_{n+1}-\varrho_{n}\|_{1}\leq$
$\displaystyle\,q\,\|\varrho_{n}-\varrho_{n-1}\|_{1},$ (3.51)
$\displaystyle\|\varrho_{n+1}-\varrho_{n}\|_{k+1}\leq$
$\displaystyle\,q\,\|\varrho_{n}-\varrho_{n-1}\|_{k+1}+\overline{C}_{{\Omega},T}\|\varrho_{n+1}-\varrho_{n}\|_{k},$

hold for all $n=2,3,..$ and $k=1,2,...,{\widehat{k}}-1$. Let’s denote by
$I_{1}$ and $I_{2}$ the two integral in the right hand side of (3.43). By
(2.12) and (2.21) a direct computation give us

$\displaystyle|I_{1}|\leq$
$\displaystyle\overline{C}_{{\Omega},T}\|\varrho_{{n+1}}-\varrho_{n}\|_{k}\,{C}_{{\Omega},T}\int_{\mathbb{R}}\widetilde{\Gamma}^{\varepsilon}({\omega},t;{\xi},T_{k})\mathrm{e}^{-M_{k}\xi^{2}}d\xi\leq$
(3.52) $\displaystyle\leq$
$\displaystyle\overline{C}_{{\Omega},T}\|\varrho_{{n+1}}-\varrho_{n}\|_{k}\,{C}_{{\Omega},T}\;\mathrm{exp}\left(-{\omega}^{2}\frac{M_{k}}{1+M_{k}(1+{\varepsilon})(\mathrm{e}^{2(T_{k+1}-T_{k})}-1)}\right)=$
$\displaystyle=$
$\displaystyle\overline{C}_{{\Omega},T}\|\varrho_{{n+1}}-\varrho_{n}\|_{k}\,{C}_{{\Omega},T}\mathrm{e}^{-M_{k+1}{\omega}^{2}}.$

In order to estimate $I_{2}$, we use again (2.12) and (2.21), together with
(3.50) and the exponential decay of $\partial_{\omega}\varrho_{n}$

$\displaystyle|I_{2}|$
$\displaystyle\leq\frac{2\pi^{\frac{3}{2}}KG\overline{C}_{{\Omega},T}}{\sqrt{M_{k+1}}}\|\varrho_{n}-\varrho_{n-1}\|_{k+1}{C}_{{\Omega},T}\int_{T_{k}}^{t}\int_{\mathbb{R}}\frac{\widetilde{\Gamma}^{\varepsilon}({\omega},t;{\xi},{\tau})\mathrm{e}^{-\overline{M}\xi^{2}}}{\sqrt{\tau}}d\xi
d\tau\leq$ (3.53) $\displaystyle\leq
2\pi^{\frac{3}{2}}KG\overline{C}_{{\Omega},T}\sqrt{\frac{t_{0}}{A}}\|\varrho_{n}-\varrho_{n-1}\|_{k+1}{C}_{{\Omega},T}\mathrm{e}^{-M_{k+1}{\omega}^{2}}=q\|\varrho_{n}-\varrho_{n-1}\|_{k+1}{C}_{{\Omega},T}\mathrm{e}^{-M_{k+1}{\omega}^{2}},$

where we have used the inequalities in (3.46). This concludes the proof of
(3.51).

We next recall that Lemma 3.4 in [2] states that (3.51) imply the inequalities

$\|\varrho_{n+1}-\varrho_{n}\|_{k}\leq
q^{n-2}\sum_{i=0}^{k-1}(\overline{C}_{{\Omega},T}(n-2))^{i}\|\varrho_{3}-\varrho_{2}\|_{k-i},$
(3.54)

for all $n=3,4,...$ and $k=1,...,{\widehat{k}}$. We then find

$\|\varrho_{n+1}-\varrho_{n}\|_{C(\overline{S^{*}_{T}})}\leq\sum_{k=1}^{{\widehat{k}}}\|\varrho_{n+1}-\varrho_{n}\|_{k}\leq\sum_{k=1}^{{\widehat{k}}}q^{n-2}\sum_{i=0}^{k-1}(\overline{C}_{{\Omega},T}(n-2))^{i}\|\varrho_{3}-\varrho_{2}\|_{k-i}.$
(3.55)

By using the elementary inequality
$(\overline{C}_{{\Omega},T}(n-2))^{i}\leq(n-2)^{\widehat{k}}(1+\overline{C}_{{\Omega},T})^{{\widehat{k}}}$
and

$\sum_{k=1}^{{\widehat{k}}}\sum_{i=0}^{k-1}\|\varrho_{3}-\varrho_{2}\|_{k-i}=\sum_{k=1}^{{\widehat{k}}}\sum_{j=1}^{k}\|\varrho_{3}-\varrho_{2}\|_{j}\leq\sum_{k=1}^{{\widehat{k}}}\sum_{j=1}^{{\widehat{k}}}\|\varrho_{3}-\varrho_{2}\|_{j}={\widehat{k}}\sum_{j=1}^{{\widehat{k}}}\|\varrho_{3}-\varrho_{2}\|_{j},$
(3.56)

we finally have

$\|\varrho_{n+1}-\varrho_{n}\|_{C(\overline{S^{*}_{T}})}\leq
q^{n-2}(n-2)^{\widehat{k}}(1+\overline{C}_{{\Omega},T})^{{\widehat{k}}}{\widehat{k}}\sum_{j=1}^{\widehat{k}}\|\varrho_{3}-\varrho_{2}\|_{j}.$
(3.57)

Then, if we set

$V:=(1+\overline{C}_{{\Omega},T})^{{\widehat{k}}}{\widehat{k}}\sum_{j=1}^{\widehat{k}}\|\varrho_{3}-\varrho_{2}\|_{j},$
(3.58)

and we notice that $V$ does not depend on $n$, we eventually obtain

$\sum_{n=3}^{+\infty}\|\varrho_{n+1}-\varrho_{n}\|_{C(\overline{S^{*}_{T}})}\leq
V\sum_{n=3}^{+\infty}q^{n-2}(n-2)^{\widehat{k}}=V\sum_{n=1}^{+\infty}q^{n}n^{\widehat{k}}<\infty.$
(3.59)

This accomplishes the proof of (3.30).

From (3.30) and (3.28) it follows that $\varrho$ satisfies the identity
(3.22), it has property (E) in the set $S_{T}^{*}$, and

$\varrho({\omega},\vartheta,\Omega,0)=\varrho_{0}({\omega},\vartheta,\Omega),\qquad\mathrm{for\;every}\
({\omega},\vartheta,\Omega)\in\mathbb{R}^{3}.$ (3.60)

Moreover, ${\varrho}$ is a classical solution to
$\mathscr{L}_{{\Omega},{\varrho}}{\varrho}=0$ in $S_{T}^{*}$, thanks to the
bounds (3.5) in Lemma 3.1, that hold for $\varrho_{n}$, uniformly with respect
to $n$. This completes the proof of Proposition 3.2. $\square$

We now prove that, under the additional assumption that $g$ has compact
support, there is only one solution satisfying the property (E).

###### Proposition 3.3

Let us suppose that $g$ has support in the set $[-{\Omega}_{1},{\Omega}_{1}]$,
and let $\varrho_{1}$ and $\varrho_{2}$ be two classical solutions to the
Cauchy problem (1.16). If $\varrho_{1}$ and $\varrho_{2}$ satisfy property
(E), then $\varrho_{1}=\varrho_{2}$.

Proof. Suppose that $\varrho_{1}$ and $\varrho_{2}$ are two solutions to the
Cauchy problem (1.16), both verifying property (E), with constants
$\overline{M},C_{{\Omega},T}$. We define
$\overline{\varrho}:=\varrho_{1}-\varrho_{2}$, and we note that the following
quantities are finite for every ${\tau},t\in[0,T[$

$\displaystyle\widetilde{\varphi}({\tau}):=\int_{\mathbb{R}}\max_{\vartheta\in\mathbb{R},{\Omega}\in[-{\Omega}_{1},{\Omega}_{1}]}\left|\frac{\overline{\varrho}({\omega},\vartheta,{\Omega},{\tau})}{C_{{\Omega},T}}\right|d{\omega},\qquad{\varphi}(t):=\sup_{0\leq{\tau}\leq
t}\widetilde{\varphi}({\tau}).$ (3.61)

Moreover, $\overline{\varrho}$ is a classical solution to

$\begin{dcases}\partial^{2}_{\omega\omega}\overline{\varrho}+\omega\partial_{\omega}\overline{\varrho}-\omega\partial_{\vartheta}\overline{\varrho}+\overline{\varrho}-\partial_{t}\overline{\varrho}=-\left((\Phi_{\varrho_{2}}-\Phi_{\varrho_{1}})\partial_{\omega}\varrho_{2}-\Phi_{\varrho_{1}}\partial_{\omega}(\varrho_{2}-\varrho_{1})\right),\\\
\overline{\varrho}({\omega},\vartheta,\Omega,0)=0.\end{dcases}$ (3.62)

Arguing as in the proof of Theorem 1.3, we see that $\overline{\varrho}$ can
be represented as

$\overline{\varrho}({\omega},\vartheta,\Omega,t)=\int_{0}^{t}\int_{\mathbb{R}^{2}}\overline{{\Gamma}}({\omega},\vartheta,t;\xi,\eta,{\tau})\left((\Phi_{\varrho_{2}}-\Phi_{\varrho_{1}})\partial_{\xi}\varrho_{2}-\Phi_{\varrho_{1}}\partial_{\xi}(\varrho_{2}-\varrho_{1})\right)(\xi,{\eta},{\Omega},{\tau})d\xi
d{\eta}d{\tau},$ (3.63)

where $\overline{{\Gamma}}$ is the fundamental solution of

$\overline{\mathscr{L}}=\partial^{2}_{\omega\omega}+\omega\partial_{\omega}-\omega\partial_{\vartheta}+1-\partial_{t}.$
(3.64)

Our aim is to obtain an estimate for ${\varphi}(t)$, using the representation
formula (3.63), which will implies ${\varphi}(t_{0})=0$ for a sufficiently
small $t_{0}$. Moreover, $t_{0}$ only depends on the operator
$\overline{\mathscr{L}}$ and on the quantities appearing in property (E), then
the argument can be repeated in the interval $[t_{0},2t_{0}]$, then in
$[2t_{0},3t_{0}]$, and so on, hence we conclude that
$\overline{\varrho}({\omega},\vartheta,\Omega,t)=0$ for every
$({\omega},\vartheta,\Omega,t)\in\mathbb{R}^{3}\times[0,T[$.

For this purpose, we first recall the constant $G$ introduced in (3.32), and
we note that the following estimate

$\displaystyle|\Phi_{\varrho_{2}}({\eta},{\tau})-$
$\displaystyle\Phi_{\varrho_{1}}({\eta},{\tau})|\leq$ (3.65) $\displaystyle
K\int_{\mathbb{R}}g(\Omega)\int_{\mathbb{R}}\int_{0}^{2\pi}|\sin({\eta}-\vartheta^{\prime})(\varrho_{2}(\omega^{\prime},\vartheta^{\prime},\Omega,{\tau})-\varrho_{1}(\omega^{\prime},\vartheta^{\prime},\Omega,{\tau}))|d\vartheta^{\prime}d\omega^{\prime}d\Omega\leq$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\leq
2\pi KG{\varphi}(t),\qquad{\eta}\in\mathbb{R},{\tau}\in[0,t],$

plainly follows from the definition of $\Phi_{\varrho_{1}}$ and
$\Phi_{\varrho_{2}}$. Then, from this bound, the exponential decay property of
the derivative $\partial_{\omega}\varrho_{2}$ and the bound (3.27) applied to
$\Phi_{\varrho_{1}}$ we find that

$\displaystyle\int_{\mathbb{R}}|\overline{\varrho}({\omega},\vartheta,$
$\displaystyle\Omega,t)|d{\omega}\leq$ (3.66) $\displaystyle 2\pi
KG{\varphi}(t)\int_{0}^{t}\int_{\mathbb{R}^{3}}\overline{{\Gamma}}({\omega},\vartheta,t;\xi,\eta,{\tau})\frac{C_{{\Omega},T}}{\sqrt{{\tau}}}\mathrm{e}^{-{\overline{M}\xi\,^{2}}}d{\eta}d{\omega}d\xi
d{\tau}+$
$\displaystyle(|\Omega|+K)\int_{0}^{t}\int_{\mathbb{R}^{3}}|\partial_{\xi}\overline{{\Gamma}}({\omega},\vartheta,t;\xi,\eta,{\tau})|(\varrho_{2}-\varrho_{1})(\xi,{\eta},{\Omega},{\tau})d{\omega}d{\eta}d\xi
d{\tau}=I_{1}+I_{2}.$

By a direct computation we find that

$I_{1}\leq
C_{{\Omega},T}\;\frac{\pi^{\frac{3}{2}}KG}{\sqrt{\overline{M}}}\sqrt{t}{\varphi}(t),$
(3.67)

Moreover, by (2.21), together with definition (3.61) and bounds (3.16),
(3.31), we obtain

$I_{2}\leq C_{{\Omega},T}\overline{C}\frac{\sqrt{t}}{2}{\varphi}(t),$ (3.68)

for some positive constant $\overline{C}$ that depends only on ${\Omega}_{1}$.
Thus, since the previous estimates gives us

${\varphi}(t)\leq{\varphi}(t)\left(\frac{\pi^{\frac{3}{2}}KG}{\sqrt{\overline{M}}}\sqrt{t}+\overline{C}\frac{\sqrt{t}}{2}\right),$
(3.69)

we can state that there exists $t_{0}$ such that, if $T\leq t_{0}$, we have
${\varphi}(t)\leq c\,{\varphi}(t)$ for some $c<1$, and therefore
${\varphi}(t)=0$.

Finally, by iterating this arguments in time intervals $[kt_{0},(k+1)t_{0}]$
the thesis follows. $\square$

Proof.of Theorem 1.3 The proof of the Theorem follows from the results we have
established in Lemma 3.1, Proposition 3.2 and Proposition 3.3. The continuity
dependence of the solution with respect to ${\Omega}$ is proved in [7].
$\square$

##  4 Finite difference method

The purpose of this Section is to develop a numerical scheme in the spirit of
the analytical framework introduced in Section 2. We propose therefore a
finite difference scheme where we discretize the Lie derivative instead of the
classical derivatives with respect to $\vartheta$ and $t$. In particular, we
are following the pattern described in [8] and [20], where an accurate
analysis concerning the discretization of degenerate Kolmogorov operators has
been made. The discretization we would like to carry out would make us
approximate
$Y\varrho=\omega\partial_{\omega}\varrho-\omega\partial_{\vartheta}\varrho-\partial_{t}\varrho$
with the following quotient

$\frac{\varrho({\omega}\mathrm{e}^{\delta},\omega-\omega\mathrm{e}^{\delta}-\vartheta,t-{\delta})-\varrho({\omega},\vartheta,{\Omega},t)}{{\delta}}.$
(4.1)

The main difficulty when setting an appropriate grid is represented by the
factor $\mathrm{e}^{\delta}$, which arises from the term
${\omega}\partial_{\omega}$ in the Lie derivative. This problem can be
overcome by observing that, thanks to the continuity w.r.t. the derivative
$\partial_{\omega}$, the solution is also continuous with respect to the Lie
derivative $Y_{r}=-\omega\partial_{\vartheta}-\partial_{t}$, whose
discretization is easier.

In our analysis we assume that the natural frequency distribution $g$ is
compactly supported in the set $[-{\Omega}_{1},{\Omega}_{1}]$, so that the
solution we approximate is unique, according to Proposition 3.3.

We list here the various approximations that we use for the numerical scheme:

$\displaystyle\partial^{2}_{{\omega}{\omega}}\varrho({\omega},\vartheta,{\Omega},t)=\frac{\varrho({\omega}+\Delta{\omega},\vartheta,{\Omega},t)-2\varrho({\omega},\vartheta,{\Omega},t)+\varrho({\omega}-\Delta{\omega},\vartheta,{\Omega},t)}{(\Delta{\omega})^{2}}+o({\Delta}{\omega})^{2},$
(4.2)
$\displaystyle\partial_{{\omega}}\varrho({\omega},\vartheta,{\Omega},t)=\frac{\varrho({\omega}+\Delta{\omega},\vartheta,{\Omega},t)-\varrho({\omega}-\Delta{\omega},\vartheta,{\Omega},t)}{2(\Delta{\omega})}+o({\Delta}{\omega})^{2},$
$\displaystyle
Y_{r}\varrho({\omega},\vartheta,{\Omega},t)=\frac{\varrho({\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t-{\Delta}t)-\varrho({\omega},\vartheta,{\Omega},t)}{{\Delta}t}+o({{\Delta}t}).$

Finally, we approximate the nonlinear term by

$-{\Omega}-K\int_{-{\Omega}_{1}}^{{\Omega}_{1}}\int_{0}^{2\pi}\int_{-G_{\omega}}^{G_{\omega}}\varrho({\omega}^{\prime},\vartheta^{\prime},{\Omega},t-{\Delta}t)\sin(\vartheta-\vartheta^{\prime})g({\Omega}^{\prime})d{\omega}^{\prime}d\vartheta^{\prime}d{\Omega}^{\prime}+o({{\Delta}t}).$
(4.3)

This approximation introduces the constant $G_{\omega}$ to overcome the
problem of an unlimited interval in ${\omega}$. We observe that, thanks to the
exponential decay, we can make this last approximation truly negligible.
Finally, we approximate this integral by the method of rectangles, and we
denote as $\widetilde{\Phi}_{\varrho}$ this last approximation. As in the
classical case, the various remainders depend on the $L^{\infty}$ norm of the
solution and of some of its derivatives.

In the following, in order to provide a comprehensive analysis of the model,
we will include the cases where the inertial term $m$ and the noise term $D$
are not necessarily equal to one, in order to investigate the changes in the
solution as these parameters vary.

We define the approximate operator $\mathscr{L}_{a}$ as

$\displaystyle\mathscr{L}_{a}\varrho({\omega},\vartheta,$
$\displaystyle{\Omega},t):=$ (4.4) $\displaystyle=$
$\displaystyle\frac{D}{m^{2}}\frac{\varrho({\omega}+\Delta{\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t)-2\varrho({\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t)+\varrho({\omega}-\Delta{\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t)}{(\Delta{\omega})^{2}}$
$\displaystyle+$
$\displaystyle\frac{1}{m}\left({\omega}+\widetilde{\Phi}_{\varrho}(\vartheta-{\omega}{\Delta}t,{\Omega},t)\right)\frac{\varrho({\omega}+\Delta{\omega},\vartheta-{\omega}{\Delta}t,,{\Omega},t)-\varrho({\omega}-\Delta{\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t)}{2(\Delta{\omega})}$
$\displaystyle+$
$\displaystyle\frac{\varrho({\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t)-\varrho({\omega},\vartheta,{\Omega},t+{\Delta}t)}{{\Delta}t}+\frac{1}{m}\varrho({\omega},\vartheta-{\omega}{\Delta}t,{\Omega},t).$

The previous approximations make $\mathscr{L}_{a}$ consistent with
$\mathscr{L}_{\varrho}$; indeed when $G_{\omega}=+\infty$ we have:

$\mathscr{L}_{\varrho}\varrho({\omega},\vartheta,{\Omega},t)-\mathscr{L}_{a}\varrho({\omega},\vartheta,{\Omega},t)=o(({\Delta}{\omega})^{2}+{{\Delta}t}).$
(4.5)

In order to define the corresponding grid, let us examine the domain in which
we solve the problem: as anticipated above, we suppose that ${\omega}$ belongs
to the bounded interval $[-G_{\omega},G_{\omega}]$; moreover, by the
periodicity in $\vartheta$ we can suppose that this variable belongs to
$[0,2\pi]$ and then we work with periodic conditions. Thus, the natural grid
where we discretize our problem is

$\displaystyle G_{r}=$
$\displaystyle\left\\{i{{\Delta}{\omega}},j{{\Delta}{\omega}}{{\Delta}t},k{\Delta}{\Omega},n{{\Delta}t},\right.$
(4.6) $\displaystyle\left.-\frac{G_{\omega}}{{\Delta}{\omega}}\leq
i\leq\frac{G_{\omega}}{{\Delta}{\omega}},\;\;0\leq
j\leq\frac{2\pi}{{\Delta}{\omega}{\Delta}t},\;0\leq
n\leq\frac{T}{{\Delta}t},\;-\frac{{\Omega}_{1}}{{\Delta}{\Omega}}\leq
k\leq\frac{{\Omega}_{1}}{{\Delta}{\Omega}}\right\\},$ $\displaystyle G_{0}=$
$\displaystyle
G_{r}\cap\\{n=0\\},\;\;\;\;\;G_{b}=G_{r}\cap\left\\{|i|=\frac{G_{\omega}}{{\Delta}{\omega}}\right\\},\;\;\;\;\;G_{in}=G_{r}\setminus(G_{0}\cup
G_{b}).$

We conclude with one last remark. As we have shown in Lemma 3.1, if the
initial datum $\varrho_{0}$ is normalized for every ${\Omega}$, then the same
condition holds true for the solution $\varrho$; in order to achieve this, we
introduce the following normalization condition

$({\Delta}{\omega})^{2}({\Delta}t)\sum_{\underline{i},\underline{j}}\varrho^{n}_{\underline{i},\underline{j},k}=1,\qquad\forall\;\;0\leq
n\leq\frac{T}{{\Delta}t},\;-\frac{{\Omega}_{1}}{{\Delta}{\Omega}}\leq
k\leq\frac{{\Omega}_{1}}{{\Delta}{\Omega}},$ (4.7)

The approximate Cauchy problem is

$\begin{cases}\mathscr{L}_{a}\varrho=0,\;\;\;\;\;&\mathrm{in}\;G_{in},\\\
\varrho=\varrho_{0},\;\;\;\;\;&\mathrm{in}\;G_{0},\\\
\varrho=0,\;\;\;\;\;&\mathrm{in}\;G_{b},\\\ \end{cases}$ (4.8)

with $\varrho$ that verifies (4.7) and such that
$\varrho({\omega},\vartheta,{\Omega},t)=\varrho({\omega},\vartheta+2\pi,{\Omega},t)$.
While the first two equations are the classical approximation of the Cauchy
problem (1.16), the last is intended to approximate the exponential decay
condition.

As in the heat equation case, we need to introduce some stability conditions.

###### Theorem 4.1

Suppose that the grid $G_{r}$ is such that

${\Delta}{\omega}\leq\frac{\sqrt{2D}}{m},\qquad{\Delta}t\leq\frac{m^{2}({\Delta}{\omega})^{2}}{2D-m({\Delta}{\omega})^{2}},\qquad
G_{\omega}\leq\frac{2D}{m{\Delta}{\omega}}-{\Omega}_{1}-K,$ (4.9)

then every solution to

$\begin{cases}\mathscr{L}_{a}\varrho\leq 0,\;\;\;\;\;&\mathrm{in}\;G_{in},\\\
\varrho\geq 0,\;\;\;\;\;&\mathrm{in}\;G_{0},\\\
\varrho=0,\;\;\;\;\;&\mathrm{in}\;G_{b},\end{cases}$ (4.10)

that verifies (4.7) and such that
$\varrho({\omega},\vartheta,{\Omega},t)=\varrho({\omega},\vartheta+2\pi,{\Omega},t)$,
is nonnegative in $G_{r}$.

Proof. The proof runs by induction. Let us set

$\varrho^{n}_{i,j,k}=\varrho(i{\Delta}{\omega},j{\Delta}{\omega}{\Delta}t,k{\Delta}{\Omega},n{\Delta}t).$
(4.11)

As $\varrho\geq 0$ in $G_{0}$, the condition holds for $n=0$. Let now suppose
true the claim for $n$.

We notice that $\mathscr{L}_{a}\varrho\geq 0$ can be read as

$\displaystyle\varrho^{n+1}_{i,j,k}$
$\displaystyle\geq\left(1-\frac{2D{\Delta}t}{m^{2}({\Delta}{\omega})^{2}}+\frac{{\Delta}t}{m}\right)\varrho^{n}_{i,j-i,k}$
(4.12)
$\displaystyle+\left(\frac{D{\Delta}t}{m^{2}({\Delta}{\omega})^{2}}-\frac{{\Delta}t}{2{\Delta}{\omega}m}(i\Delta{\omega}+\widetilde{\Phi}_{\varrho}((j-i){\Delta}{\omega},k{\Delta}{\Omega},n{\Delta}t))\right)\varrho^{n}_{i-1,j-i,k}$
$\displaystyle+\left(\frac{D{\Delta}t}{m^{2}({\Delta}{\omega})^{2}}+\frac{{\Delta}t}{2{\Delta}{\omega}m}(i{\Delta}{\omega}+\widetilde{\Phi}_{\varrho}((j-i){\Delta}{\omega},k{\Delta}{\Omega},n{\Delta}t)\right)\varrho^{n}_{i+1,j-i,k},$

where

$\widetilde{\Phi}_{\varrho}((j-i){\Delta}{\omega},k{\Delta}{\Omega},n{\Delta}t)=-k{\Delta}{\Omega}-K{\Delta}{\Omega}({\Delta}{\omega})^{2}({\Delta}t)\sum_{\underline{i},\underline{j},\underline{k}}\varrho^{n}_{\underline{i},\underline{j},\underline{k}}g(\underline{k})\sin(j-i+\underline{i}).$
(4.13)

We employ the normalization condition (4.7) and the normalization of $g$ to
note that $|\widetilde{\Phi}_{\varrho}|\leq K+{\Omega}_{1}$, then; if (4.9)
holds, it follows $\varrho^{n+1}_{i,j,k}\geq 0$. $\square$

We illustrate now some numerical results. The following quantities have been
analysed

$\displaystyle|r(t)|=\left|\int_{0}^{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}\vartheta}\varrho({\omega},\vartheta,\Omega,t)g(\Omega)d\Omega
d{\omega}d\vartheta\right|,$ (4.14)
$\displaystyle|s(t)|=\left|\int_{0}^{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}{\omega}}\varrho({\omega},\vartheta,\Omega,t)g(\Omega)d\Omega
d{\omega}d\vartheta\right|.$

The first one represents the phase coherence, and gives us information about
the phase synchrony, while the latter is the analogous for frequencies.

We have considered identical oscillators with natural frequency ${\Omega}=0$,
and with the following initial conditions

$\varrho_{0}({\omega},\vartheta)=\frac{\mathrm{e}^{-{{\omega}^{2}}}(\sin(\vartheta)+1)}{N({\omega}+1)},\;\;\;\;\;N=\int_{\mathbb{R}}\int_{0}^{2\pi}\frac{\mathrm{e}^{-{\omega}^{2}}(\sin(\vartheta)+1)}{({\omega}+1)}d{\omega}d\vartheta.$
(4.15)

As we could have expected, we see in Figure 2 that an increase in the coupling
strength $K$ leads to a greater phase synchrony, while in Figure 2 we see that
the frequency synchronization seems to be asymptotically independent of $K$.

Figure 1: Time evolution of the phase coherence, $m=1,D=1,T=10,\Delta
t=0.0317$.

Figure 2: Time evolution of the frequency coherence, $m=1,D=1,T=10,\Delta
t=0.0317$.

Conversely, changes in the inertial term $m$ seem to have the opposite nature:
indeed, in Figure 4 we see that the asymptotic behaviour of phase
synchronization does not change; on the other side, in Figure 4 we notice that
the behaviour of frequency synchronization significantly depends on this
parameter, as one can observe by comparing this case with the one in Figure 2.
In particular, we see that the asymptotic behaviour of the frequency coherence
increases as $m$ increases.

Figure 3: Time evolution of the phase coherence, $D=1,K=6,T=10,\Delta
t=0.0079$.

Figure 4: Time evolution of the frequency coherence, $D=1,K=6,T=10,\Delta
t=0.0079$.

Regarding changes in the noise term $D$, we see what is expected in the
classical Kuramoto model. We remind indeed (see for instance [23]) that in the
original model ($m=D=0$) whenever we have identical oscillators the phase
synchrony is total (_i.e._ $|r|=1$). Moreover, these oscillators synchronize
at the same frequency (therefore $|s|=1$). Our numerical results are in good
agreement with the theoretical ones: indeed, in Figure 6 and Figure 6 we see
that the phase and frequency coherence increase for decreasing $D$.

Figure 5: Time evolution of the phase coherence, $m=1,K=6,T=10,\Delta
t=0.0317$.

Figure 6: Time evolution of the frequency coherence, $m=1,K=6,T=10,\Delta
t=0.0317$.

Figure 7: Time evolution of the phase coherence, $m=1,D=1,K=4,T=10,\\\ \Delta
t=0.053$.

Figure 8: Time evolution of the frequency coherence, $m=1,D=1,K=4,T=10,\\\
\Delta t=0.053$.

Finally, in Figure 8 and Figure 8 we compare two different initial conditions

$\displaystyle\varrho_{0,1}({\omega},\vartheta)=\frac{\mathrm{e}^{-{\omega}^{2}}(\sin(\frac{\vartheta}{2})+1)}{N_{1}({\omega}+1)},\;\;\;\;\;N_{1}=\int_{\mathbb{R}}\int_{0}^{2\pi}\frac{\mathrm{e}^{-{\omega}^{2}}(\sin(\frac{\vartheta}{2})+1)}{({\omega}+1)}d{\omega}d\vartheta,$
(4.16)
$\displaystyle\varrho_{0,2}({\omega},\vartheta)=\frac{\mathrm{e}^{-{\omega}^{2}}(\sin(\vartheta)+1)}{N_{2}({\omega}+1)},\;\;\;\;\;N_{2}=\int_{\mathbb{R}}\int_{0}^{2\pi}\frac{\mathrm{e}^{-{\omega}^{2}}(\sin(\vartheta)+1)}{({\omega}+1)}d{\omega}d\vartheta,$

and we observe two quite different behaviours, suggesting great susceptibility
to initial conditions.

We conclude with one last remark. These numerical results are in agreement
with the ones in [1], where the finite size system (1.13) was analysed. The
point we are making is that we have used a different method here: indeed, in
[1] the authors used a Monte Carlo method (thus a stochastic method) in the
finite size case (with $N=30000$ oscillators), while we have employed a finite
difference method to analyse (1.14), that is the limit of $\eqref{spig}$ as
$N\rightarrow\infty$.

## References

  * [1] J. A. Acebroan, R. Spigler, Adaptive frequency model for phase-frequency synchronization in large populations of globally coupled nonlinear oscillators. Phys. Rev. Lett. 81, 14 September 1998, 229–232.
  * [2] D. R. Akhmetov, M. M. Lavrentiev Jr., R. Spigler, Nonlinear integroparabolic equations on unbounded domain: Existence of classical solutions with special properties. Siberian Math. J. 42 (2001), 495–516.
  * [3] D. R. Akhmetov, M. M. Lavrentiev Jr., R. Spigler, Regularizing a nonlinear integroparabolic Fokker-Planck equation with space-periodic solutions: Existence of strong solutions. Siberian Math. J. 42 (2001), 693–714.
  * [4] D. R. Akhmetov, M. M. Lavrentiev Jr., R. Spigler, Existence and uniqueness of classical solutions to certain nonlinear integrodifferential Fokker-Planck-type equations. Electron. J. Differential Equations, Vol. 2002 (2002), No. 24, pp. 1–17.
  * [5] F. Anceschi, S. Polidoro, A survey on the classical theory for Kolmogorov equation. Le Matematiche, 2020, 75.1: 221–258.
  * [6] M. Di Francesco, A. Pascucci, On a class of degenerate parabolic equations of Kolmogorov type. AMRX Appl. Math. Res. Express, (2005), pp. 77–116.
  * [7] M. Di Francesco, A. Pascucci, A continuous dependence result for ultra-parabolic equations in option pricing. J. Math. Anal. Appl. (2007), doi:10.1016/j.jmaa.2007.03.031
  * [8] M. Di Francesco, P. Foschi, A. Pascucci, Analysis of an uncertain volatility model. Journal of Applied Mathematics and Decision Sciences, 2006, 2006.
  * [9] B. Ermentrout, An adaptive model for synchrony in the firefly Pteroptyx malaccae. J. Math. Biol. 29, no. 6 (1991), 571–585.
  * [10] L. Hörmander, Hypoelliptic second order differential equations. Acta Math., 119 (1967), pp. 147–171.
  * [11] N. A. Kolmogorov, Zufallige Bewegungen. (Zur Theorie der Brownschen Bewegung). Ann. of Math., Vol. 35-1 (1934), pp. 116–117.
  * [12] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence. Springer, Berlin, 1984.
  * [13] M. M. Lavrentiev Jr., R. Spigler, Uniform and optimal estimates for solutions to singularly perturbed parabolic equations. J. Evol. Equ. 7 (2007), 347–372.
  * [14] M. M. Lavrentiev Jr., R. Spigler, Existence and uniqueness of solutions to the Kuramoto-Sakaguchi parabolic integrodifferential equation. Differential Integral Equations 13 (2000), 649–667.
  * [15] M. M. Lavrentiev Jr., R. Spigler, Time-independent estimates and a comparison theorem for a nonlinear integroparabolic equation of the Fokker-Planck type. Differential Integral Equations 17 (2004), no. 5-6, 549–570.
  * [16] M. M. Lavrentiev Jr.,R. Spigler, A. Tani, Existence, uniqueness, and regularity for the Kuramoto-Sakaguchi equation with unboundedly supported frequency distribution. Differential Integral Equations,27, No. 9-10 (2014), 879–892.
  * [17] S. Pagliarani, M. Pignotti, Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups. http://arxiv.org/abs/1707.01422v2
  * [18] C. S. Peskin, Mathematical aspects of heart physiology. Courant Institute of Mathematical Sciences, 1975.
  * [19] S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type. Matematiche (Catania), 49 (1994), pp. 53–105 (1995).
  * [20] S. Polidoro, C. Mogavero, A finite difference method for a boundary value problem related to the Kolmogorov equation. Calcolo, 1995, 32: 193-205.
  * [21] H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields. Progr. Theoret. Phys. 79 (1988) 39
  * [22] R. Spigler, The mathematics of Kuramoto models which describe synchronization phenomena. Matematica, Cultura e Società. Rivista dell’Unione Matematica Italiana, Serie 1, Vol. 1 (2016), n.2, p. 123–132
  * [23] S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D: Nonlinear Phenomena, vol. 143, pagg. 1-20, 2000
  * [24] S. H. Strogatz, R. E. Mirollo, Stability of incoherence in a population of coupled oscillators. J. Statist. Phys. 63 (1991) 613.
  * [25] H. Tanaka, A. Lichtenberg, S. Oishi, Self-synchronization of coupled oscillators with hysteretic responses. Physica D 100(1997),279-300
  * [26] K. Wiesenfeld, P. Colet, S. H. Strogatz, Synchronization transitions in a disordered Josephson series array. Physical review letters 76.3 (1996): 404.
  * [27] K. Wiesenfeld, P. Colet, S. H. Strogatz, Frequency locking in Josephson arrays: Connection with the Kuramoto model., Physical Review E 57.2 (1998): 1563.


# Diagnosing Catastrophe: Large Parts of Accuracy Loss in
Continual Learning Can Be Accounted for by Readout Misalignment

Daniel Anthes Sushrut Thorat Peter König Tim C. Kietzmann
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Institute of Cognitive Science, Osnabrück University

Figure 1: A: Classification accuracy averaged over the ten tasks sampled from
CIFAR100. Prior to averaging, task performance trajectories are temporally
aligned to task onset such that the x-axis reflects performance after t
additional tasks have been learned. The shaded area around each line indicates
the standard error computed over five repetitions of the procedure. B: Mean
class representations for a task split. Task mean representations from all
phases are projected to a shared two dimensional space using multidimensional
scaling Torgerson (1952). Shown are representation vectors directly after
learning the task, after learning 5, and after learning 9 additional tasks
(left to right). C: Mean performance over all tasks at task onset ($t=0$).
Network size does not have an effect on how well tasks are learned initially.
D: Performance loss measured as the difference between performance at $t=0$
and the mean over performances measured at all $t>0$. Standard error for
additional networks is computed over three simulations.

## Abstract

Unlike primates, training artificial neural networks (ANNs) on changing data
distributions leads to a rapid decrease in performance on old tasks. This
phenomenon is commonly referred to as catastrophic forgetting. In this paper,
we investigate the representational changes that underlie this performance
decrease and identify three distinct processes that together account for the
phenomenon. The largest component is a misalignment between hidden
representations and readout layers. Misalignment occurs due to learning on
additional tasks and causes internal representations to shift.
Representational geometry is partially conserved under this misalignment and
only a small part of the information is irrecoverably lost. All types of
representational changes scale with the dimensionality of hidden
representations. These insights have implications for deep learning
applications that need to be continuously updated, but may also aid aligning
ANN models to the rather robust biological vision.

> Keywords: continual learning; catastrophic forgetting; artificial neural
> networks; representations; lifelong learning

## Introduction

Our world is inherently sequential. Adapted to this, humans are successful in
continuously learning new skills over their lifetime. However, most state-of-
the-art training procedures for artificial neural networks (ANNs) rely on data
being independent and identically distributed. In settings where the data
distribution changes, networks have been reported to rapidly forget previous
knowledge Parisi . (2019); Hadsell . (2020). This phenomenon is commonly
termed catastrophic forgetting French (1999); McCloskey Cohen (1989).

A number of factors influence the degree to which performance decreases in
sequential learning scenarios: the dimensionality of representations Mirzadeh
. (2022), pretraining Ramasesh . (2022), objective function S. Li . (2022);
Davari . (2022) and task similarity Ramasesh . (2020). However, the changes to
the task-relevant representations during continual learning remain to be fully
characterized (see davari2022probing for first steps). In this work, we
characterize changes in representational geometry and their contribution to
the observed decrease in performance. We find that rather than forgetting,
much of the degraded performance can be explained by a misalignment of
representations and the readouts of the network.

## Analysis

Our model system is a standard four layer convolutional network The training
procedure, task and network architecture are identical to zenke2017continual.
We study catastrophic forgetting in the task-incremental scenario Van de Ven
Tolias (2019), initializing a new classification head every time a novel task
is encountered. After pretraining the network on CIFAR10 Krizhevsky Hinton
(2009), we sequentially train on ten equal task splits from CIFAR-100. We
repeat this procedure $5$ times controlling for the effects of task similarity
by randomly assigning each class to a task Ramasesh . (2020).

We characterise the information present throughout learning by training
diagnostic readouts for all tasks after every phase of training. A drop in
performance, despite adjusted readout, constitutes a loss of task relevant
information. This scenario constitutes true forgetting. Contrary to this,
performance loss attributed to misalignment is computed by the difference in
performance between the original readout ($t=0$) and the newly trained
diagnostic readouts at every phase of training. Third, to estimate the extent
to which misalignment is due to rotation, translation, and uniform scaling of
an otherwise static geometry, we align representations for each task after
each training phase to the representations immediately after learning the task
($t=0$) with a geometry-preserving Procrustes transformation Gower (1975).

Finally, as increasing layer width has been shown to alleviate catastrophic
forgetting Mirzadeh . (2022), we vary the width of the final hidden layer to
investigate how the different components of representational change are
modulated by network capacity.

## Results

As expected, we observe effects of catastrophic forgetting, i.e. a rapid drop
in performance of the original readouts as the network is trained on
additional tasks (Fig. 1A, ‘continual’ at T$>$0). Notably, however,
performance of diagnostic readouts decreases much less, indicating that the
discriminability of the old classes is indeed preserved, i.e. there is little
“actual” forgetting. The primary cause of decreased performance is readout
misalignment, the extent of which is shown by the large difference between
‘continual’ performance and performance measured at the diagnostic readouts
(Fig. 1A, ‘diagnostic’), in line with similar previous analyses Davari .
(2022).

Does misalignment preserve the original representational geometry? If so, we’d
expect that Procrustes alignment should yield performance as good as the
linear diagnostic readouts. We observe that aligning representations accounts
for approximately half of the performance difference between continual and
diagnostic readouts (Fig. 1 A, ‘procrustes’). Therefore, misalignment can be
characterized as a combination of geometry preserving and deforming changes of
representations.

An open question that remains from our and previous work is whether the
comparably good performance of the diagnostic readout is explained by transfer
learning based on features learned for earlier tasks in the sequence. Indeed,
we observe that the features learned for previously encountered tasks transfer
to unseen tasks (‘Feature Transfer’ in Fig. 1). Yet, transfer cannot fully
explain the performance observed with diagnostic readouts, as a clear
discontinuity in the diagnostic readout performance trajectory from before to
after training a new task ($t=0$) can be seen. This suggests that newly
learned features better support the new task. This additional information
stays preserved in the network over learning of multiple additional tasks, as
evidenced by the fact that diagnostic readout performance stays above the
performance measured at $t=-1$ for the subsequent phases ($t>0$).

Finally, characterizing the influence of network size in continual learning
with our new analysis techniques, we find that varying the width of the final
hidden layer attenuates all three measures of representational change. Yet, we
still observe small amounts of changes to the representational geometry and
misalignment with the readouts of the respective networks (Fig. 1 C & D).

## Discussion

In characterizing representational changes in a neural network during
continual learning, we observed that misalignment of the pre-readout
representations with the task readouts explains large parts of performance
degradation that is commonly referred to as ‘catastrophic forgetting’.
Interestingly, only a small amount of performance cannot be linearly read out
and is irrecoverably ‘forgotten’.

Many algorithms addressing catastrophic forgetting rely on restricting
learning at synapses that encode information for previous tasks Zenke .
(2017); Kirkpatrick . (2017) or regularize learning of representations for new
tasks Z. Li Hoiem (2017) in order to not lose information relevant for the
previous tasks. We argue that information in hidden layers is largely
preserved, even without restricting learning trajectories or placing
constraints on representations the network is allowed to learn. This is
especially prominent in larger networks. We hypothesize that catastrophic
forgetting may instead be efficiently addressed by solving the problem of
readout misalignment without influencing the learning of new tasks (See also:
Lesort . (2021)). Indeed, there may be benefits to not restricting learning of
representations more than necessary, as restrictions to the learning dynamics
of the network may lead to decreased plasticity or sub-optimal solutions over
long sequences of tasks.

Lastly, the primate visual system is successfully able to learn new tasks
without exhibiting forgetting of old tasks. If we are to use ANNs as models of
biological vision, then the discrepancies in the learning dynamics of the two
systems remain to be addressed. Future work will test the currently described
analysis framework for characterising representational changes in continual
learning on biological data to further understand where, how, and when the
visual system copes with the newly arriving information.

## Acknowledgments

The project was financed by the Deutsche Forschungsgemeinschaft (DFG, research
training group “Computational Cognition”, GRK2340), as well as the European
Research Council (ERC, TIME, Project 101039524). Compute resources used for
this project are funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation), project number 456666331.

## References

  * Davari . (2022) davari2022probingDavari, M., Asadi, N., Mudur, S., Aljundi, R. Belilovsky, E.  2022\.  Probing representation forgetting in supervised and unsupervised continual learning Probing representation forgetting in supervised and unsupervised continual learning.  Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Proceedings of the ieee/cvf conference on computer vision and pattern recognition ( 16712–16721).
  * French (1999) french1999catastrophicFrench, RM.  1999\.  Catastrophic forgetting in connectionist networks Catastrophic forgetting in connectionist networks.  Trends in cognitive sciences34128–135.
  * Gower (1975) gower1975generalizedGower, JC.  1975\.  Generalized procrustes analysis Generalized procrustes analysis.  Psychometrika4033–51.
  * Hadsell . (2020) hadsell2020embracingHadsell, R., Rao, D., Rusu, AA. Pascanu, R.  2020\.  Embracing change: Continual learning in deep neural networks Embracing change: Continual learning in deep neural networks.  Trends in cognitive sciences24121028–1040.
  * Kirkpatrick . (2017) kirkpatrick2017overcomingKirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, AA.Hadsell, R.  2017\.  Overcoming catastrophic forgetting in neural networks Overcoming catastrophic forgetting in neural networks.  Proceedings of the national academy of sciences114133521–3526.
  * Krizhevsky Hinton (2009) krizhevsky2009learningKrizhevsky, A. Hinton, G.  2009\.  Learning multiple layers of features from tiny images Learning multiple layers of features from tiny images.
  * Lesort . (2021) lesort2021continualLesort, T., George, T. Rish, I.  2021\.  Continual learning in deep networks: an analysis of the last layer Continual learning in deep networks: an analysis of the last layer.  arXiv preprint arXiv:2106.01834.
  * S. Li . (2022) li2022energyLi, S., Du, Y., van de Ven, G. Mordatch, I.  2022\.  Energy-based models for continual learning Energy-based models for continual learning.  Conference on Lifelong Learning Agents Conference on lifelong learning agents ( 1–22).
  * Z. Li Hoiem (2017) li2017learningLi, Z. Hoiem, D.  2017\.  Learning without forgetting Learning without forgetting.  IEEE transactions on pattern analysis and machine intelligence40122935–2947.
  * McCloskey Cohen (1989) mccloskey1989catastrophicMcCloskey, M. Cohen, NJ.  1989\.  Catastrophic interference in connectionist networks: The sequential learning problem Catastrophic interference in connectionist networks: The sequential learning problem.  Psychology of learning and motivation Psychology of learning and motivation ( 24, 109–165).  Elsevier.
  * Mirzadeh . (2022) mirzadeh2022wideMirzadeh, SI., Chaudhry, A., Yin, D., Hu, H., Pascanu, R., Gorur, D. Farajtabar, M.  2022\.  Wide neural networks forget less catastrophically Wide neural networks forget less catastrophically.  International Conference on Machine Learning International conference on machine learning ( 15699–15717).
  * Parisi . (2019) parisi2019continualParisi, GI., Kemker, R., Part, JL., Kanan, C. Wermter, S.  2019\.  Continual lifelong learning with neural networks: A review Continual lifelong learning with neural networks: A review.  Neural networks11354–71.
  * Ramasesh . (2020) ramasesh2020anatomyRamasesh, VV., Dyer, E. Raghu, M.  2020\.  Anatomy of catastrophic forgetting: Hidden representations and task semantics Anatomy of catastrophic forgetting: Hidden representations and task semantics.  arXiv preprint arXiv:2007.07400.
  * Ramasesh . (2022) ramasesh2022effectRamasesh, VV., Lewkowycz, A. Dyer, E.  2022\.  Effect of scale on catastrophic forgetting in neural networks Effect of scale on catastrophic forgetting in neural networks.  International Conference on Learning Representations. International conference on learning representations.
  * Torgerson (1952) torgerson1952multidimensionalTorgerson, WS.  1952\.  Multidimensional scaling: I. Theory and method Multidimensional scaling: I. theory and method.  Psychometrika174401–419.
  * Van de Ven Tolias (2019) van2019threeVan de Ven, GM. Tolias, AS.  2019\.  Three scenarios for continual learning Three scenarios for continual learning.  arXiv preprint arXiv:1904.07734.
  * Zenke . (2017) zenke2017continualZenke, F., Poole, B. Ganguli, S.  2017\.  Continual learning through synaptic intelligence Continual learning through synaptic intelligence.  International conference on machine learning International conference on machine learning ( 3987–3995).