# Steady states of an Elo-type rating model
for players of varying strength

Bertram Düring  Mathematics Institute, University of Warwick, Coventry CV4
7AL, United Kingdom  Josephine Evans  Mathematics Institute, University of
Warwick, Coventry CV4 7AL, United Kingdom  Marie-Therese Wolfram  Mathematics
Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

###### Abstract

In this paper we study the long-time behaviour of a kinetic formulation of an
Elo-type rating model for a large number of interacting players with variable
strength. The model results in a non-linear mean-field Fokker-Planck equation
and we show the existence of steady states via a Schauder fixed point
argument. Our proof relies on the study of a related linear equation using
hypocoercivity techniques.

Keywords: Nonlinear Fokker-Planck equation, Elo rating model, kinetic model,
steady states, hypocoercivity
Mathematics Subject Classification: 35Q84, 35B40, 47D07, 35Q91, 35K15

##  1 Introduction

In 2015 Jabin and Junca [31] introduced a kinetic version of the Elo rating
model for two player zero sum games. The Elo model was originally introduced
by the Hungarian physicist Arpad Elo to rank chess players, but variants of it
are nowadays used in bingo, football, basketball and American football. In
this model players are characterised by their strength $\rho$, which is an
unobservable characteristic, and their rating $R$, which is observable. In the
ideal situation the rating $R$ of a player converges to their strength $\rho$
over time as ratings are updated.

We start by recalling the kinetic version of the Elo rating model [31].
Consider $N\in\mathbb{N}$ players who participate in a sequence of two-player
zero sum games. Each player is characterised by their respective rating
$R_{\ell}$ (which is observable) and strength $\rho_{\ell}$ (which is
unobservable), $\ell\in\\{1,\dots,N\\}$. When two players
$i,j\in\\{1,\dots,N\\}$ play a game, their ratings are updated after the
encounter using the following binary interaction rule:

$\displaystyle R_{i}^{*}$
$\displaystyle=R_{i}+K\left(S_{ij}-b(R_{i}-R_{j})\right),$ (1a) $\displaystyle
R_{j}^{*}$ $\displaystyle=R_{j}+K\left(-S_{ij}-b(R_{j}-R_{i})\right),$ (1b)

where $K$ is a positive constant and $S_{ij}$ the outcome of the game. The
random variable $S_{ij}$ takes values $\pm 1$ and, in average, outcomes are
assumed to depend on the difference in the underlying strength, that is
$\mathbb{E}(S_{ij})=b(\rho_{i}-\rho_{j})$. Note that this assumption can be
generalised to include, for example, draws, then $S_{ij}\in\\{-1,0,1\\}$ or to
consider continuous random variables $S_{ij}$ on the interval $[-1,1]$. The
function $b$ is usually set to

$\displaystyle b(z)=\textrm{tanh}(cz)\quad\text{ with }c\in\mathbb{R}.$

In general $b$ is assumed to be an odd, sufficiently smooth function. Hence,
in (1) the expected outcome of the game based on the difference in ratings,
that is $b(R_{i}-R_{j})$, is compared to the actual score and the ratings are
adjusted accordingly.
Jabin and Junca showed that the distribution of players $f=f(\rho,R,t)$
satisfies the following Fokker-Planck equation in the quasi-invariant limit in
[31]:

$\displaystyle\partial_{t}f(\rho,R,t)=\partial_{R}\left(a[f]f(\rho,R,t)\right)\quad\text{
in }\Omega\times(0,\infty),$ (2)

with a given initial distribution $f(\rho,R,t)=f_{0}(\rho,R)$, and
$\Omega\subseteq\mathbb{R}^{2}$. The operator $a$ is given by

$\displaystyle
a[f]=\int_{\Omega}\left(b(\rho-\rho^{\prime})-b(R-R^{\prime})\right)f(r^{\prime},\rho^{\prime},t)\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}.$
(3)

We observe that the distribution of players with respect to their ratings is
translated by the ‘velocity’ $a$, which depends on the difference between the
expected outcome and the actual score (integrated against the agent
distribution).
Note that interactions (1) are translation-invariant on $\mathbb{R}$, since
they depend on the difference between ratings only. Similarly, if the initial
datum $f_{0}$ of the Fokker-Planck equation (2) is shifted by constants
$\rho_{0}$ and $R_{0}$ in $\mathbb{R}^{+}$, that is
$g_{0}=f_{0}(\rho+\rho_{0},R+R_{0})$, then
$g(\rho,R,t)=f(\rho+\rho_{0},R+R_{0},t)$ is the solution to (2). Using energy
arguments, Jabin and Junca [31] show that solutions $f$ to (2) concentrate on
the diagonal as $t\rightarrow\infty$. Hence, the observable ratings $R$ are
guaranteed to converge to the unobservable strengths $\rho$, giving
justification for the validity of the kinetic Elo rating model in the many-
agents and long-time limit.

Originating in statistical mechanics, in particular in rarefied gas dynamics,
Boltzmann-type and Fokker-Planck-type equations (and other kinetic models)
have found new applications in socio-economic applications in the past two
decades, see [37] for an overview. Applications aside from Elo-type rating
models [31, 21, 17] include wealth distribution in societies [11, 22, 19, 20],
opinion formation [36, 18, 3, 12, 23, 24], compartmental epidemiology [15, 4]
and others.

A generalisation of the original Elo model (2) with variable underlying
strength $\rho$ was proposed and investigated by Düring et al. [21]. Here the
players’ strength changes in encounters and is influenced by random
fluctuations (introducing an additional diffusion term). We outline the
details of the modelling in Section 2.1, but note that the corresponding
player distribution satisfies the following nonlinear Fokker-Planck equation,

$\displaystyle\partial_{t}f(\rho,R,t)=\partial_{R}\left(a[f]f(\rho,R,t)\right)+\gamma\partial_{\rho}\left(a_{1}[f]f(\rho,R,t)\right)+\frac{\sigma^{2}}{2}\partial_{\rho}^{2}f(\rho,R,t)\quad\text{
in }\Omega\times(0,\infty),$ (4)

where $\gamma>0$ and

$\displaystyle
a_{1}[f]=\int_{\Omega}b(\rho-\rho^{\prime})f(\rho^{\prime},r^{\prime},t)\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}.$

In [21] existence of weak solutions to (4) was proved (for
$\Omega\subset\mathbb{R}^{2}$) and numerical experiments illustrated the
dynamics of the model. Some heuristic arguments on the long-time behaviour of
solutions to (4) were given, but no rigorous analysis carried out. In this
paper we present an existence results of steady states to equation (4). Since
the diffusion part in (4) is singular, the equation is degenerate parabolic.
Degenerate Fokker-Planck equations frequently, despite their lack of
coercivity, exhibit exponential convergence to equilibrium, a behaviour which
has been referred to by Villani as hypocoercivity [38]. This was subsequently
extended [16] to a wide variety of kinetic equations.

For linear kinetic Fokker-Planck equations most existing quantitative
equilibration results are confined to equations with explicit steady states,
or even with linear (in the variables) drift terms, where the whole semigroup
can be written explicitly. It is sometimes possible to work with non-explicit
steady states as in [9]. This result requires precise bounds on the non-
explicit steady state. The other option is to work in a perturbative setting
around an equation where the steady state is know as in [7]. In the present
situation of (4), however, there is no obvious choice of equation to perturb
around. The non-explicitness of the steady state encountered here is similar
to the theory of non-equilibrium steady states in thermodynamics which appears
in the kinetic theory of gasses, see the review article [25] and references
therein.

Linking the linear theory of hypocoercivity to non-linear equations is usually
done through a linear stability analysis. This provides another motivation for
finding the steady state as it gives us something to linearise around. The
Vlasov Poisson Fokker-Planck equation is structurally similar to equation (4)
studied here. The linearised problem for this equation is studied in [2].
Often for non-linear kinetic equations the challenge in this is to link the
(typically small) spaces in which we can study the linearised equations with
(typically larger) spaces in which we expect the fully non-linear equation to
be well-posed. This was done for the Boltzmann equation in [29]. As our
equation is well-posed in spaces with exponential weights (due to the
boundedness of the non-linear drift terms) we do not expect the same kind of
problems to appear here. Studying the long-time behaviour of the fully non-
linear equation (4) is in general very challenging. This problem is strongly
linked to the problem of uniqueness for a steady state. We would need to have
an entropy function which works for data far from the steady state. For the
Boltzmann equation and the kinetic Fokker-Planck equation this role is played
by the Boltzmann entropy. There are very few fully non-linear results giving
conditional convergence to equilibrium results [13, 14].

This paper is organised as follows: we discuss the underlying modeling
assumption of the generalised Elo model proposed by Düring et al. [21] in
Section 2.1 and illustrate the dynamics of solutions with computational
experiments in Section 2.2. Section 3 presents the main contribution of the
paper – the existence of steady solutions. We conclude by discussing the link
to hypocoercivity and future research directions in Section 4.

##  2 The kinetic Elo rating system with learning effects

In this section we recall the underlying modeling assumptions and present
computational results, that motivate and guide the presented analytical
results.

###  2.1 Modelling

We start by discussing the generalisation of the kinetic model proposed by
Düring et al. [21]. We recall that the rating $R$ should ideally correspond to
the underlying strength $\rho$, giving a way to rank and compare players,
whose intrinsic strength is not observable.

While the original Elo model assumes that the underlying players’ strength is
constant, Düring et al. proposed that players’ strength changes over time in
various ways: (i) gain of strength by learning from encounters, which depends
on the strength difference between the players; (ii) gain or loss of self-
confidence due to winning or being defeated in a game; and (iii) random
performance fluctuations. In particular, they generalised the interaction rule
(1) to

$\displaystyle R_{i}^{*}$
$\displaystyle=R_{i}+K\left(S_{ij}-b(R_{i}-R_{j})\right),$ (5a) $\displaystyle
R_{j}^{*}$ $\displaystyle=R_{j}+K\left(-S_{ij}-b(R_{j}-R_{i})\right),$ (5b)
$\displaystyle\rho_{i}^{*}$ $\displaystyle=\rho_{i}+\gamma
h(\rho_{j}-\rho_{i})+\eta,$ (5c) $\displaystyle\rho_{j}^{*}$
$\displaystyle=\rho_{j}+\gamma h(\rho_{i}-\rho_{j})+\tilde{\eta},$ (5d)

where $\eta$ and $\tilde{\eta}$ are independent and identically distributed
random variables with mean zero and variance $\sigma^{2}$ (which account for
daily fluctuations in the individual performance). The function $h$ models
individual learning. Düring et al. proposed the following form (with
parameters $\alpha,\beta\geq 0$)

$\displaystyle h(\rho_{i}-\rho_{j})=\alpha h_{1}(\rho_{i}-\rho_{j})+\beta
h_{2}(\rho_{i}-\rho_{j},S_{ij}),$

where $h_{1}$ corresponds to the increase of strength and knowledge from
encounters,

$\displaystyle h_{1}(z)=1+b(z).$ (6)

Since $h_{1}$ is positive, both players are able to learn and improve in each
game, depending on the difference in strengths, and with a player with lower
strength benefiting more. The function $h_{2}$ models the gain or loss of
self-confidence if a player wins a game or is defeated in it – it is either
positive or negative depending on the actual outcome of the game and the
expected outcome based on the difference in the players’ strength. Then the
distribution of players $f$ satisfies the following Fokker-Planck equation in
the quasi-invariant limit:

$\displaystyle\partial_{t}f(\rho,R,t)=\partial_{R}\left(a[f]f(\rho,R,t)\right)+\partial_{\rho}\left(c[f]f(\rho,R,t)\right)+\frac{\sigma^{2}}{2}\partial_{\rho}^{2}f(\rho,R,t)\quad\text{
in }\Omega\times(0,\infty),$

where we assumed that the initial data $f_{0}$ is normalised, the integral
operator $a$ is given by (3) and

$\displaystyle c[f]=\int_{\Omega}\left[\alpha h_{1}(\rho^{\prime}-\rho)+\beta
h_{2}(\rho^{\prime}-\rho)\right]f(\rho^{\prime},R^{\prime},t)\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}.$
(7)

Since the positive function $h_{1}$ results in a continuous increase of the
players’ strength $\rho$ over time, Düring et al. studied a suitably shifted
problem, which has a steady state. To this end they define

$\displaystyle H(\rho,R,t)=\alpha t,$

and the function

$\displaystyle g(\rho,R,t):=f(\rho+H(\rho,R,t),R,t).$ (8)

Then the function $g$ satisfies the following Fokker-Planck equation

$\displaystyle\partial_{t}g(\rho,R,t)=\partial_{R}\left(a[g]g(\rho,R,t)\right)+\partial_{\rho}\left(\tilde{c}[g]g(\rho,R,t)\right)+\frac{\sigma^{2}}{2}\partial_{\rho}^{2}g(\rho,R,t)\quad\text{
in }\Omega\times(0,\infty),$ (9)

where

$\displaystyle\tilde{c}[g]=\int_{\Omega}\left[\alpha
b(\rho^{\prime}-\rho)+\beta
h_{2}(\rho^{\prime}-\rho)\right]g(\rho^{\prime},R^{\prime},t)\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}.$

Equation (4) corresponds to (9) with $\beta=0$, i.e. players improve their
strength by participating in games, but do not loose or gain confidence due to
wins and losses. The presented analysis investigates the steady states of this
suitably shifted variant of the Elo model.

###  2.2 Numerical simulations

In the following we present several computational experiments which motivate
and corroborate our analysis.

We simulate (4) on the unit square with no-flux boundary conditions, using
Strang splitting and an upwind finite volume discretisation in $\rho$ and $R$
proposed in [21]. The domain is discretised into squares of length
$h=\frac{1}{800}$, the discrete time steps are set to $\Delta t=2\times
10^{-6}$. We assume that agents are initially uniformly distributed, hence
$f(\rho,R,t=0)\equiv 1$.

Figure 1 shows the computationally obtained steady state of players
$f_{\infty}$ for $\frac{\sigma^{2}}{2}=0.05$ after $2\times 10^{5}$ time
steps. We observe the formation of a smoothed peak at $(\rho,R)=(0.5,0.5)$
(the centre of mass). The smaller the diffusivity, the more concentrated
$f_{\infty}$ (converging to the expected Delta Dirac steady state in the case
$\sigma^{2}=0$).

Figure 1: Steady state player distribution $f_{\infty}$.

Figure 2 illustrates the evolution of different weighted relative energies,
which we will investigate in Section 3. In particular, we consider

$\displaystyle E(f;f_{\infty})=\int_{\mathbb{R}^{2}}\phi(\rho,R)\lvert
f(\rho,R,t)-f_{\infty}(\rho,R)\rvert\,\mathrm{d}\rho\mathrm{d}R$ (10)

with a weight function
$\phi=\phi_{\beta}=\exp\left(\beta\sqrt{1+4\rho^{2}/\gamma+2\rho R+\gamma
R^{2}}\right)$ or $\phi=f^{-1}_{\infty}$. We note that $\phi_{\beta}$ will be
used in the analysis later. We compute the relative energies with respect to
the computationally obtained steady state density $f_{\infty}$ (we see in
Figure 2 that the solution has indeed equilibrated).

(a) Weighted relative energies for $\phi=\phi(\beta,\gamma)$.

(b) Weighted relative energy for $\phi=f^{-1}_{\infty}$

Figure 2: Decay of the relative energy (10) for different weights functions
$\phi=\phi(\beta,\gamma)$ defined by (12) and for $\phi=f_{\infty}^{-1}$.

The evolution of (10) shows a staircase like decay – a phenomena often
observed in linear kinetic Fokker Planck equations [1] and the Boltzmann
equation [28].

##  3 Existence of steady states

###  3.1 Main Results

The previous discussion motivates the main result of this paper – an existence
proof of steady states of the following equation:

$\partial_{t}f=-\partial_{R}\left(a[f]f\right)+\partial_{\rho}\left(\frac{\sigma^{2}}{2}\partial_{\rho}f+\gamma
a_{1}[f]f\right)\quad\text{ in }\mathbb{R}^{2}\times(0,\infty),$ (11)

with $a[\mu](\rho,R)=a_{1}[\mu](\rho)-a_{2}[\mu](R)$, where

$\displaystyle a_{1}[\mu](\rho)$
$\displaystyle=\int_{\mathbb{R}^{2}}b(\rho-\rho^{\prime})\mu(\rho^{\prime},R^{\prime})\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime},$
$\displaystyle a_{2}[\mu](R)$
$\displaystyle=\int_{\mathbb{R}^{2}}b(R-R^{\prime})\mu(\rho^{\prime},R^{\prime})\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}.$

Equation (11) is supplemented with initial condition
$f(\rho,R,0)=f_{0}(\rho,R)$ with
$\int_{\mathbb{R}^{2}}f_{0}\,\mathrm{d}\rho\mathrm{d}R=1$. Note that (11)
corresponds to (9) with $\beta=0$ and $\alpha=\gamma$.

We make the following assumption on the function $b$.

######  Assumption 3.1.

We assume that $b$ is an odd, smooth function on $\mathbb{R}$ and that there
exist constants $\alpha>0$ and $C>0$ such that

$|1-b(|z|)|\leq Ce^{-\alpha\langle z\rangle}.$

We also recall the function $\phi_{\beta}$ which is used in the analysis,

$\phi_{\beta}=\exp\left(\beta\sqrt{1+4\rho^{2}/\gamma+2\rho R+\gamma
R^{2}}\right).$ (12)

The following theorem states the main result of the paper.

######  Theorem 3.1.

Let Assumption 3.1 be satisfied. Then there exists a steady solution
$f_{\infty}$ to (11), which is a probability measure on $\mathbb{R}^{2}$.
Furthermore, $f_{\infty}$ is a smooth function with a bounded exponential
moment.

We will prove existence of a steady state using Schauder’s fixed point
argument. In doing so we consider the following linearised transport equation
for fixed probability measure $\mu$,

$\partial_{t}f=-\partial_{R}\left(a[\mu]f\right)+\partial_{\rho}\left(\frac{\sigma^{2}}{2}\partial_{\rho}f+\gamma
a_{1}[\mu](\rho)f\right).$ (13)

We will prove the existence of a steady state using the following steps:

  * •

Step 1: Define a map $G(\mu)$, from the set of probability measures on
$\mathbb{R}^{2}$ to itself by setting $G(\mu)$ to be the unique steady state
of (13).

  * •

Step 2: Show that $G(\mu)$ is well-defined by a Harris’s theorem argument.
This step will also give us some bounds on $G(\mu)$. We will define a scale of
exponential moments of the form
$\int_{\mathbb{R}^{2}}\mu\phi_{\beta}\mathrm{d}\rho\mathrm{d}R$ and get bounds
on $\int_{\mathbb{R}^{2}}G(\mu)\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R$ for
$\beta$ sufficiently small.

  * •

Step 3: We show that under Assumption 3.1 our bounds on $G(\mu)$ will allow us
to show that if

$\int_{\mathbb{R}^{2}}\mu\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R\leq M,$

then

$\int_{\mathbb{R}^{2}}G(\mu)\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R\leq
CM^{\eta},$

where $\eta$ is a constant we can find explicitly in terms of $\alpha$ from
Assumption 3.1 and $\beta$. When $\beta$ is small enough we will have $\eta<1$
which allows us to find a convex set of measures which is preserved by the
function $G$.

  * •

Step 4: We then need to choose a topology to show that $G$ is a continuous map
with a compact image. The results of step 2 will provide us with a natural
topology and allow us to relate continuity of $G$ with respect to $\mu$ to
continuity of the semigroup associated to the linear equation (13) with
respect to $\mu$. We show continuity of the semigroup in Wasserstein distance
and then convert this to continuity in our strong topology using an
interpolation argument and the regularising nature of (13). We have weak
compactness as a result of our moment estimates.

  * •

Step 5: The above allows us to apply Schauder’s fixed point theorem to the map
$G$. This gives us the existence of a steady state to the non-linear equation.
Furthermore, from our study of $G$ and the fact that the steady state is a
fixed point of $G$ this also give us that the steady state has a smooth
density (from the regularising estimates in step 4) and
$\int_{\mathbb{R}^{2}}f\phi\,\mathrm{d}\rho\mathrm{d}R\leq M$ (from the bounds
found in step 3).

####  3.1.1 Discussion of the linear equation (13)

Before we begin the main technical proofs we comment that the analysis relies
strongly on an understanding of both the steady states and convergence to the
steady states for (13). This is a linear Fokker-Planck equation and is in some
ways very similar to equations which have been studied using hypocoercivity
theory, particularly by Villani in [38].

A key example of an equation to which we can apply the theory in [38] is the
kinetic Fokker-Planck equation of the following form:

$\partial_{t}f=-v\partial_{x}f+\partial_{x}\phi(x)\partial_{v}f+\partial_{v}\left(\frac{\sigma^{2}}{2}\partial_{v}f+vf\right).$
(14)

Here the spatial variable $x$ corresponds to the ratings $R$ and the velocity
$v$ to the underlying strength $\rho$. We can make a comparison of the
different terms in this equation with terms in (13).

  * •

$\sigma^{2}\partial^{2}_{v}f$ and $\sigma^{2}\partial_{\rho}^{2}f$ both
correspond to linear second order diffusions which are only present in one of
the two variables.

  * •

The transport terms in $x$ and $R$, that is $-v\partial_{x}$ and
$-\partial_{R}(a_{1}[\mu](\rho)f)$, correspond to mixing between the diffused
and undiffused variables.

  * •

The transport terms in $v$ and $\rho$, that is $\partial_{v}(vf)$ and
$\gamma\partial_{\rho}(a_{1}[\mu](\rho)f)$, correspond to confining effects in
the diffused variables.

  * •

The terms $\partial_{x}\phi(x)\partial_{v}f$ and
$\partial_{R}(a_{2}[\mu](R)f)$ both correspond to confining terms in the
undiffused variables. We should note here that these two terms are less
directly comparable.

For both equations we expect the term that mixes between the diffused and
undiffused variables to allow the smoothing and spreading effect from the
diffusion operator to affect the non-diffused variables. This is the key
effect in both hypoellipticity and hypocoercivity. We expect the combination
of this with the presence of confining terms in both variables to allow us to
show convergence to equilibrium.

However, there are differences between (13) and (14) which mean the study of
their long time behaviour is very different. The most important of these
differences is is that for (14) we can write down an explicit steady state
$\exp(-\phi(x)-v^{2}/2\sigma).$ This is not the case for (13). This does not
just affect the amount of information we have about the steady state it also
vastly reduces the tools we have for studying the convergence to equilibrium.
Almost all hypocoercivity theories including [38] and [16] require us to work
in spaces weighted against the equilibrium state and to use explicit knowledge
of this state when working in these spaces. There are a few works where they
are able to apply these theorems to equations with non-explicit steady states
in a perturbative setting [7, 30]. The only work we are aware of in a non-
perturbative setting is [9]. We note that Harris’s theorem in a good tool in
this situation and has been successfully applied to linear kinetic equations
with non-explicit steady states in [10, 6, 27, 26].

Another additional challenge when working with (13) as compared to (14) is
that the term that mixes the diffused and undiffused variables is weaker in
(13). Specifically, in the kinetic Fokker-Planck equation we will see a
diffusive effect similar to what would be produced by
$[v\partial_{x},\partial_{v}]^{2}=\partial^{2}_{x}$ at times of order $t^{2}$.
Working analogously for (13) we expect to generate a diffusion in the $R$
variables at second order in time similar to what would be produced by the
operator
$[a_{1}[\mu](\rho)\partial_{R},\partial_{\rho}]^{2}=(\partial_{\rho}a_{1}[\mu](\rho))^{2}\partial^{2}_{R}$.
But since $\partial_{\rho}a_{1}[\mu](\rho)\rightarrow 0$ exponentially fast as
$|\rho|\rightarrow\infty$ we see that the diffusion effect in the $R$
variables becomes weaker and weaker as $|\rho|\rightarrow\infty.$ Similar
effects were observed in case of a special relativistic kinetic Fokker-Planck
equation, see [8]. We expect this weak mixing to be especially challenging in
our setting as our confining terms are bounded. This means we expect to have
exponential rather than Gaussian concentration of the steady states. The
combination of these two effects means that even if we had an explicit form
for the steady states the Poincaré inequalities needed in the theory of [38]
would not be valid. In this respect we are also helped by using Harris’s
theorem. As we will see in more detail in the proof verifying the assumptions
of Harris’s theorem only requires us to verify the mixing property of the
semigroup on a compact set. This means we do not need to worry that our mixing
effect becomes 0 as $|\rho|\rightarrow\infty.$

####  3.1.2 Topologies and spaces

We will use the following function spaces, topologies and theorems in the main
proof. We work in the space $\mathcal{P}_{\beta}$ defined by

######  Definition 3.1.

We define $\mathcal{P}_{\beta}$ to be the space of probability measures on
$\mathbb{R}^{2}$ which have the property that

$\int_{\mathbb{R}^{2}}\phi_{\beta}(\rho,R)\mu(\mathrm{d}\rho,\mathrm{d}R)<\infty.$

Here $\phi_{\beta}$ is as defined in (12).

We consider this as a subspace of the space $\mathcal{M}_{\beta}$ defined by

######  Definition 3.2.

We define $\mathcal{M}_{\beta}$ to be the space of signed measures on
$\mathbb{R}^{2}$ which satisfy

$\int_{\mathbb{R}^{2}}\phi_{\beta}(\rho,R)|\mu|(\mathrm{d}\rho,\mathrm{d}R)<\infty.$

We can make $\mathcal{M}_{\beta}$ a Banach space with the following norm.

######  Definition 3.3.

We define a norm $\|\cdot\|_{\beta}$ on $\mathcal{M}_{\beta}$ by

$\|\mu\|_{\beta}=\int_{\mathbb{R}^{2}}\phi_{\beta}(\rho,R)|\mu|(\mathrm{d}\rho,\mathrm{d}R)<\infty.$

We consider two topologies on $\mathcal{P}_{\beta}$: the topology defined by
the norm $\|\cdot\|_{\beta}$ and the topology defined by the notion of weak
convergence of measures. We recall Prokhorov’s theorem.

######  Theorem 3.2 (Prokhorov’s Theorem).

Suppose that $S$ is a metric space and $\mathcal{P}(S)$ is the space of
probability measures on $S$, then a collection of probability measures
$\mathscr{C}$ is sequentially compact in the topology of weak convergence of
measures if and only if it is tight. Where we say the set $\mathscr{C}$ is
tight if for every $\epsilon>0$ there exists a compact set
$K_{\epsilon}\subset S$ such that for any $\mu\in\mathscr{C}$ we have

$\mu(S\setminus K_{\epsilon})<\epsilon.$

######  Lemma 3.3.

Any subset $\mathscr{C}$ of $\mathcal{P}_{\beta}$ which is bounded with
respect to the $\|\cdot\|_{*}$ norm is sequentially compact in the topology of
weak convergence of measures.

###### Proof.

By Prokhorov’s theorem it is sufficient to prove that such a set is tight.
Suppose that for every $\mu\in\mathscr{C}$ we have

$\int_{\mathbb{R}^{2}}\phi(\rho,R)\,\mu(\mathrm{d}\rho,\mathrm{d}r)\leq M.$

Then we know that $\phi^{-1}$ converges to $0$ as
$\|(\rho,R)\|\rightarrow\infty$ so for every $\epsilon$ there exists a compact
set $K_{\epsilon}$ such that

$\phi^{-1}(\rho,R)<\epsilon/M,\;\forall(\rho,R)\notin K_{\epsilon}.$

Using this we have

$\displaystyle\int_{\mathbb{R}^{2}\setminus
K_{\epsilon}}\mu(\mathrm{d}\rho,\mathrm{d}R)$
$\displaystyle=\int_{\mathbb{R}^{2}}\left(\phi(\rho,R)^{-1}1_{\mathbb{R}^{2}\setminus
K_{\epsilon}}\right)\left(\phi(\rho,R)\right)\mu(\mathrm{d}\rho,\mathrm{d}R)$
$\displaystyle\leq\frac{\epsilon}{M}\int_{\mathbb{R}^{2}}\phi(\rho,R)\mu(\mathrm{d}\rho,\mathrm{d}R)$
$\displaystyle\leq\epsilon.$

∎

Finally, we define the compact set used in Schauder’s fixed point theorem.

######  Definition 3.4.

We write $\mathscr{C}_{M,\beta}$ to be the set

$\mathscr{C}_{M,\beta}:=\\{\mu\,|\,\mu\in\mathcal{P}_{\beta},\|\mu\|_{*}\leq
M\\},$

for positive constants $\beta,M$. We note that by the results above
$\mathscr{C}_{M,\beta}$ is compact in the topology of weak convergence of
measures for any $M,\beta$.

###  3.2 Schauder fixed point argument

In order to apply Schauder’s fixed point argument we first need to define a
function which will have a steady state of $f_{\infty}$ as a fixed point.

######  Definition 3.5.

For fixed $\mu$, a probability measure, we write $\mathcal{S}_{t}^{\mu}$ to be
the linear semigroup associated to the evolution governed by equation (13).
Furthermore, when this equation has a unique steady state we write $G(\mu)$ to
be this steady state.

####  3.2.1 The function $G$ is well-defined

In this section we prove the following proposition.

######  Proposition 3.1.

For any probability measure $\mu$ on $\mathbb{R}^{2}$ there exists a unique
steady state to equation (13). Therefore, the function $G$ is well-defined.

We prove this proposition after showing the following lemma first.

######  Lemma 3.4.

For any $\mu$ there exists $z_{1}$ such that $|a_{1}[\mu](\rho)|>1/2,$
whenever $|\rho|>z_{1}$, and $z_{2}$ such that $|a_{2}[\mu](R)|>1/2$, whenever
$|R|>z_{2}$.

###### Proof.

We have that $b(\rho-\rho^{\prime})\rightarrow c$ as $\rho\rightarrow\infty$.
So by dominated convergence $a_{1}[\mu](\rho)\rightarrow c$ as
$\rho\rightarrow\infty$ and $a_{1}[\mu](\rho)\rightarrow-c$ as
$\rho\rightarrow\infty$. The same behavior holds true for the operator
$a_{2}[\mu](R)$, but this time as a function of $R$. We can also differentiate
these functions to see that they are monotonically increasing. This gives the
result. ∎

As a consequence of the proof techniques in the proposition we also find the
following result.

######  Lemma 3.5.

For $\beta$ sufficiently small there exists some positive constants
$C,\lambda$ such that for any two initial data
$f_{1},f_{2}\in\mathcal{P}_{\beta}$ we have

$\|S_{t}^{\mu}(f_{1}-f_{2})\|_{\beta}\leq Ce^{-\lambda
t}\|f_{1}-f_{2}\|_{\beta}$

Then there exits a constant $D>0$, not depending on $\mu$, such that

$\int_{\mathbb{R}^{2}}G(\mu)\phi\,\mathrm{d}\rho\mathrm{d}R\leq
D\sup_{|(\rho^{\prime},R^{\prime})|\leq
r(\mu)}\phi(\rho^{\prime},R^{\prime}).$

Here, the function $r(\mu)$ is defined such that
$|(\rho^{\prime},R^{\prime})|<r(\mu)$ implies that $|\rho|<z_{1}$ and
$|R|\leq\max\\{1,2\gamma\\}z_{2}$ and $|(\rho,R)|\leq z_{3}$.

In this section we work with stochastic tools. Before we begin doing this we
relate the linear equation (13) to a Markov process. In particular, equation
(13) is the Kolmogorov forward equation (equation which evolves the law
forward in time) related to the SDE for the continuous in time stochastic
processes $R_{t}$ and $\rho_{t}$,

$\displaystyle\mathrm{d}R_{t}$
$\displaystyle=a[\mu](\rho_{t},R_{t})\,\mathrm{d}t,$ (15a)
$\displaystyle\mathrm{d}\rho_{t}$
$\displaystyle=-a_{1}[\mu](\rho_{t})\,\mathrm{d}t+\sigma\,\mathrm{d}B_{t},$
(15b)

where $B$ is a Brownian motion.

We prove both Proposition 3.1 and Lemma 3.5 by applying Harris’s theorem. We
use the version of Harris’s theorem found in [32], which we restate here.
First we need to state their assumptions. The theorem is for a Markov chain
$X_{t}$ with transition kernel $P_{t}(x,A)$. Then the semigroup associated to
our PDE will be given by $(S_{t}^{\mu}f_{0})(A)=\int_{A}\int
P_{t}(x,y)f_{0}(x)\,\mathrm{d}x\mathrm{d}y=\int_{A}f_{t}(y)\,\mathrm{d}y.$ We
also have our PDE (13) is written $\partial_{t}f=\mathcal{L}f,$ where
$\mathcal{L}$ is the generator of $S_{t}^{\mu}.$ Then $\mathcal{L}^{*}$ is the
formal adjoint of $\mathcal{L},$ and the generator of the semigroup
$(S^{\mu*}_{t}\phi)(x)=\int\mathcal{P}_{t}(x,\mathrm{d}y)\phi(y).$

######  Assumption 3.2.

The transition kernel has to satisfy the following two assumptions:

  * •

For any compact set $K$, there is some $y_{*}$, such that for any $\delta>0$
there exists $t(\delta)$ such that
$P_{t(\delta)}(x,B(y_{*},\delta))>0\quad\forall x\in K.$

  * •

For every $t$ the transition kernel possesses a density $p_{t}(x,y)$ which is
jointly continuous in $x,y$ everywhere.

The second assumption is as follows.

######  Assumption 3.3.

For some fixed $T$ there exists a non-negative function $V$ with
$V(x)\rightarrow\infty$ as $|x|\rightarrow\infty$ and two constants
$\alpha\in(0,1)$ and $C\geq 0$ such that

$\mathbb{E}(V(X_{(n+1)T})\,|\,X_{nT})\leq\alpha V(X_{nT})+C.$

It is standard to verify this assumption by proving that

$\mathcal{L}^{*}V\leq-\lambda V+C,$

where $\mathcal{L}^{*}$ is the formal adjoint of the generator of the
semigroup and $\lambda,C>0.$

We then have the Theorem

######  Theorem 3.6 (Harris’s theorem, as in [32]).

If we have a Markov process $X_{t}$ that satisfies both Assumption 3.2 and
Assumption 3.3. Then there exists a unique steady state probability measure
for the Markov semigroup $P_{t}$. Furthermore, we have that there exists
constants $C>0$ and $\lambda>0$ such that

$\|P_{t}(f_{1}-f_{2})\|_{V}\leq Ce^{-\lambda t}\|f_{1}-f_{2}\|_{V}.$

where $\|f\|_{V}=\int V(x)|f|(\mathrm{d}x).$

Before verifying the above assumptions we note that a more standard way of
writing Harris’s theorem is to replace Assumption 3.2 by the assumption that
the semigroup $S_{t}^{\mu}$ has a uniformly over $x_{0}$ in any compact set,
lower bound of the form

$\mathcal{S}_{t}^{\mu}\delta_{x_{0}}\geq\alpha\nu,$

where $\alpha\in(0,1)$ and $\nu$ is a probability measure. For our equation an
assumption of this form could be verified by applying the result of [5]. This
would also be a more quantitative result. However, precisely applying the
Theorem of Bally and Kohatsu-Higa would take a lot of time relative to the
less quantitative results given here.

We proceed by showing that the semigroup $\mathcal{S}_{t}^{\mu}$ satisfies
Assumption 3.2 and 3.3 for any $\mu$.

######  Lemma 3.7.

The linear semigroup $S_{t}^{\mu}$ satisfies Assumption 3.2 for any $\mu$.

Before proving this lemma, we state two useful theorems from stochastic
calculus.

######  Theorem 3.8 (Malliavin’s Theorem, see [33], Section 7.5).

Given a $d$-dimensional SDE in Stratonovich form

$dX_{t}=V_{0}(X_{t})\,\mathrm{d}t+\sum_{k=1}^{m}V_{k}(X_{t})\circ\mathrm{d}B^{k}_{t},$

with $m$ independent Brownian motions. Furthermore, we define the set of
vector fields

$\mathcal{V}_{0}=\\{V_{0}\\},\quad\mathcal{V}_{n+1}=\mathcal{V}_{n}\cup\\{[V_{k},U]\,:\,k=1,\dots,m,\,U\in\mathcal{V}_{n}\\}.$

Then if there exists an $m$ such that $\mathcal{V}_{m}$ spans $\mathbb{R}^{2}$
at each $x$, then the stochastic process $X_{t}$ has a jointly continuous
transition kernel.

######  Theorem 3.9 (Strook-Varadhan support theorem, see [35]).

Given a $d$-dimensional SDE in Stratonovich form

$dX_{t}=V_{0}(X_{t})\,\mathrm{d}t+\sum_{k=1}^{m}V_{k}(X_{t})\circ\mathrm{d}B^{k}_{t},\quad
X_{0}=x,$

the support of the law of $X_{t}$ is the closure of the set of points reached
at time $t$ by the ODE

$\frac{\mathrm{d}x(t)}{\mathrm{d}t}=V_{0}(x(t))+\sum_{k=1}^{m}V_{k}(x(t))\frac{\mathrm{d}h_{k}(t)}{\mathrm{d}t},$

when we let the $h_{k}$ range over the set of continuously differentiable
functions.

###### Proof of Lemma 3.7.

This part of the proof is very similar in spirit to the proof of the analagous
result for the Langevin equation found in [32]. The second part of the
assumption is an immediate consequence of the hypoellipticity of the equation.
We can see that the SDE 15 satisfies the assumptions to apply Malliavin’s
version of Hörmander’s theorem. The vector field $\partial_{\rho}$ and
$[\partial_{\rho},a[\mu]\partial_{R}]=(\partial_{\rho}a[\mu](\rho))\partial_{R}$
span the tangent space at every point $(\rho,R)$ as $\partial_{\rho}a[\mu]>0.$

For the first part, as in Higham, Stuart and Mattingly [32], we use the
Strook-Varadhan support theorem [35]. Given this theorem we fix the point
$(\rho_{*},R_{*})$, which depends on $\mu$, chosen so that
$a_{1}[\mu](\rho_{*})=a_{2}[\mu](R_{*})=0.$ Then fix $\delta$ and a compact
set $\mathcal{K}$. We have the control system

$\left(\begin{array}[]{c}\mathrm{d}R/\mathrm{d}t\\\
\mathrm{d}\rho/\mathrm{d}t\end{array}\right)=\left(\begin{array}[]{c}a_{1}[\mu](\rho)-a_{2}[\mu](R)\\\
-\gamma a_{1}[\mu](\rho)+\sigma\mathrm{d}v/\mathrm{d}t\end{array}\right).$

As $a_{1}[\mu](\rho)$ is a smooth function this has the same reachable sets as
the control system

$\left(\begin{array}[]{c}\mathrm{d}R/\mathrm{d}t\\\
\mathrm{d}\rho/\mathrm{d}t\end{array}\right)=\left(\begin{array}[]{c}a_{1}[\mu](\rho)-a_{2}[\mu](R)\\\
\mathrm{d}\tilde{v}/\mathrm{d}t\end{array}\right).$ (16)

Now for any $\rho_{0}$ we can find $\tilde{v}$ such that
$\rho_{0}+\tilde{v}(1)=\rho_{*}$ and $\mathrm{d}\tilde{v}/\mathrm{d}t=0$ for
$t>1$ and $\tilde{v}$ is smooth. With this control we have that $R(1)$ is
somewhere in a ball of radius $c$ around $R_{0}$, so if $R_{0}$ was in the
original compact set $K$, $R(1)$ is now in a new compact set, $K^{\prime}$.
Now there exist a $T=T(\delta,K)$ such that after time $T$ the ODE

$\frac{\mathrm{d}R}{\mathrm{d}t}=-a_{2}[\mu](R)$

when started inside $K^{\prime}$ will be in $B(R_{*},\delta)$. Therefore, our
control $\tilde{v}$ will move the equivalent control system to a point in
$B((\rho_{*},R_{*}),\delta)$ after time $T+1$. Consequently, the Strook-
Varadhan support theorem 3.9 implies that

$P((\rho_{t},R_{t})\in B(\rho_{*},R_{*},\delta))>0$

if $t>T+1.$ ∎

We now move onto the second assumption.

######  Lemma 3.10.

The function $\phi_{\beta}$ is a Foster-Lyapunov function for the semigroup
$S_{t}^{\mu}$, for $\beta$ sufficiently small. That is to say if $\mathcal{L}$
is the generator associated to $S_{t}^{\mu},$

$\mathcal{L}^{*}\phi_{\beta}\leq-\lambda\phi_{\beta}+A1_{|(\rho,R)|\leq B},$

for some strictly positive constants $\lambda,A,B,$ again for $\beta$
sufficiently small.

###### Proof.

First we compute

$\displaystyle\frac{\mathcal{L}^{*}\phi_{\beta}}{\phi_{\beta}}$
$\displaystyle=\frac{\beta\left(-3a_{1}[\mu](\rho)\rho-\gamma
a_{2}[\mu](R)R-a_{2}[\mu](R)\rho\right)}{\sqrt{1+4\rho^{2}/\gamma+2\rho
R+\gamma R^{2}}}$ $\displaystyle\quad+\frac{\sigma^{2}}{2}\left(\frac{3\beta
R^{2}}{(1+4\rho^{2}/\gamma+2\rho R+\gamma
R^{2})^{3/2}}+\frac{\beta^{2}(R+4\rho/\gamma)^{2}}{1+4\rho^{2}/\gamma+2\rho
R+\gamma R^{2}}\right)$

This implies the following bound from above

$\displaystyle\frac{\mathcal{L}^{*}\phi_{\beta}}{\phi_{\beta}}$
$\displaystyle\leq\frac{\beta\left(-3a_{1}[\mu](\rho)\rho-\gamma
a_{2}[\mu](R)R+c|\rho|\right)}{\sqrt{1+4\rho^{2}/\gamma+2\rho R+\gamma
R^{2}}}+\frac{2\sigma^{2}\beta}{\gamma}\left(\frac{1}{\sqrt{1+4\rho^{2}/\gamma+2\rho
R+\gamma R^{2}}}+\beta\right).$

Using Lemma 3.4 we have that whenever $|\rho|>z_{1}$ and $|R|>z_{1}$ that

$-3a_{1}[\mu](\rho)\rho-\gamma
a_{2}[\mu](R)R+c|\rho|\leq-c|\rho|/2-c\gamma|R|/2.$

Furthermore, if $|\rho|>z_{1}$ and $|R|<z_{2}$, then

$-3a_{1}[\mu](\rho)\rho-\gamma a_{2}[\mu](R)R+c|\rho|\leq-c|\rho|/2.$

And if $|\rho|<z_{1}$ and $|R|>\max\\{1,2/\gamma\\}z_{2}$, we have

$-3a_{1}[\mu](\rho)\rho-\gamma a_{2}[\mu](R)R+c|\rho|\leq-\gamma c|R|/4.$

Therefore, there exists some $\Lambda$ which doesn’t depend on $\mu$ such that
if $|\rho|>z_{1}$ or $|R|>\max\\{1,2/\gamma\\}z_{2}$, we have

$\frac{\mathcal{L}^{*}\phi_{\beta}}{\phi_{\beta}}\leq-\beta\Lambda++\frac{2\sigma^{2}\beta}{\gamma}\left(\frac{1}{\sqrt{1+4\rho^{2}/\gamma+2\rho
R+\gamma R^{2}}}+\beta\right).$

Therefore, if $\beta$ is small enough and
$|\rho|>z_{1},|R|>\max\\{1,2/\gamma\\}z_{2}$ and $\rho,R$ large enough so that
the second term is small, specifically

$|(\rho,R)|\geq
z_{3}:=\frac{\frac{\Lambda\gamma}{4\sigma^{2}}-\beta}{\sqrt{2/\gamma+\gamma/2}},$

then

$\frac{\mathcal{L}^{*}\phi_{\beta}}{\phi_{\beta}}\leq-\beta\Lambda/2.$

Therefore, we have

$\mathcal{L}^{*}\phi_{\beta}\leq-\frac{1}{2}\beta\Lambda\phi_{\beta}+\beta\left(\sqrt{\gamma/3}c+2\sigma^{2}\frac{(1+\beta)}{\gamma}+\frac{\Lambda}{2}\right)\sup_{|\rho|\leq
z_{1},|R|\leq\max\\{1,2\gamma\\}z_{2},|(\rho,R)|\leq
z_{3}}\phi_{\beta}(\rho^{\prime},R^{\prime}).$

These calculations show that $\phi_{\beta}$ satisfies the conditions to be a
Foster-Lyapunov function when $\beta$ is sufficiently small. ∎

We now prove both Proposition 3.1 and Lemma 3.5 simultaneously.

###### Proof of Proposition 3.1 and Lemma 3.5.

Using Lemma 3.7 and Lemma 3.10 we have that the Markov process defined by (15)
satisfies the condition of Harris’s theorem with Lyapunov function
$\phi_{\beta}$. This gives the existence and uniqueness of a steady state
measure for this SDE, and gives the convergence result in the first part of
Lemma 3.5. For the other bound in Lemma 3.5 we recall the final inequality
from the proof of Lemma 3.10,

$\mathcal{L}^{*}\phi_{\beta}\leq-\frac{1}{2}\beta\Lambda\phi_{\beta}+\beta\left(\sqrt{\gamma/3}c+2\sigma^{2}\frac{(1+\beta)}{\gamma}+\frac{\Lambda}{2}\right)\sup_{|\rho|\leq
z_{1},|R|\leq\max\\{1,2\gamma\\}z_{2}}\phi_{\beta}(\rho^{\prime},R^{\prime}).$

These calculations show that when $\beta$ is sufficiently small, we have, for
$G(\mu)$ being the steady state,

$\int_{\mathbb{R}^{2}}G(\mu)\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R\leq\frac{2}{\Lambda}\left(\sqrt{\gamma/3}c+2\sigma^{2}\frac{(1+\beta)}{\gamma}+\frac{\Lambda}{2}\right)\sup_{|(\rho^{\prime},R^{\prime})|\leq
r(\mu)}\phi_{\beta}(\rho^{\prime},R^{\prime}).$

This bound only depends on $\mu$ through its explicit dependence on
$z_{1},z_{2}$. ∎

####  3.2.2 Finding a set which is preserved by the function $G$

In this section we prove the following proposition.

######  Proposition 3.2.

Let $M$ be sufficiently large, $\beta$ sufficiently small and $b$ satisfies
Assumption 3.1. Then the sets $\mathcal{C}_{M,\beta}$, defined in Definition
3.4 are preserved by the map $G$.

We begin with a lemma.

######  Lemma 3.11.

Let $\mu\in\mathscr{C}_{M,\beta}$ and $b$ satisfies Assumption 3.1. Then we
can give explicit expressions for the constants $z_{1}$ and $z_{2}$ appearing
in Lemma 3.4 for $M$ large. These are

$\displaystyle z_{1}=\log(4MC^{\prime})/\delta\text{ and
}z_{2}=\log(4MC^{\prime})/\delta^{\prime},$ (17)

with constants $\delta$ and $\delta^{\prime}$ given by

$\displaystyle\delta=\frac{2\alpha\beta\sqrt{3}}{\alpha\sqrt{\gamma}+\beta\sqrt{3}},\text{
and
}\delta^{\prime}=\frac{2\alpha\beta\sqrt{3\gamma}}{\alpha\sqrt{\gamma}+\beta\sqrt{3}}.$

We recall that $\beta$ is a constant which is fixed in the proof of Lemma
3.10.

###### Proof.

We begin with

$\displaystyle|1-a_{1}[\mu](\rho)|$
$\displaystyle=|\int_{\mathbb{R}^{2}}(1-b(\rho-\rho^{\prime}))\mu(\rho^{\prime},R^{\prime})\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}|$
$\displaystyle\leq\int_{\mathbb{R}^{2}}|1-b(\rho-\rho^{\prime})|\mu(\rho^{\prime},R^{\prime})\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}$
$\displaystyle=\int_{\mathbb{R}^{2}}|1-b(\rho-\rho^{\prime})|\frac{1}{\phi(\rho^{\prime},R^{\prime})}\phi(\rho^{\prime},R^{\prime})\mu(\rho^{\prime},R^{\prime})\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}$
$\displaystyle\leq\sup_{\rho^{\prime},R^{\prime}}\left(|1-b(\rho-\rho^{\prime})|\frac{1}{\phi(\rho^{\prime},R^{\prime})}\right)\int_{\mathbb{R}^{2}}\phi(\rho^{\prime},R^{\prime})\phi(\rho^{\prime},R^{\prime})\,\mathrm{d}\rho^{\prime}\mathrm{d}R^{\prime}$
$\displaystyle\leq
CM\sup_{\rho^{\prime},R^{\prime}}\left(e^{-\alpha\langle\rho-\rho^{\prime}\rangle}\frac{1}{\phi(\rho^{\prime},R^{\prime})}\right)$
$\displaystyle\leq
CM\sup_{\rho^{\prime}}\left(e^{-\alpha\langle\rho-\rho^{\prime}\rangle}\exp\left(-\beta\sqrt{1+\frac{3\rho^{\prime
2}}{\gamma}}\right)\right)=:MF(\rho).$

Furthermore,

$F(\rho)\leq C^{\prime}\exp\left(-\delta\langle\rho\rangle\right),$

with

$\delta=\frac{\sqrt{12}\alpha\beta}{\alpha\sqrt{\gamma}+\beta\sqrt{3}}.$

This means that if $\langle\rho\rangle>\log(4MC^{\prime})/\delta$, then
$|a_{1}[\mu](\rho)|>1/2.$
In a similar way, we have

$\displaystyle\lvert 1-a_{2}[\mu](R)\rvert$ $\displaystyle\leq
CM\sup_{\rho^{\prime},R^{\prime}}\left(e^{-\alpha\langle
R-R^{\prime}\rangle}\frac{1}{\phi(\rho^{\prime},R^{\prime})}\right)$
$\displaystyle\leq
CM\sup_{R^{\prime}}\left(\exp\left(-\alpha\langle\rho-\rho^{\prime}\rangle-\beta\sqrt{1+3\gamma
R^{2}/4}\right)\right)=:M\tilde{F}(R).$

Again,

$\tilde{F}(R)\leq C^{\prime}\exp\left(-\delta^{\prime}\langle
R\rangle\right),$

with

$\delta^{\prime}=\frac{4\alpha\beta\sqrt{3\gamma}}{2\alpha+\beta\sqrt{3\gamma}}.$

Therefore,

$|a_{2}[\mu](R)|>1/2,$

when

$\langle R\rangle\geq\frac{1}{\delta^{\prime}}\log\left(2C^{\prime}M\right).$

When $M$ is large, we can set

$z_{1}=\log(4MC^{\prime})/\delta\text{ and
}z_{2}=\log(4MC^{\prime})/\delta^{\prime}.$

∎

We now move on to the proof of proposition 3.2.

###### Proof of Proposition 3.2.

We begin by recalling part of the result of Lemma 3.5 that

$\int_{\mathbb{R}^{2}}G(\mu)\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R\leq\frac{2}{\Lambda}\left(\sqrt{\gamma/3}c+2\sigma^{2}\frac{(1+\beta)}{\gamma}+\frac{\Lambda}{2}\right)\sup_{|(\rho^{\prime},R^{\prime})|\leq
r(\mu)}\phi_{\beta}(\rho^{\prime},R^{\prime}).$

When $M$ is sufficiently large, $z_{1},z_{2}$ limit the choice of $r(\mu)$.
Let us work in the case $2/\gamma\leq 1$. We therefore are interested in

$\displaystyle\sup_{|\rho|<z_{1},|R|<\max\\{1,2/\gamma\\}z_{2}}\phi_{\beta}(\rho,R)\leq\exp\left(\beta\sqrt{1+\frac{4}{\gamma}z^{2}_{1}+2z_{1}z_{2}+\gamma
z_{2}^{2}}\right)$
$\displaystyle=\exp\left(\sqrt{\beta^{2}+\log\left(\frac{2C^{\prime}M}{c}\right)^{2}\left(\frac{4(\alpha\sqrt{\gamma}+\beta\sqrt{3})^{2}}{12\gamma\alpha^{2}}+2\frac{(\alpha\sqrt{\gamma}+\beta\sqrt{3})(2\alpha+\beta\sqrt{3\gamma})}{24\alpha^{2}\sqrt{\gamma}}+\gamma\frac{(2\alpha+\beta\sqrt{3\gamma})^{2}}{48\alpha^{2}\gamma}\right)}\right)$
$\displaystyle=\exp\left(\sqrt{\log\left(\frac{2C^{\prime}M}{c}\right)^{2}\frac{7}{12}+o(\beta)}\right)$
$\displaystyle\leq CM^{\eta},$

where $\eta=\sqrt{7/12}+o(\beta)$. We then choose $\beta$ so that $\eta<1.$
Therefore, we have for some constant $C$,

$\int_{\mathbb{R}^{2}}\mu\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R\leq
M\Rightarrow\int_{\mathbb{R}^{2}}G(\mu)\phi_{\beta}\,\mathrm{d}\rho\mathrm{d}R\leq
CM^{\eta}.$

Therefore, if $M$ is large enough we will map the set $\mathscr{C}_{M,\beta}$
onto itself. ∎

####  3.2.3 Continuity of the function $G$

In this section we prove the following proposition.

######  Proposition 3.3.

The function $G$ from $\mathcal{P}_{\beta}$ to $\mathcal{P}_{\beta}$ is
continuous with respect to the topology induced by the norm
$\|\cdot\|_{\beta},$ and in the topology of weak convergence of measures.

First let us describe the main intuition. We start by showing that we can turn
the question of continuity of $G$ into a question of continuity of the
semigroup $S_{t}^{\mu}$ with respect to $\mu$. We will see below that it is
straightforward to show that if $\mu_{1}$ and $\mu_{2}$ are close in total
variation then we can show that $S_{t}^{\mu_{1}}\nu$ and $S_{t}^{\mu_{2}}\nu$
are close in Wasserstein-1 distance by a stability estimate on the SDEs. As
$S_{t}^{\mu}$ is a regularising semi-group for every $\mu$, and depends
continuously on $\mu$, we expect Wasserstein closeness of
$S_{t}^{\mu_{1}},S_{t}^{\mu_{2}}$ to imply closeness in the $\|\cdot\|_{*}$
norm. We would like to prove an inequality like

$W_{1}(\nu_{1},\nu_{2})\leq\|\nu_{1}-\nu_{2}\|_{\beta}^{a}H(\nu_{1},\nu_{2}),$

where $H(\nu_{1},\nu_{2})$ is a quantity that depends on some norm of
$\nabla\nu_{1},\nabla\nu_{2}$. We were not able to prove such an inequality,
wich is why we use moments to control the tail behaviour.

We start by stating a sequence of lemmas to control and relate the different
distances to each other.

######  Lemma 3.12.

There exists a function $C(t)$ which is finite for $t$ sufficiently large so
that

$\|G(\mu_{1})-G(\mu_{2})\|_{\beta}\leq
C(t)\|S_{t}^{\mu_{1}}-S_{t}^{\mu_{2}}\|_{\beta}^{\prime}.$

###### Proof.

We use the fact that for any $t$ we have $S_{t}^{\mu}G(\mu)=G(\mu).$ We also
have from the Harris’s theorem result that there exists $\lambda,D$ which
depend on $\mu_{1}$ such that

$\|S_{t}^{\mu_{1}}(f-g)\|_{\beta}\leq De^{-\lambda t}\|f-g\|_{\beta}.$

Using these two facts we have

$\displaystyle\|G(\mu_{1})-G(\mu_{2})\|_{\beta}$
$\displaystyle=\|S_{t}^{\mu_{1}}G(\mu_{1})-S_{t}^{\mu_{2}}G(\mu_{2})\|_{\beta}$
$\displaystyle\leq\|S_{t}^{\mu_{1}}(G(\mu_{1})-G(\mu_{2}))\|_{\beta}+\|(S_{t}^{\mu_{1}}-S_{t}^{\mu_{2}})G(\mu_{2})\|_{\beta}$
$\displaystyle\leq De^{-\lambda
t}\|G(\mu_{1})-G(\mu_{2})\|_{\beta}+\|(S_{t}^{\mu_{1}}-S_{t}^{\mu_{2}})G(\mu_{2})\|_{\beta}.$

Rearranging this we have

$(1-De^{-\lambda
t})\|G(\mu_{1})-G(\mu_{2})\|_{\beta}\leq\|(S_{t}^{\mu_{1}}-S_{t}^{\mu_{2}})G(\mu_{2})\|_{\beta}.$

∎

Next we show that in Wasserstein distances, we can control the difference
between $S_{t}^{\mu_{1}}$ and $S_{t}^{\mu_{2}}$.

######  Lemma 3.13.

There exists a constant $C>0$ such that for any $\nu$ we have

$W_{1}(S_{t}^{\mu_{1}}\nu,S_{t}^{\mu_{2}}\nu)\leq
Ce^{Ct}\left(\|a_{1}[\mu_{1}]-a_{1}[\mu_{2}]\|_{\infty}+\|a_{2}[\mu_{1}]-a_{2}[\mu_{2}]\|_{\infty}\right)$

###### Proof.

We show this by construction of an explicit coupling of $S_{t}^{\mu_{1}}\nu$
and $S_{t}^{\mu_{2}}\nu$. Let $(R_{0},\rho_{0})$ be distributed with law $\nu$
and construct two stochastic processes $(R^{(1)}_{t},\rho^{(1)}_{t})$ and
$(R^{(2)}_{t},\rho^{(2)}_{t})$ as solutions to the SDEs

$\displaystyle\mathrm{d}R^{(i)}_{t}$
$\displaystyle=a[\mu_{i}](R^{(i)}_{t},\rho^{(i)}_{t})\,\mathrm{d}t,$
$\displaystyle\mathrm{d}\rho^{(i)}_{t}$ $\displaystyle=-\gamma
a_{1}[\mu_{i}](\rho^{(i)}_{t})\,\mathrm{d}t+\sigma\,\mathrm{d}W_{t},$

where both SDEs have the same initial data $(R_{0},\rho_{0})$ and the same
Brownian motion $(W_{t})_{t\geq 0}.$ Then the law of
$((R^{(1)}_{t},\rho^{(1)}_{t}),(R^{(2)}_{t},\rho^{(2)}_{t}))$ defines a
coupling of $S_{t}^{\mu_{1}}\nu$ and $S_{t}^{\mu_{2}}\nu$. We can also compute

$\displaystyle\mathrm{d}\left(|R^{(1)}_{t}-R^{(2)}_{t}|+|\rho^{(1)}_{t}-\rho^{(2)}_{t}|\right)$
$\displaystyle\leq\operatorname{sign}(R^{(1)}_{t}-R^{(2)}_{t})\left(a[\mu_{1}](R^{(1)}_{t},\rho^{(1)}_{t})-a[\mu_{2}](R^{(2)}_{t},\rho^{(2)}_{t})\right)\mathrm{d}t$
$\displaystyle\quad-\gamma\operatorname{sign}(\rho^{(1)}_{t}-\rho^{(2)}_{t})\left(a_{1}[\mu_{1}](R^{(1)}_{t},\rho^{(1)}_{t})-a_{1}[\mu_{2}](R^{(2)}_{t},\rho^{(2)}_{t})\right)\mathrm{d}t$
$\displaystyle\leq
C\left(\|a^{\prime}[\mu_{1}]\|_{\infty}+\|a_{1}^{\prime}[\mu_{1}]\|_{\infty}\right)\left(|R^{(1)}_{t}-R^{(2)}_{t}|+|\rho^{(1)}_{t}-\rho^{(2)}_{t}|\right)\mathrm{d}t$
$\displaystyle\quad+C\left(\|a_{1}[\mu_{1}]-a_{1}[\mu_{2}]\|_{\infty}+\|a_{2}[\mu_{1}]-a_{2}[\mu_{2}]\|_{\infty}\right)\mathrm{d}t.$

Integrating this and using the fact that the initial conditions are the same
we get

$\left(|R^{(1)}_{t}-R^{(2)}_{t}|+|\rho^{(1)}_{t}-\rho^{(2)}_{t}|\right)\leq
Ce^{Ct}\left(\|a_{1}[\mu_{1}]-a_{1}[\mu_{2}]\|_{\infty}+\|a_{2}[\mu_{1}]-a_{2}[\mu_{2}]\|_{\infty}\right).$

Taking expectations then gives our result. ∎

We now need to prove a sequence of lemmas relating our distances to each
other.

######  Lemma 3.14.

For any $Z$, we have that

$\|f-g\|_{*}\leq\left(\int(f+g)\phi^{1+a}\right)\sup_{|(\rho,R)|>Z}\phi(\rho,R)^{-a}+\left(\int_{|(\rho,R)|<Z}|f-g|\right)\sup_{|(\rho,R)|<Z}\phi(\rho,R).$

###### Proof.

For every $Z$ we have

$\|f-g\|_{*}=\int_{|(\rho,R)|\leq Z}|f-g|\phi+\int_{(\rho,R)|>Z}|f-g|\phi.$

We then bound the first term by

$\left(\int_{|(\rho,R)|<Z}|f-g|\right)\sup_{|(\rho,R)|<Z}\phi(\rho,R).$

We then bound the second term by

$\left(\int_{|(\rho,R)|\geq
Z}|f-g|\phi^{1+a}\phi^{-a}\right)\leq\left(\int(f+g)\phi^{1+a}\right)\sup_{|(\rho,R)|>Z}\phi^{-a}.$

∎

######  Lemma 3.15.

For any $Z$ we have some constant $C>0$ such that

$\int_{|(\rho,R)|\leq Z}|f-g|\,\mathrm{d}\rho\mathrm{d}R\leq C\left(\\\
\int_{|(\rho,R)|\leq Z}(|\nabla f|^{2}+|\nabla
g|^{2})\,\mathrm{d}\rho\mathrm{d}R\right)^{1/4}\left(W_{1}(f,g)\right)^{1/2}$

###### Proof.

Let $\eta$ be a smooth mollifying function. We have that

$\int_{|(\rho,R)|\leq
Z}|f-g|\,\mathrm{d}\rho\mathrm{d}R=\sup_{\|\psi\|_{\infty}\leq
1}\int_{|(\rho,R)|\leq Z}(f-g)\psi\,\mathrm{d}\rho\mathrm{d}R.$

We also have,

$W_{1}(f,g)=\sup_{\|\psi\|_{Lip}\leq
1}\int(f-g)\psi\,\mathrm{d}\rho\mathrm{d}R$

Using these formulations, let us take $\psi$ to have $\|\psi\|_{\infty}\leq
1$. Furthermore, letting $\eta^{\epsilon}=\eta(z/\epsilon)/\epsilon^{2}$, we
can see

$\displaystyle\int_{|(\rho,R)|\leq Z}(f-g)\psi\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=$ $\displaystyle\int_{|(\rho,R)|\leq
Z}f(\psi-\eta^{\epsilon}*\psi)\,\mathrm{d}\rho\mathrm{d}R+\int_{|(\rho,R)|\leq
Z}(f-g)\eta^{\epsilon}*\psi\,\mathrm{d}\rho\mathrm{d}R+\int_{|(\rho,R)|\leq
Z}g(\psi-\eta^{\epsilon}*\psi)\,\mathrm{d}\rho\mathrm{d}R$ $\displaystyle\leq$
$\displaystyle\|\eta^{\epsilon}*\psi\|_{Lip}W_{1}(f,g)+\int_{|(\rho,R)|\leq
Z}\psi\left(f-\eta^{\epsilon}*f+g-\eta^{\epsilon}*g\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\leq$
$\displaystyle\|\eta^{\epsilon}*\psi\|_{Lip}W_{1}(f,g)+\int_{z\leq
Z}\int_{z^{\prime}}\psi(z)\eta^{\epsilon}(z^{\prime})\left(f(z)+g(z)-f(z-z^{\prime})-g(z-z^{\prime})\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\leq$
$\displaystyle\|\eta^{\epsilon}*\psi\|_{Lip}W_{1}(f,g)+\|\eta^{\epsilon}(z)|z|\|_{2}\left(\int_{z\leq
Z}\left(|\nabla f|^{2}+|\nabla
g|^{2}\right)\,\mathrm{d}\rho\mathrm{d}R\right)^{1/2}$ $\displaystyle\leq$
$\displaystyle C\epsilon^{-1}W_{1}(f,g)+C\epsilon\left(\int_{z\leq
Z}\left(|\nabla f|^{2}+|\nabla
g|^{2}\right)\,\mathrm{d}\rho\mathrm{d}R\right)^{1/2}.$

Optimising over $\epsilon$ gives

$\int_{|(\rho,R)|\leq Z}(f-g)\psi\,\mathrm{d}\rho\mathrm{d}R\leq
CW_{1}(f,g)^{1/2}\left(\int_{z\leq Z}\left(|\nabla f|^{2}+|\nabla
g|^{2}\right)\,\mathrm{d}\rho\mathrm{d}R\right)^{1/4}.$

This implies our result. ∎

Combining these two lemmas we have the following lemma.

######  Lemma 3.16.

For any radially decreasing weight $w(z)$ and any $a>0$ we have,

$\|f-g\|_{*}\leq\left(\int(f+g)\phi^{1+a}\right)\sup_{|(\rho,R)|>Z}\phi(\rho,R)^{-a}+CW_{1}(f,g)^{1/2}\left(\int
w(z)(|\nabla f|^{2}+|\nabla
g|^{2})\right)^{1/4}\sup_{|(\rho,R)|<Z}\phi(\rho,R).$

This implies

$\|f-g\|_{\beta}\leq
C\|f+g\|_{\beta(1+a)}^{1/(1+a)}W_{1}(f,g)^{a/2(1+a)}\left(\int
w(z)\left(|\nabla f|^{2}+|\nabla g|^{2}\right)\right)^{a/4(1+a)}.$

This section is inspired partly from [10]. We extend the regularisation
estimates from this paper by including the weight
$\partial_{\rho}a_{1}[\mu](\rho)$ in front of the terms $\partial_{R}f$, this
allows us to deal with the weak mixing term.

######  Lemma 3.17.

For any weight function $m$, with $|\nabla m|\leq Cm$,
$\partial^{2}_{\rho^{2}}m\leq Cm$, we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}mf^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\leq
C\int_{\mathbb{R}^{2}}mf^{2}\,\mathrm{d}\rho\mathrm{d}R-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\leq
C\int_{\mathbb{R}^{2}}mf^{2}\mathrm{d}\rho\mathrm{d}R+C\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R-\int_{\mathbb{R}^{2}}m\partial_{\rho}a_{1}\partial_{\rho}f\partial_{R}f\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial^{2}_{\rho^{2}}f)^{2}\,\mathrm{d}\rho\mathrm{d}R,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}m\partial_{\rho}f\partial_{R}f\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\leq
C\int_{\mathbb{R}^{2}}mf^{2}\,\mathrm{d}\rho\mathrm{d}R+C\int
m|\partial_{\rho}f\partial_{R}f|\,\mathrm{d}\rho\mathrm{d}R-\int_{\mathbb{R}^{2}}m\partial_{\rho}a_{1}(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial^{2}_{\rho^{2}}f)(\partial^{2}_{\rho,R}f)\,\mathrm{d}\rho\mathrm{d}R,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}m(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\leq
C\int_{\mathbb{R}^{2}}mf^{2}\mathrm{d}\rho\mathrm{d}R+C\int_{\mathbb{R}^{2}}m(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial^{2}_{\rho
R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R.$

###### Proof.

These inequalities are all essentially calculations. Let us start with the
first one,

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}mf^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}mf\left(-\partial_{R}((a_{1}-a_{2})f)+\frac{\sigma^{2}}{2}\partial_{\rho^{2}}^{2}f+\gamma\partial_{\rho}(a_{1}f)\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}\left((a_{1}-a_{2})\partial_{R}(mf)f-\frac{\sigma^{2}}{2}\partial_{\rho}f\partial_{\rho}(mf)-\gamma
a_{1}f\partial_{\rho}(mf)\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}\left((a_{1}-a_{2})\partial_{R}m-\gamma
a_{1}\partial_{\rho}m\right)f^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}\left((a_{1}-a_{2})m\partial_{R}(f^{2})-\gamma
a_{1}m\partial_{\rho}(f^{2})\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\int_{\mathbb{R}^{2}}\frac{\sigma^{2}}{2}\partial_{\rho}m\partial_{\rho}(f^{2})\mathrm{d}\rho\mathrm{d}R-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=\int_{\mathbb{R}^{2}}\left((a_{1}-a_{2})\partial_{R}m+\partial_{R}a_{2}m-\gamma
a_{1}\partial_{\rho}m+\gamma\partial_{\rho}a_{1}m+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}m\right)f^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}m\partial_{\rho}f\partial_{\rho}\left(-\partial_{R}((a_{1}-a_{2})f)+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}f+\gamma\partial_{\rho}(a_{1}f)\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}m\partial_{\rho}f\left(-\partial_{\rho}a_{1}\partial_{R}f+\partial_{R}a_{2}\partial_{\rho}f+(a_{2}-a_{1})\partial^{2}_{\rho
R}f\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+2\int_{\mathbb{R}^{2}}m\partial_{\rho}f\left(\frac{\sigma^{2}}{2}\partial^{3}_{\rho^{3}}f+\gamma\partial^{2}_{\rho^{2}}a_{1}f+2\gamma\partial_{\rho}a_{1}\partial_{\rho}f+\gamma
a_{1}\partial^{2}_{\rho^{2}}f\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=-2\int_{\mathbb{R}^{2}}m\partial_{\rho}a_{1}(\partial_{\rho}f)(\partial_{R}f)\mathrm{d}\rho\mathrm{d}R+\int_{\mathbb{R}^{2}}\left(m\partial_{R}a_{2}+(a_{1}-a_{2})\partial_{R}m)\right)(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial^{2}_{\rho^{2}}f)^{2}\,\mathrm{d}\rho\mathrm{d}R+\frac{\sigma^{2}}{2}\int_{\mathbb{R}^{2}}(\partial^{2}_{\rho^{2}})(\partial_{\rho}f)^{2}-\gamma\int_{\mathbb{R}^{2}}\partial_{\rho}(m\partial^{2}_{\rho^{2}}a_{1})f^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}(4\gamma\partial_{\rho}a_{1}m-\gamma\partial_{\rho}(a_{1}m))(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=-2\int_{\mathbb{R}^{2}}m\partial_{\rho}a_{1}\partial_{\rho}f\partial_{R}f\,\mathrm{d}\rho\mathrm{d}R-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial^{2}_{\rho^{2}}f)^{2}\,\mathrm{d}\rho\mathrm{d}R-\gamma\int_{\mathbb{R}^{2}}\partial_{\rho}(m\partial^{2}_{\rho^{2}}a_{1})f^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}\left(m(\partial_{R}a_{2}+3\gamma\partial_{\rho}a_{1})+(a_{1}-a_{2})\partial_{R}m\mathrm{d}\rho\mathrm{d}R-\gamma
a_{1}\partial_{\rho}m+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}m\right)(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}m\partial_{\rho}f\partial_{R}f\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=\int_{\mathbb{R}^{2}}m\partial_{\rho}f\partial_{R}\left(-\partial_{R}((a_{1}-a_{2})f)+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}f+\gamma\partial_{\rho}(a_{1}f)\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}m\partial_{R}f\partial_{\rho}\left(-\partial_{R}((a_{1}-a_{2})f)+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}f+\gamma\partial_{\rho}(a_{1}f)\right)\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=\int_{\mathbb{R}^{2}}m\left(\partial^{2}_{R^{2}}a_{2}f\partial_{\rho}f+3\partial_{R}a_{2}\partial_{R}f\partial_{\rho}f-\partial_{\rho}a_{1}(\partial_{R}f)^{2}-(a_{1}-a_{2})(\partial_{\rho}f\partial^{2}_{R^{2}}f+\partial_{R}f\partial^{2}_{\rho
R}f)\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\frac{\sigma^{2}}{2}\int_{\mathbb{R}^{2}}m\left(\partial_{\rho}f\partial^{2}_{\rho^{2}R}f+\partial_{R}f\partial^{3}_{\rho^{3}}f\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\gamma\int_{\mathbb{R}^{2}}m\left(\partial_{\rho}a_{1}\partial_{\rho}f\partial_{R}f+a_{1}\partial_{\rho}f\partial^{2}_{\rho
R}f+\partial^{2}_{\rho^{2}}a_{1}f\partial_{R}f+2\partial_{\rho}a_{1}\partial_{\rho}f\partial_{R}f+a_{1}\partial_{R}f\partial_{\rho^{2}}f\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2}}\left(\partial_{R^{2}}^{2}a_{2}\partial_{\rho}m\mathrm{d}\rho\mathrm{d}R+\partial_{\rho^{2}}^{2}a_{1}\partial_{R}m\right)f^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\int_{\mathbb{R}^{2}}m\partial_{\rho}a_{1}(\partial_{R}f)^{2}\mathrm{d}\rho\mathrm{d}R-\sigma^{2}\int_{\mathbb{R}^{2}}m\partial^{2}_{\rho^{2}}f\partial^{2}_{\rho
R}f\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}\left(2m\partial_{R}a_{2}+(a_{1}-a_{2})\partial_{R}m+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}m+2\gamma\partial_{\rho}a_{1}m-\gamma
a_{1}\partial_{\rho}m\right)\partial_{\rho}f\partial_{R}f\,\mathrm{d}\rho\mathrm{d}R,$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbb{R}^{2}}m(\partial_{R}f)^{2}\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}m\partial_{R}f\partial_{R}\left(-\partial_{R}((a_{1}-a_{2})f)+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}f+\gamma\partial_{\rho}(a_{1}f)\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=2\int_{\mathbb{R}^{2}}m(\partial_{R}f)\left(\partial^{2}_{R^{2}}a_{2}f+2\partial_{R}a_{2}\partial_{R}f-(a_{1}-a_{2})\partial^{2}_{R^{2}}f\right)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\sigma^{2}\int_{\mathbb{R}^{2}}m\partial_{R}f\partial^{3}_{R\rho^{2}}f\,\mathrm{d}\rho\mathrm{d}R+2\gamma\int_{\mathbb{R}^{2}}m\partial_{R}f(\partial_{\rho}a_{1}\partial_{R}f+a_{1}\partial^{2}_{\rho
R}f)\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle=-\int_{\mathbb{R}^{2}}\partial_{R}(m\partial^{2}_{R^{2}}a_{2})f^{2}m\,\mathrm{d}\rho\mathrm{d}R-\sigma^{2}\int_{\mathbb{R}^{2}}m(\partial^{2}_{\rho
R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad\int_{\mathbb{R}^{2}}\left(3m\partial_{R}a_{2}+(a_{1}-a_{2})\partial_{R}m+\frac{\sigma^{2}}{2}\partial^{2}_{\rho^{2}}m+\gamma
m\partial_{\rho}a_{1}-\gamma
a_{1}\partial_{\rho}m\right)(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R.$

∎

Using this lemma we can prove a first regularisation result.

######  Lemma 3.18.

Suppose that $m$ satisfies the conditions from the previous lemma, then we can
choose $A_{1},A_{2},A_{3}$ such that $\mathcal{F}(t,f)$ defined by

$\displaystyle\mathcal{F}(t,f):=\int_{\mathbb{R}^{2}}\left[m\left(f^{2}+A_{1}t^{2}(\partial_{\rho}f)^{2}+A_{2}t^{4}(\partial_{\rho}a_{1})(\partial_{\rho}f)(\partial_{R}f)+A_{3}t^{6}(\partial\rho
a_{1})^{2}(\partial_{R}f)^{2}\right)\right]\,\mathrm{d}\rho\mathrm{d}R,$

is a decreasing quantity for $t$ sufficiently small. Specifically, that there
exists a $t_{*}$ such that for $t\leq t_{*}$ there is some $\Lambda$ with

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}(t,f_{t})\leq\int_{\mathbb{R}^{2}}f^{2}m\,\mathrm{d}\rho\mathrm{d}R-\Lambda\int_{\mathbb{R}^{2}}m\left((\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R+t^{4}(\partial_{\rho}a_{1})^{2}(\partial_{R}f)^{2}\right)\,\mathrm{d}\rho\mathrm{d}R.$

###### Proof.

We can differentiate $\mathcal{F}$ using the identities from the previous
lemma. This gives

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}(t,f_{t})$
$\displaystyle\leq
C(1+A_{1}t^{2}+A_{2}t^{4}+A_{3}t^{6})\int_{\mathbb{R}^{2}}f^{2}m\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}m(2A_{1}t+A_{1}t^{2}C-\sigma^{2})(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}m(\partial_{\rho}a_{1})(4A_{2}t^{3}+A_{2}t^{4}C-A_{1}t^{2})|(\partial_{\rho}f)(\partial_{R}f)|\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}m(\partial_{\rho}a_{1})^{2}(6A_{3}t^{5}+CA_{3}t^{6}-A_{2}t^{4})(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\sigma^{2}\int_{\mathbb{R}^{2}}m\left(A_{1}t^{2}(\partial^{2}_{\rho^{2}}f)^{2}+A_{2}t^{4}\partial_{\rho}a_{1}(\partial^{2}_{\rho^{2}}f)(\partial^{2}_{\rho
R}f)+A_{3}t^{6}(\partial_{\rho}a_{1})^{2}(\partial^{2}_{\rho
R}f)^{2}\right)\,\mathrm{d}\rho\mathrm{d}R.$

If $A_{2}\leq 4A_{1}A_{3}$, then the last line will vanish. Splitting the
third line up with Young’s inequality will give

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}(t,f_{t})$
$\displaystyle\leq
C(1+A_{1}t^{2}+A_{2}t^{4}+A_{3}t^{6})\int_{\mathbb{R}^{2}}f^{2}m\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}m\left(2A_{1}t+A_{1}t^{2}C-\sigma^{2}+\frac{1}{2\eta}\left|4A_{2}t+A_{2}t^{2}C-A_{1}\right|\right)(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad+\int_{\mathbb{R}^{2}}m\left(6A_{3}t^{5}+CA_{3}t^{6}-A_{2}t^{4}-+\frac{\eta}{2}\left|4A_{t}t^{5}+A_{2}Ct^{6}-A_{1}t^{4}\right|\right)(\partial_{\rho}a_{1})^{2}(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$

Now if we choose $\eta=A_{1}/A_{2}$ we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}(t,f_{t})$
$\displaystyle\leq
C(t)\int_{\mathbb{R}^{2}}f^{2}m\,\mathrm{d}\rho\mathrm{d}R-(\sigma^{2}-A_{1}^{2}/2A_{2}+o(t))\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-\left(\frac{A_{2}}{2}t^{4}+o(t^{5})\right)\int_{\mathbb{R}^{2}}(\partial_{\rho}a_{1})^{2}(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R.$

Therefore, if we set $A_{1}=A_{2}=A_{3}=\sigma^{2}$ we will have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{F}(t,f_{t})$
$\displaystyle\leq
C(1+A_{1}t^{2}+A_{2}t^{4}+A_{3}t^{6})\int_{\mathbb{R}^{2}}f^{2}m\,\mathrm{d}\rho\mathrm{d}R-(\sigma^{2}/2+o(t))\int_{\mathbb{R}^{2}}m(\partial_{\rho}f)^{2}\,\mathrm{d}\rho\mathrm{d}R$
$\displaystyle\quad-(\sigma^{2}/2+o(t))t^{3}\int_{\mathbb{R}^{2}}m(\partial_{\rho}a_{1})^{2}(\partial_{R}f)^{2}\,\mathrm{d}\rho\mathrm{d}R.$

∎

###### Proof of proposition 3.3.

We know that

$\int_{\mathbb{R}^{2}}f^{2}m\,\mathrm{d}\rho\mathrm{d}R=\int_{\mathbb{R}^{2}}|f\phi||fm\phi^{-1}|\,\mathrm{d}\rho\mathrm{d}R\leq\int_{\mathbb{R}^{2}}|f|\phi\|fm\phi^{-1}\|_{\infty}\,\mathrm{d}\rho\mathrm{d}R.$

Now, if $\mu\in\mathcal{C}_{M,\beta}$ then $G(\mu)$ is a $C^{\infty}$ density
by Hörmander’s theorem; and, in particular, it is in $L^{\infty}$ and since
$m\phi^{-1}$ is in $L^{\infty}$ this means that
$\|G(\mu)m\phi^{-1}\|_{\infty}<\infty$, though we can’t get uniform estimates
on this over $\mathcal{C}_{M,\beta}$.

Integrating the result of the previous lemma gives

$\mathcal{F}(t,f_{t})\leq C(t)\|f_{t}\sqrt{m}\|_{2}^{2}.$

This implies that

$\frac{\mathrm{d}}{\mathrm{d}t}\|f_{t}\sqrt{m}\|^{2}_{2}\leq
C\|f_{t}\sqrt{m}\|_{2}^{2}.$

Furthermore, if $\mu_{1},\mu_{2}\in\mathcal{C}_{M,a(1+\beta)}$ then

$\displaystyle\|S_{t}^{\mu_{1}}G(\mu_{2})-S_{t}^{\mu_{2}}G(\mu_{2})\|_{\beta}$
$\displaystyle\quad\leq
C\|S_{t}^{\mu_{1}}G(\mu_{2})+S_{t}^{\mu_{1}}G(\mu_{2})\|^{1/(1+a)}W_{1}(S_{t}^{\mu_{1}}G(\mu_{2}),S_{t}^{\mu_{1}}G(\mu_{2}))^{a/2(1+a)}\|m(\partial_{\rho}a_{1})^{2}(\nabla
f+\nabla g)\|_{2}^{a/4(1+a)}$ $\displaystyle\quad\leq
C(t)M^{1/(1+a)}\left(\|a_{1}[\mu_{1}]-a_{1}[\mu_{2}]\|_{\infty}+\|a_{2}[\mu_{1}]-a_{2}[\mu_{2}]\|_{\infty}\right)^{a/2(1+a)}\|G(\mu_{2})m\|_{2}^{a/2(1+a)}$

We know that $\|G(\mu_{2})m\|_{2}<\infty$ as

$\|G(\mu_{2})m\|_{2}^{2}\leq\|G(\mu_{2})\phi_{\beta}\|_{1}\|G(\mu_{2})m\phi_{\beta}^{-1}\|_{\infty},$

and we know the second term is finite as $G(\mu_{2})$ is a continuous
probability density so is in $L^{\infty}.$ We also have that

$|a_{1}[\mu_{1}](\rho)-a_{1}[\mu_{2}](\rho)|\leq\int_{\mathbb{R}^{2}}|b(\rho-\rho^{\prime})||\mu_{1}(\rho,R)-\mu_{2}(\rho,R)|\,\mathrm{d}\rho\mathrm{d}R\leq\|b\|_{\infty}\|\mu_{1}-\mu_{2}\|_{TV}\leq
C\|\mu_{1}-\mu_{2}\|_{\beta}.$

Therefore,

$\|S_{t}^{\mu_{1}}G(\mu_{2})-S_{t}^{\mu_{2}}G(\mu_{2})\|_{\beta}\leq
C(t)\|\mu_{1}-\mu_{2}\|_{\beta}^{a/1(1+a)}\|G(\mu_{2})m\|_{2}^{a/1(1+a)}.$

Hence,

$\|G(\mu_{1})-G(\mu_{2})\|_{\beta}\leq
C(t)\|\mu_{1}-\mu_{2}\|_{\beta}^{a/2(1+a)}\|G(\mu_{2})m\|_{2}^{a/2(1+a)}.$

This gives strong continuity of the map $G$ from $\mathcal{C}_{M,(1+a)\beta}$
to itself.

We also have that if $\mu_{n}\rightarrow\mu_{\infty}$ weakly then
$a_{i}[\mu_{n}](\rho)\rightarrow a_{i}[\mu_{\infty}](\rho),$ so we also have
that in this case

$\|G(\mu_{n})-G(\mu_{\infty})\|_{\beta}\rightarrow 0.$

Which implies convergence in total variation and hence that $G(\mu_{n})$
converges weakly towards $G(\mu_{\infty})$.

∎

####  3.2.4 Putting the steps together

We conclude by gathering the results of the previous section to prove our main
Theorem 3.1. First we state Schauder fixed point theorem as can be found in
[34, Theorem 2.3.7].

######  Theorem 3.19 (Schauder Fixed Point Theorem).

Let $S$ be a non-empty, convex closed subset of a Hausdorff topological vector
space and $F$ a mapping of $S$ into itself so that $F(S)$ is compact, then $F$
has a fixed point.

###### Proof of Theorem 3.1.

We apply Schauder’s fixed point theorem to the map $G$, the set
$\mathscr{C}_{M,\beta}$ with $M$ large enough and $\beta$ small enough so that
Propositions 3.1, 3.2 and 3.3 are valid. We work in the topology of weak
convergence of measure so that the set $\mathscr{C}_{M,\beta}$ is clearly
convex and is compact thanks to Lemma 3.3. The function $G$ is well-defined
thanks to Proposition 3.1, continuous thanks to Proposition 3.3. This allows
us to verify all the conditions of Schauder’s fixed point theorem. Lastly we
note that as we have shown that $G(\mu)$ has $\|G(\mu)\|_{\beta}\leq M$
(Proposition 3.1) and is smooth thanks to Malliavin’s theorem then this
properties are also true for our steady state. ∎

##  4 Conclusion and future research directions

In this paper we investigated the existence of steady states to a nonlinear
Fokker-Planck equation, describing the evolution of player ratings competing
in zero sum games. The existence result is based on Schauder’s fixed point
theorem and investigating the behaviour of a corresponding linear problem
using hypercoercivity techniques. To this point we are not able to prove
uniqueness or say anything about the trend to equilibrium. Hence the most
natural next steps in understanding the long-time behaviour of this equation
are:

  * •

Uniqueness of the steady state. It seems likely that for the equation studied
here the steady state will be unique. However, it is challenging to prove. The
kinetic structure means that we cannot view the steady state as the minimiser
of a convex energy. The non-explicitness for the function $G(\mu)$ makes it
challenging to work with. Furthermore, it is difficult to see how we could
convert our fixed point argument into a contraction mapping argument.

  * •

Linear and non-linear stability of the steady state. The linear stability of
the steady state found in this paper seems to be a much more tractable
problem. The numerical results presented in Section 2.2 suggest that the
solution to the non-linear equation is converging exponentially fast towards
its steady state in spaces weighted by $\phi_{\beta}.$ This is also close to a
typical situation for the application of hypocoercivity theory. However,
significant challenges remain. These are essentially the same as the those for
the linear equation (13). We are not able to use Harris’s theorem for the
linearised equation as Harris’s theorem requires the preservation of
positivity.

## Acknowledgements

JE acknowledges partial support from the Leverhulme Trust, Grant ECF-2021-134.
The author(s) would like to thank the Isaac Newton Institute for Mathematical
Sciences, Cambridge, for support and hospitality during the programme
Frontiers in Kinetic Theory: connecting microscopic and macroscopic scales
where work on this paper was undertaken.

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