# Convergence to the uniform distribution of moderately self-interacting
diffusions on compact Riemannian manifolds

Simon Holbach111Institut für Mathematik, Johannes Gutenberg-Universität Mainz,
Staudingerweg 9, 55099 Mainz, Germany (s.holbach@uni-mainz.de), Olivier
Raimond222MODAL’X, Université Paris Nanterre, 200 Avenue de la République,
92000 Nanterre, France<EMAIL_ADDRESS>
Abstract: We consider a self-interacting diffusion $X$ on a smooth compact
Riemannian manifold $\mathbb{M}$, described by the stochastic differential
equation

$dX_{t}=\sqrt{2}dW_{t}(X_{t})-\beta(t)\nabla V_{t}(X_{t})dt,$

where $\beta$ is suitably lower-bounded and grows at most logarithmically, and

$V_{t}(x)=\frac{1}{t}\int_{0}^{t}V(x,X_{s})ds$

for a suitable smooth function $V\colon\mathbb{M}^{2}\to\mathbb{R}$ that makes
the term $-\nabla V_{t}(X_{t})$ self-repelling. We prove that the normalized
occupation measure $\mu_{t}$ of $X$ converges almost surely in total variation
to the uniform distribution $\mathcal{U}$, and we provide a polynomial rate of
convergence. The key to this result is showing that $\mu_{e^{t}}$ shadows the
solution to the measure valued ordinary differential equation

$\dot{\nu}_{t}=-\nu_{t}+\mathcal{U}.$

This work complements and extends results from [13] and [4].

Keywords: self-interacting diffusion, ergodicity, asymptotic pseudotrajectory,
measure-valued ordinary differential equation, stochastic calculus

Mathematics Subject Classification (2020): 60K35

Acknowledgments: S.H. acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation)–Project No.
233630050–TRR 146 “Multiscale Simulation Methods for Soft Matter Systems”.
This research has been conducted within the Fédération Parisienne de
Modélisation Mathématique (FP2M)–CNRS FR 2036. This research has been
conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01).

##  1 Introduction and main result

Let $\mathbb{M}$ a smooth compact Riemannian manifold, and let
$\mathcal{M}(\mathbb{M})$ denote the space of finite signed Borel measures on
$\mathbb{M}$. For any smooth function $V\colon\mathbb{M}^{2}\to\mathbb{R}$ and
any $\mu\in\mathcal{M}(\mathbb{M})$ we write

$V_{\mu}(x):=\mu(V(x,\cdot)):=\int_{\mathbb{M}}V(x,y)\mu(dy)\quad\text{for all
$x\in\mathbb{M}$.}$ (1)

A stochastic process $X$ is called a self-interacting diffusion on
$\mathbb{M}$, if it satisfies an equation of the type

$dX_{t}=\sqrt{2}dW_{t}(X_{t})-\beta(t)\nabla V_{\mu_{t}}(X_{t})dt,$ (2)

where $V\colon\mathbb{M}^{2}\to\mathbb{R}$ is smooth, $\nabla$ denotes the
surface gradient on $\mathbb{M}$, $\beta\colon[0,\infty)\to\mathbb{R}$ is a
continuous function,

$\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}ds$

is the normalized occupation measure of the process $X$ up until time $t$, and
$W$ is a standard Brownian vector field on $\mathbb{M}$.

Here, $V$ describes the type of self-interaction and $\beta$ is the temporal
weight that is given to the self-interaction mechanism. The asymptotic
behavior of $X$ has been studied for different choices of $V$ and $\beta$, and
we provide a summary of corresponding results and methods in Section 1.1
below. Let us now describe the assumptions under which we work in the present
article.

######  Assumption 1.1.

There is a smooth function $v\colon\mathbb{M}\to\mathbb{R}^{N}$ with
$\left\|v\right\|_{\infty}=1$ and

$\int_{\mathbb{M}}v(x)dx=0$ (3)

such that

$V(x,y)=v(x)\cdot v(y)\quad\text{for all $x,y\in\mathbb{M}$,}$

where $\cdot$ denotes the euclidean inner product.

With this choice of $V$, (1) yields

$V_{\mu_{t}}(x)=\mu_{t}(v)\cdot
v(x)=\left(\frac{1}{t}\int_{0}^{t}v(X_{s})ds\right)\cdot v(x).$

Hence, the drift term $-\nabla V_{\mu_{t}}(X_{t})$ is self-repelling in the
sense that it tends to drive $v(X_{t})$ away from the temporal mean of
$(v(X_{s}))_{s\in[0,t]}$. This interpretation is particularly intuitive in the
case where

$\mathbb{M}=\mathbb{S}^{n}=\\{x\in\mathbb{R}^{n+1}:\left|x\right|=1\\}$

and $v(x)=x$, so that

$V(x,y)=x\cdot y=\cos(d(x,y))\quad\text{for all $x,y\in\mathbb{S}^{n}$,}$

where $d$ is the geodesic distance on $\mathbb{S}^{n}$.

Next, we explain our assumptions on $\beta$. Here and everywhere else in this
article, $C$ and $t_{0}$ denote finite, positive, deterministic constants, the
exact value of which is unimportant and may change from one step to the next
with no indication.

######  Assumption 1.2.

The function $\beta\colon[0,\infty)\to\mathbb{R}$ is differentiable and there
are $a\in(0,\infty)$ and $\gamma\in(0,1]$ such that

$\left|\beta(t)\right|\leq a\log
t\quad\text{and}\quad\left|\beta^{\prime}(t)\right|\leq
Ct^{-\gamma}\quad\text{for all $t\geq t_{0}$.}$

Assumption 1.2 allows the weight $\beta(t)$ of the self-repelling drift
$-\nabla V_{\mu_{t}}$ to increase to infinity, but not fast enough to fully
compensate the normalization $\frac{1}{t}$ of the occupation measure. The
generic case that we will usually have in mind is that of

$\beta(t)=b\log(t+1)\quad\text{with $b>0$.}$

Other valid choices include $\beta(t)=b\log(\log(t+e))$ or simply $\beta\equiv
b$. Note that we cap $\gamma$ at 1 only for simplicity, as $\gamma>1$ provides
no meaningful improvement for the estimates that are relevant to our proofs
and only results in awkward case distinctions. Also note that the
normalization $\left\|v\right\|_{\infty}=1$ in Assumption 1.1 makes sure that
the parameter $a$ is actually meaningful and cannot simply be hidden in $v$.

Assumption 1.2 does not require $\beta$ to be non-negative, but we will assume
a suitable lower bound in Assumption 1.3 below. Before we can state it
precisely, we need to introduce some notation. Let $m\in\mathbb{R}^{N}$.
Setting

$Z(m)=\int_{\mathbb{M}}e^{-m\cdot v(x)}dx,$

we define a probability measure $\Pi(m)$ on $\mathbb{M}$ via

$\Pi(m)(dx)=\frac{e^{-m\cdot v(x)}}{Z(m)}dx.$

We also interpret $\Pi$ as a function on $\mathcal{M}(\mathbb{M})$ by setting

$\Pi(\mu):=\Pi(\mu(v))\quad\text{for all $\mu\in\mathcal{M}(\mathbb{M})$.}$
(4)

Furthermore, for any probability measure $\mu$ on $\mathbb{M}$ and any
$f\in(\mathbb{L}^{2}(\mu))^{N}$ we write

$\operatorname{Cov}_{\mu}(f)=\big{(}\mu(f_{i}f_{j})-\mu(f_{i})\mu(f_{j})\big{)}_{i,j\in\\{1,\ldots,N\\}}\in\mathbb{R}^{N\times
N}.$

######  Assumption 1.3.

There is a $\beta_{0}\geq 0$ such that

$\beta(t)\geq-\beta_{0}>-\frac{1}{\Lambda}\quad\text{for all $t\geq t_{0}$,}$
(5)

where

$\Lambda=\sup_{m\in\mathbb{R}^{N}}\left(\sup_{x\in\mathbb{R}^{N},\,\left|x\right|=1}x^{T}\operatorname{Cov}_{\Pi(m)}(v)x\right).$

Assumption 1.3 means that while the weight factor in front of the self-
repelling drift $-\nabla V_{\mu_{t}}$ is in fact allowed to turn negative, we
do require a suitable lower bound that depends on $v$. In other words, we
allow a limited amount of self-attraction. Clearly,

$0<\Lambda\leq\left\|v\right\|_{\infty}^{2},$ (6)

so (5) makes sense. In particular, $\beta\equiv b$ is covered in our setting
if and only if $b>-1/\Lambda$.

We always equip $\mathcal{M}(\mathbb{M})$ with the total variation norm

$\left\|\mu\right\|=\sup\\{\mu(f):f\in\mathcal{F},\,\left\|f\right\|_{\infty}\leq
1\\}\quad\text{for all $\mu\in\mathcal{M}(\mathbb{M})$,}$

which, due to density arguments, is the same whether we take $\mathcal{F}$ as
$C(\mathbb{M})$ or $C^{\infty}(\mathbb{M})$ or the real-valued measurable
functions on $\mathbb{M}$ or just the indicator functions of Borel-measurable
subsets of $\mathbb{M}$. Our main result is the following:

######  Theorem 1.4.

Let $\mathcal{U}$ denote the uniform distribution on $\mathbb{M}$. There is a
constant $\kappa\in(0,\infty)$ such that if the Assumptions 1.1, 1.2, and 1.3
hold with $2a\kappa<\gamma$, then

$\limsup_{t\to\infty}\frac{\log\left\|\mu_{t}-\mathcal{U}\right\|}{\log
t}\leq-\eta\quad\text{almost surely,}$ (7)

where

$\eta=\min\left\\{\frac{\gamma}{2}-a\kappa,1-\Lambda\beta_{0}\right\\}>0.$

In particular, Theorem 1.4 yields weak convergence

$\mu_{t}\xrightarrow{w}\mathcal{U}\quad\text{almost surely for $t\to\infty$},$

i.e. the ergodicity property

$\frac{1}{t}\int_{0}^{t}f(X_{s})ds\xrightarrow{t\to\infty}\mathcal{U}(f)\quad\text{almost
surely for all $f\in C(\mathbb{M})$}.$

However, Theorem 1.4 is much stronger than this, as it also includes some
uniformity with respect to $f$ and a polynomial convergence rate.

######  Example 1.5 (Weak self-interaction).

Let

$\beta\equiv b>-\frac{1}{\Lambda}.$

Depending on the sign of $b$, this corresponds either to a self-repelling
($b>0$) or self-attracting ($b<0$) diffusion, and since $\beta$ is constant in
time, we speak of weak self-interaction.

Then $\beta^{\prime}\equiv 0$ and for any $a>0$ we have $|\beta(t)|\leq a\log
t$ for all $t\geq t_{0}=e^{b/a}$. Hence, Theorem 1.4 can be applied with
$\gamma=1$, $\beta_{0}=\max\\{0,-b\\}$ and any $a>0$, and we get that

$\limsup_{t\to\infty}\frac{\log\left\|\mu_{t}-\mathcal{U}\right\|}{\log
t}\leq-\min\left\\{\frac{1}{2},1-\Lambda|b|\right\\}\quad\text{almost
surely.}$

In the particular case $\mathbb{M}=\mathbb{S}^{n}$ and $v(x)=x$, this
strengthens part (i) of [4, Theorem 4.5]. In Section 3, we provide a more
detailed investigation of this connection and also a proof that we can
actually weaken Assumption 1.3 in this situation (Proposition 3.4).

######  Example 1.6 (Moderate self-repulsion).

Let

$\beta(t)=b\log(t+1)\quad\text{with $b>0$.}$

In this situation, $\beta$ is positive and increases to infinity, but slower
than the normalization factor $t$ in $\mu_{t}$, so we speak of moderate self-
repulsion.

Then $|\beta^{\prime}(t)|\leq Ct^{-1}$ and for any $a>b$ there is a $t_{0}>0$
such that $|\beta(t)|\leq a\log t$ for all $t\geq t_{0}$. Hence, Theorem 1.4
can be applied whenever $b<\frac{1}{2\kappa}$, and we get that

$\limsup_{t\to\infty}\frac{\log\left\|\mu_{t}-\mathcal{U}\right\|}{\log
t}\leq-\left(\frac{1}{2}-b\kappa\right)\quad\text{almost surely.}$

###  1.1 Context

The asymptotic behavior of self-interacting diffusions has been studied with
various degrees of generality concerning the state space $\mathbb{M}$ and the
type of self-interaction governed by $V$. The weight $\beta(t)$ is usually
chosen as either $\beta\equiv b$ or $\beta(t)=bt$, which are sometimes
referred to as weak and strong self-interaction respectively. In this sense,
the prototypical case $\beta(t)=b\log(t+1)$ of Assumption 1.2 corresponds to
moderate self-interaction.

There are a number of case studies and some general results for
$\mathbb{M}=\mathbb{R}^{n}$ (e.g. [9, 12, 11, 8]), but our focus lies on
compact state spaces. Some of the results in the following summary are more
general than others, but all of them are valid for $\mathbb{M}=\mathbb{S}^{n}$
and include the important case $V(x,y)=\cos(d(x,y))$ (self-repulsion) or
$V(x,y)=-\cos(d(x,y))$ (self-attraction).333Of course, it is somewhat
arbitrary to include the sign that distinguishes repulsion from attraction in
$V$ instead of $\beta$, but this choice reinforces the interpretation of $V$
as the type of interaction and $\beta$ as a weight (even though our
Assumptions 1.2 and 1.3 do not require $\beta$ to be non-negative). The
general expectation is that a diffusion with sufficient self-attraction will
asymptotically be concentrated around (or even converge to) some limit random
variable $X_{\infty}\in\mathbb{M}$, while a self-repelling diffusion (on a
compact state space) will quite contrarily be uniformly distributed in the
limit.

We will now give a brief overview of the methods that have been used in these
different cases, the corresponding results are summarized in Table 1.

The case of weak self-interaction has been studied the most thoroughly, in
particular in the series of papers [4, 5, 6, 7]. Under mild conditions on $V$,
the authors link $\mu_{t}$ to a measure-valued ordinary differential equation
and use this to precisely describe its asymptotic behavior for some specific
choices of $V$. The same approach is used in [13] to treat the case of
moderate self-interaction. This turns out to be more delicate and only the
self-attracting case is solved satisfyingly. The case of moderate self-
repulsion is the content of the present paper, and we use a similar strategy.
A detailed explanation of the general idea of this method is presented in
Section 1.2 below, including a discussion of the differences between [4],
[13], and the present paper. The case of strong interaction has been studied
with different methods. In [3], the authors rewrite (2) as a time-homogeneous
proper stochastic differential equation for an extended variable
$(X_{t},Y_{t})\in\mathbb{S}^{n}\times\mathbb{R}^{d}$ and prove that in the
self-repelling case it is Harris recurrent and exponentially ergodic, where
the invariant distribution in restriction to $X_{t}$ is the uniform
distribution. Almost sure convergence in the self-attracting case is proved in
[10], again with arguments that involve the shadowing of an ordinary
differential equation, but in a completely different way than in [4], [13], or
the present paper.

$\mathbb{M}=\mathbb{S}^{n}$ |  strong $\beta(t)=bt$ |  moderate $\beta(t)=b\log(t+1)$ |  weak $\beta(t)=b$
---|---|---|---
self-attraction $V(x,y)=-\cos(d(x,y))$ |  $X_{t}\to X_{\infty}$ a.s. ([10]) |  $\mu_{t}\xrightarrow{w}\delta_{X_{\infty}}$ a.s. ([13]) |  $\mu_{t}\xrightarrow{w}\mu_{\infty}(b,n)$ a.s. ([4, 5, 6, 7], this paper)
self-repulsion $V(x,y)=\cos(d(x,y))$ |  $\mu_{t}\xrightarrow{w}\mathcal{U}$ a.s. ([3, 2] for $n=1$) |  $\left\|\mu_{t}-\mathcal{U}\right\|\to 0$ a.s. (this paper) |  $\left\|\mu_{t}-\mathcal{U}\right\|\to 0$ a.s. ([4, 5, 6, 7], this paper)
Table 1: An (incomplete) overview of the literature on the asymptotics of
self-interacting diffusions on $\mathbb{S}^{n}$. In the case of weak self-
attraction, we have $\mu_{\infty}(b,n)=\mathcal{U}$ if $b\leq n+1$, and else
$\mu_{\infty}(b,n)$ is a Gaussian distribution centered around a random
$X_{\infty}\in\mathbb{S}^{n}$, but with a deterministic variance that depends
on $b$ and $n$. The present paper is primarily focussed on moderate and weak
self-repulsion, but also contains a partial improvement of this result on weak
self-attraction.

Of course, it seems plausible that increasing the strength of the repulsion or
attraction by increasing the weight $\beta$ should not change the results
qualitatively. If $\tilde{\beta}$ is asymptotically larger than $\beta$, and
the self-repelling diffusion with weight $\beta$ is asymptotically uniformly
distributed, then the same should be true for the self-repelling diffusion
with weight $\tilde{\beta}$. Similarly, if the self-attracting diffusion with
weight $\beta$ converges to a random variable $X_{\infty}$ in some sense, then
the self-attracting diffusion with weight $\tilde{\beta}$ should converge to
some $\tilde{X}_{\infty}$. However, such comparison theorems are not
available, and they do not seem to be within reach. In view of these
considerations, it also seems counter-intuitive that we need $a$ to be
sufficiently small in Theorem 1.4 and that the rate of convergence decreases
when $a$ increases. This can be thought of as a technical assumption that is
an "artifact" of our method.

######  Remark 1.7.

The factor $\sqrt{2}$ in front of $dW_{t}(X_{t})$ in (2) is absent both in
[13] (where this equation is mentioned explicitly only in the abstract) and in
[4] (where the corresponding equation is the first one of the article).

In [13] this factor $\sqrt{2}$ is hidden in the vector fields $e_{i}$, so that
(2) is entirely equivalent to [13, (1)] and the results from [13] are
compatible with our setting with no adjustment of the parameters.

In [4] however, this is not the case and this factor $\sqrt{2}$ leads to a
factor $2$ in [4, (11)] when compared to our definition of $\Pi$ in (4), and
also to a factor $1/2$ in the definition of $A_{\mu}$ just below [4, (2)] when
compared to our definition in (12). This has to be taken into account when
comparing our results with those from [4].

###  1.2 Outline of proof

In order to get a grip of the long time behavior of

$\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}ds,$

we calculate its time evolution. First, we have

$\partial_{t}\mu_{t}=\frac{1}{t}\left(-\mu_{t}+\delta_{X_{t}}\right),$

where the derivative is to be understood as the derivative of a real function
pointwise in all $f\in C(\mathbb{M})$. In order to eliminate the factor
$\frac{1}{t}$, we look at the dynamics on an exponential time scale, i.e.

$\partial_{t}\mu_{e^{t}}=-\mu_{e^{t}}+\delta_{X_{e^{t}}}.$

If $t>0$ is large, the distribution of $X_{t}$ should be close to the
equilibrium of the current drift potential $\beta(t)V_{\mu_{t}}(x)$, i.e. to
the probability distribution $\Pi(\beta(t)\mu_{t})$ as defined in (4). If our
intuition of the process is correct, $\Pi(\beta(t)\mu_{t})$ on the other hand
should be close to the uniform distribution $\mathcal{U}$ on $\mathbb{M}$.
Therefore, we set

$\varepsilon^{1}_{t}:=\delta_{X_{e^{t}}}-\Pi(\beta(e^{t})\mu_{e^{t}})$

and

$\varepsilon^{2}_{t}:=\Pi(\beta(e^{t})\mu_{e^{t}})-\mathcal{U},$

so that

$\partial_{t}\mu_{e^{t}}=-\mu_{e^{t}}+\mathcal{U}+\varepsilon^{1}_{t}+\varepsilon^{2}_{t}.$
(8)

If $\varepsilon^{1}_{t}$ and $\varepsilon^{2}_{t}$ are asymptotically
negligible in a suitable sense, then the trajectory $t\mapsto\mu_{e^{t}}$
shadows that of a solution to the measure-valued ordinary differential
equation

$\dot{\nu}_{t}=-\nu_{t}+\mathcal{U},$ (9)

so that in particular $\mu_{t}$ converges to $\mathcal{U}$. The details of
this argument are given in Section 2.3 below, but first we devote Sections 2.1
and 2.2 to the required analysis of the asymptotics of $\varepsilon^{1}$ and
$\varepsilon^{2}$.

Our proof strategy is inspired by [4] and [13]. In these works, the authors
study the equation

$\partial_{t}\mu_{e^{t}}=-\mu_{e^{t}}+\Pi(\beta(e^{t})\mu_{e^{t}})+\varepsilon^{1}_{t},$
(10)

which will also be very useful for us in Section 2.2. The formal limit
equation corresponding to (10) is

$\dot{\nu}_{t}=-\nu_{t}+\Pi(\beta(e^{t})\nu_{t}).$ (11)

Note that under (3) we have $\Pi(\mathcal{U})=\mathcal{U}$, and so (11) can be
viewed as a variant of the limit equation (9) that is less specific to a
particular situation. For constant $\beta$ as in [4], the equation (11) is
homogeneous in time and hence after establishing that $\mu_{e^{t}}$ shadows
it, powerful results from the theory of dynamical systems can be used to study
the long-time behavior of $\mu_{t}$ for several different choices of $V$ (cf.
[4, Sections 3 and 4]). For non-constant $\beta$ as in [13], this link can
still be established under certain conditions (cf. [13, Theorems 2.1 and
2.2]), but it is not as fruitful, since (11) is no longer homogeneous in time.
In particular, [13, Theorems 2.1 and 2.2] are not used in the proof of the
convergence result for moderately self-attracting diffusions on the sphere
([13, Theorem 3.1]) which is instead proved "by hand".

The limit equation (9) is much simpler than (11), because it is tailor-made
for cases in which we expect $\mu_{t}$ to converge to the uniform distribution
(while in other cases we might not expect a deterministic limit at all).
Introducing the second error term $\varepsilon^{2}$ allows us to move the
explicit dependence of the problem on $\beta$ entirely into the error terms.

##  2 Proof of the main result

For Section 2.1, we only need the Assumptions 1.1 and 1.2 to hold. In Sections
2.2 and 2.3 on the other hand, we suppose that all of the Assumptions 1.1,
1.2, and 1.3 hold.

###  2.1 Dealing with $\varepsilon^{1}$

This subsection is mostly taken from Section 2 of [13]. We include this
detailed summary, since many of the objects and intermediate results will be
used in the next subsection.

Let us fix $\mu\in\mathcal{M}(\mathbb{M})$ and consider the time-homogeneous
stochastic differential equation with no self-interaction

$dY_{t}=\sqrt{2}dW_{t}(Y_{t})-\nabla V_{\mu}(Y_{t})dt,$

where we think of the dynamics as those arising from freezing the drift
potential in (2) at some time $t_{0}$, so that formally
$\mu=\beta(t_{0})\mu_{t_{0}}$. These dynamics can also be described via the
infinitesimal generator

$A_{\mu}f=\Delta f-\nabla V_{\mu}\cdot\nabla f\quad\text{for all
}f\in\mathcal{D}(A_{\mu})\supset C^{2}(\mathbb{M}),$ (12)

and the corresponding equilibrium is given by $\Pi(\mu)$ as defined in (4).
Let $(P_{s}^{\mu})_{s\geq 0}$ denote the transition semigroup on
$\mathbb{L}^{2}(\Pi(\mu))=\mathbb{L}^{2}(dx)$ generated by $A_{\mu}$. Then
$A_{\mu}$ satisfies a Poincaré inequality and therefore $P_{s}^{\mu}f$
converges to $\Pi(\mu)(f)$ exponentially fast with respect to the
$\mathbb{L}^{2}$-norm for any $f\in\mathbb{L}^{2}(dx)$ (see Sections 1.1 and
1.4.3 of [14]). Using this and basic semigroup theory, one can easily show
that

$Q_{\mu}f:=-\int_{0}^{\infty}P^{\mu}_{s}(f-\Pi(\mu)(f))ds\quad\text{for all
}f\in\mathbb{L}^{2}(dx),$

is well-defined and satsifies

$A_{\mu}Q_{\mu}f=Q_{\mu}A_{\mu}f=f-\Pi(\mu)(f)\quad\text{for all
}f\in\mathcal{D}(A_{\mu}),$

so $Q_{\mu}$ is "almost an inverse to $A_{\mu}$".

Now set

$A_{t}:=A_{\beta(t)\mu_{t}},\quad Q_{t}:=Q_{\beta(t)\mu_{t}}.$

Then for all $f\in C^{\infty}(\mathbb{M})$ we can rewrite

$\varepsilon^{1}_{t}(f)=f(X_{e^{t}})-\Pi(\beta(e^{t})\mu_{e^{t}})(f)=A_{e^{t}}Q_{e^{t}}f(X_{e^{t}}).$

If we set

$F^{f}_{t}(x):=\frac{1}{t}Q_{t}f(x),$ (13)

applying the change of variables $r\mapsto\log r$ and then Ito’s formula
yields

$\displaystyle\begin{split}\int_{s}^{t}\varepsilon^{1}_{r}(f)d&=\int_{e^{s}}^{e^{t}}A_{r}F^{f}_{r}(X_{r})dr\\\
&=F^{f}_{e^{t}}(X_{e^{t}})-F^{f}_{e^{s}}(X_{e^{s}})-\int_{e^{s}}^{e^{t}}\dot{F}^{f}_{r}(X_{r})dr+\left(M^{f}_{t}-M^{f}_{s}\right),\end{split}$
(14)

where $(M^{f}_{t})_{t\geq 0}$ is a martingale with

$\langle M^{f},M^{g}\rangle_{t}=\int_{1}^{e^{t}}\nabla
F^{f}_{r}(X_{r})\cdot\nabla F^{g}_{r}(X_{r})dr\quad\text{for all $f,g\in
C^{\infty}(\mathbb{M})$}$ (15)

(compare [13, Section 3.3] and [4, Remark 2.2]). Therefore, in order to
estimate the integral over $\varepsilon^{1}$, we need to estimate $F^{f}_{t}$,
$\nabla F^{f}_{t}$, and the time derivative $\dot{F}^{f}_{t}$, which can be
done with the help of the following lemma that will also be needed in Section
2.2 below in order to deal with $\varepsilon^{2}$.

######  Lemma 2.1.

There is a constant

$\kappa\in(0,\infty)$ (16)

such that for all $f\in C^{\infty}(\mathbb{M})$ and $t\geq t_{0}$ the
following estimates hold.

  1. 1.

$\left\|Q_{t}f\right\|_{\infty}\leq Ct^{a\kappa}\left\|f\right\|_{\infty}$,

  2. 2.

$\left\|\nabla Q_{t}f\right\|_{\infty}\leq C(1+a\log
t)^{1/2}t^{a\kappa}\left\|f\right\|_{\infty}$,

  3. 3.

$\left\|\partial_{t}Q_{t}f\right\|_{\infty}\leq C(\log
t)^{3/2}t^{2a\kappa-\gamma}\left\|f\right\|_{\infty}$.

###### Proof.

Thanks to Assumption 1.2, this follows from Lemmas 2.3 and 2.8 of [13]. These
lemmas mainly rely on classical results about $\log$-Sobolev and Poincaré
inequalities for the operator $A_{t}$ and an application of the Bakry-Emery
criterion. ∎

From now on, $\kappa$ will always be the constant from (16). In particular,
this is the same $\kappa$ that we use in Theorem 1.4.

######  Proposition 2.2.

If $2a\kappa<\gamma$, then almost surely

$\limsup_{t\to\infty}\frac{1}{t}\log\left(\sup_{s\geq
0}\left\|\int_{t}^{t+s}\varepsilon^{1}_{r}dr\right\|\right)\leq-\left(\frac{\gamma}{2}-a\kappa\right).$

###### Proof.

This follows from [13, Theorem 2.1], which is proved via (14), (15), and Lemma
2.1. The exact value on the right hand side is not given explicitly in [13,
Theorem 2.1], but can be deduced from a careful reading of the proofs in [13,
Section 2.2.2]. ∎

###  2.2 Dealing with $\varepsilon^{2}$

We start this section with a preparatory lemma about properties of $\Pi$.
Recall that

$\Pi(m)(dx)=\frac{e^{-m\cdot v(x)}}{Z(m)}dx\quad\text{with}\quad
Z(m)=\int_{\mathbb{M}}e^{-m\cdot v(x)}dx$

for all $m\in\mathbb{R}^{N}$.

######  Lemma 2.3.

  1. 1.

The mapping $\Pi\colon\mathbb{R}^{N}\to\mathcal{M}(\mathbb{M})$ is Lipschitz
continuous, i.e.

$\left\|\Pi(m)-\Pi(m^{\prime})\right\|\leq
C\left|m-m^{\prime}\right|\quad\text{for all
$m,m^{\prime}\in\mathbb{R}^{N}$.}$

  2. 2.

For all $m\in\mathbb{R}^{N}$, we have

$\Pi(m)(v)=-\nabla\log Z(m)$ (17)

and

$\operatorname{Cov}_{\Pi(m)}(v)=\operatorname{Hess}\log Z(m),$ (18)

where $\operatorname{Hess}$ denotes the Hessian matrix.

###### Proof.

For the first part, it is easy to check that the derivative of $\Pi$ is
bounded. The second part is a simple calculation. ∎

By combining the first part of Lemma 2.3 and the first part of Assumption 1.2,
we get

$\left\|\varepsilon^{2}_{t}\right\|=\left\|\Pi(\beta(e^{t})\mu_{e^{t}}(v))-\Pi(0)\right\|\leq
Ct\left|\mu_{e^{t}}(v)\right|.$ (19)

Lemma 2.6 below provides a speed of convergence to 0 for the right hand side,
but we first need another preparatory lemma. For the remainder of this
section, we use the shorthand notation

$m_{t}:=\mu_{e^{t}}(v)\in\mathbb{R}^{N}\quad\text{for all $t\geq 0$.}$

######  Lemma 2.4.

There is a family of strongly convex functions
$J_{t}\colon\mathbb{R}^{N}\to\mathbb{R}$ such that

$\dot{m}_{t}=-\nabla J_{t}(m_{t})+\varepsilon^{1}_{t}(v)\quad\text{for all
$t\geq 0$}$ (20)

and

$m\cdot\nabla J_{t}(m)\geq(1-\Lambda\beta_{0})\left|m\right|^{2}\quad\text{for
all $t\geq t_{0}$ and $m\in\mathbb{R}^{N}$.}$ (21)

###### Proof.

By (10), we have

$\dot{m}_{t}=-m_{t}+\Pi(\beta(e^{t})m_{t})(v)+\varepsilon^{1}_{t}(v).$

Using (17) and the fact that $\Pi(0)(v)=0$ due to (3), we see that (20) holds
with

$J_{t}\colon\mathbb{R}^{N}\to\mathbb{R},\quad
J_{t}(m)=\frac{1}{2}\left|m\right|^{2}+1_{\beta(e^{t})\neq
0}\cdot\frac{1}{\beta(e^{t})}\log Z(\beta(e^{t})m).$

Because of (18), we get

$\displaystyle\operatorname{Hess}J_{t}(m)=1_{N\times
N}+\beta(e^{t})\operatorname{Cov}_{\Pi(\beta(e^{t})m)}(v),$

and thanks to Assumption 1.3, this implies that the smallest eigenvalue of
$\operatorname{Hess}J_{t}(m)$ is at least $1-\Lambda\beta_{0}>0$. Therefore
$J_{t}$ is strongly convex, more precisely

$(m-m^{\prime})\cdot\left(\nabla J_{t}(m)-\nabla
J_{t}(m^{\prime})\right)\geq(1-\Lambda\beta_{0})\left|m-m^{\prime}\right|^{2}\quad\text{for
all $m,m^{\prime}\in\mathbb{R}^{N}$.}$

Since $\nabla J_{t}(0)=\Pi(0)(v)=0$, setting $m^{\prime}=0$ yields (21). ∎

######  Remark 2.5.

$J_{t}(m)$, $\nabla J_{t}(m)$, and $\operatorname{Hess}J_{t}(m)$ are all
continuous with respect to $t\in[0,\infty)$.

In what follows, $\eta$ will always be the constant from Theorem 1.4, i.e.

$\eta=\min\left\\{\frac{\gamma}{2}-a\kappa,1-\Lambda\beta_{0}\right\\}>0.$

######  Lemma 2.6.

If $2a\kappa<\gamma$, then

$e^{\delta t}m_{t}\xrightarrow{t\to\infty}0\quad\text{almost surely for all
$\delta<\eta$.}$

###### Proof.

1.) Set

$\tilde{m}_{t}:=m_{t}-F_{e^{t}}(X_{e^{t}})$ (22)

where

$F_{t}(x):=(F_{t}^{v_{i}}(x))_{i=1,\ldots,N}=t^{-1}(Q_{t}v_{i}(x))_{i=1,\ldots,N}$

(compare (13)). The first estimate of Lemma 2.1 implies

$|e^{\delta t}m_{t}|\leq e^{\delta
t}\big{(}|\tilde{m}_{t}|+\left|F_{e^{t}}(X_{e^{t}})\right|\big{)}\leq
e^{\delta t}|\tilde{m}_{t}|+Ce^{-(1-a\kappa-\delta)t}\quad\text{for all $t\geq
t_{0}$.}$

Since $\delta<\frac{\gamma}{2}-a\kappa<1-a\kappa$, the second summand in the
above bound vanishes for $t\to\infty$, and hence it suffices to show for this
lemma that

$e^{\delta t}\tilde{m}_{t}\xrightarrow{t\to\infty}0\quad\text{almost surely
for all $\delta<\eta$.}$ (23)

Note that we know a priori that $\left|m_{t}\right|$ has a deterministic upper
bound and then (by Lemma 2.1) so has $\left|\tilde{m}_{t}\right|$.

2.) In this step, we use a similar approach as in Sections 3.1 - 3.4 of [13]
in order to find a stochastic differential equation that is fulfilled by
$|\tilde{m}_{t}|^{2}$. In order to calculate the dynamics of $\tilde{m}_{t}$,
we first deduce from (14) that

$F_{e^{t}}(X_{e^{t}})-F_{e^{s}}(X_{e^{s}})=\int_{s}^{t}\varepsilon^{1}_{r}(v)dr+\int_{e^{s}}^{e^{t}}\dot{F}_{r}(X_{r})dr-(M^{v}_{t}-M^{v}_{s}),$
(24)

where we read $M^{v}_{t}$ as the vector $(M^{v_{i}}_{t})_{i=1,\ldots,N}$.
Combining (24) with (20) yields

$d\tilde{m}_{t}=-\nabla
J_{t}(m_{t})dt-e^{t}\dot{F}_{e^{t}}(X_{e^{t}})dt+dM^{v}_{t}.$ (25)

We set

$H_{t}=-e^{t}\dot{F}_{e^{t}}(X_{e^{t}})-\left(\nabla J_{t}(m_{t})-\nabla
J_{t}(\tilde{m}_{t})\right)$ (26)

and rewrite (25) as

$d\tilde{m}_{t}=-\nabla J_{t}(\tilde{m}_{t})dt+H_{t}dt+dM^{v}_{t}.$

Ito’s formula and (15) then yield

$d|\tilde{m}_{t}|^{2}=\left(-2\tilde{m}_{t}\cdot\nabla
J_{t}(\tilde{m}_{t})+2\tilde{m}_{t}\cdot H_{t}+e^{t}\left|\nabla
F_{e^{t}}(X_{e^{t}})\right|^{2}\right)dt+2\tilde{m}_{t}\cdot dM^{v}_{t}.$ (27)

3.) Our next goal is to find a suitable upper bound for the drift in (27). For
the rest of the proof we assume that $t\geq t_{0}$ and

$0<\alpha<2\eta=\min\left\\{\gamma-2a\kappa,2(1-\Lambda\beta_{0})\right\\}.$

Since

$\dot{F}_{t}^{v_{i}}=-\frac{1}{t^{2}}Q_{t}v_{i}+\frac{1}{t}\partial_{t}Q_{t}v_{i},$

the first and third estimates of Lemma 2.1 yield

$\left|e^{t}\dot{F}_{e^{t}}(X_{e^{t}})\right|\leq
Ce^{(a\kappa-1)t}+C(1+at)^{3/2}e^{(2a\kappa-\gamma)t}\leq Ce^{-\alpha t}.$
(28)

By the first parts of Lemmas 2.3 and 2.1 respectively,

$\left|\nabla J_{t}(m_{t})-\nabla
J_{t}(\tilde{m}_{t})\right|\leq(1+C|\beta(e^{t})|)\left|m_{t}-\tilde{m}_{t}\right|\leq
C(1+t)e^{(a\kappa-1)t}\leq Ce^{-\alpha t}.$ (29)

Combining (28), (29), and the fact that $|\tilde{m}_{t}|$ is bounded, we can
conclude that with $H$ from (26) we have

$2\tilde{m}_{t}\cdot H_{t}\leq Ce^{-\alpha t}.$ (30)

For the third summand of the drift term in (27), the second estimate of Lemma
2.1 implies

$e^{t}\left|\nabla
F_{e^{t}}(X_{e^{t}})\right|^{2}=e^{-t}\sum_{i=1}^{N}\left|(\nabla
Q_{e^{t}}v_{i})(X_{e^{t}})\right|^{2}\leq C(1+at)e^{(2a\kappa-1)t}\leq
Ce^{-\alpha t}.$ (31)

Finally, plugging (30) and (31) as well as (21) from Lemma 2.4 into (27)
yields

$d|\tilde{m}_{t}|^{2}\leq\left(-2(1-\Lambda\beta_{0})|\tilde{m}_{t}|^{2}+Ce^{-\alpha
t}\right)dt+2\tilde{m}_{t}\cdot dM^{v}_{t}.$ (32)

4.) With (32) at hand, we can now investigate the asymptotics of
$\tilde{m}_{t}$. Setting

$\xi:=2(1-\Lambda\beta_{0})$

and plugging (32) into

$d\left(e^{\xi t}|\tilde{m}_{t}|^{2}\right)=\xi e^{\xi
t}|\tilde{m}_{t}|^{2}dt+e^{\xi t}d|\tilde{m}_{t}|^{2},$

we get

$\displaystyle\begin{split}|\tilde{m}_{t}|^{2}&\leq
e^{-\xi(t-t_{0})}|\tilde{m}_{t_{0}}|^{2}+C\int_{t_{0}}^{t}e^{-\xi(t-r)-\alpha
r}dr+2\int_{t_{0}}^{t}e^{-\xi(t-r)}\tilde{m}_{r}\cdot dM^{v}_{r}\\\ &\leq
Ce^{-\alpha t}+2e^{-\xi t}N_{t},\end{split}$ (33)

where

$N_{t}=\int_{t_{0}}^{t}e^{\xi r}\tilde{m}_{r}\cdot dM^{v}_{r}.$

By (15) and (31), we have

$\langle N\rangle_{t}\leq\int_{t_{0}}^{t}e^{2\xi
r}\left|\tilde{m}_{r}\right|^{2}e^{r}\left|\nabla
F_{e^{r}}(X_{e^{r}})\right|^{2}dr\leq
C\int_{t_{0}}^{t}e^{(2\xi-\alpha)r}\left|\tilde{m}_{r}\right|^{2}dr.$ (34)

Since $\tilde{m}$ is bounded and $\alpha<2\xi$, (34) yields

$\langle N\rangle_{t}\leq Ce^{(2\xi-\alpha)t},$ (35)

so the law of the iterated logarithm implies

$\limsup_{t\to\infty}\frac{N_{t}}{e^{\left(\xi-\frac{\alpha}{2}\right)t}\log(t)}<\infty\quad\text{almost
surely.}$ (36)

Combining (33) and (36) yields

$e^{\delta t}\tilde{m}_{t}\xrightarrow{t\to\infty}0\quad\text{almost surely
for all $\delta<\frac{\alpha}{4}$.}$ (37)

This can be plugged back into (34) in order to improve (35) to

$\langle N\rangle_{t}\leq Ke^{(2\xi-(\alpha+2\delta))t}$

for any $\delta<\frac{\alpha}{4}$, where $K=K(\delta)$ is an almost surely
finite random variable. Then we use the same argument as above to improve (37)
to

$e^{\delta t}\tilde{m}_{t}\xrightarrow{t\to\infty}0\quad\text{almost surely
for all $\delta<\frac{3\alpha}{8}$.}$

Iterating this argument shows that for any $n\in\mathbb{N}$ we have

$e^{\delta t}\tilde{m}_{t}\xrightarrow{t\to\infty}0\quad\text{almost surely
for all $\delta<\frac{(2^{n}-1)\alpha}{2^{n+1}}$.}$

Since $n$ can be arbitrarily large and $\alpha$ arbitrarily close to $2\eta$,
it follows that (23) holds, and thus the proof is completed. ∎

######  Remark 2.7.

Note that Lemma 2.6 can be interpreted as a statement on the polynomial decay
of the drift potential $\beta(t)V_{\mu_{t}}$ of (2). More precisely, for
$2a\kappa<\gamma$ it implies that almost surely

$\limsup_{t\to\infty}\frac{\log\left\|\beta(t)V_{\mu_{t}}\right\|_{\infty}}{\log
t}\leq-\eta.$

We can now prove the main result of this section.

######  Proposition 2.8.

If $2a\kappa<\gamma$, then almost surely

$\limsup_{t\to\infty}\frac{1}{t}\log\left\|\varepsilon^{2}_{t}\right\|\leq-\eta.$

###### Proof.

This follows from (19) and Lemma 2.6. ∎

###  2.3 Putting the pieces together

The unique solution to (9) with the initial value
$\nu_{0}\in\mathcal{M}(\mathbb{M})$ is given by

$\Phi(t,\nu_{0})=e^{-t}\nu_{0}+(1-e^{-t})\mathcal{U}\quad\text{for all $t\geq
0$,}$ (38)

and the mapping

$\Phi\colon[0,\infty)\times\mathcal{M}(\mathbb{M})\to\mathcal{M}(\mathbb{M}),\quad(t,\nu_{0})\mapsto\Phi(t,\nu_{0}),$

defines a semiflow on the normed space $\mathcal{M}(\mathbb{M})$ in the sense
of [1, p. 9].

######  Proposition 2.9.

If $2a\kappa<\gamma$, then almost surely $(\mu_{e^{t}})_{t\geq 0}$ is a
$(-\eta)$-pseudotrajectory of the semiflow $\Phi$, i.e. almost surely

$\limsup_{t\to\infty}\frac{1}{t}\log\left(\sup_{s\in[0,T]}\left\|\mu_{e^{t+s}}-\Phi(s,\mu_{e^{t}})\right\|\right)\leq-\eta\quad\text{for
all $T>0$}.$ (39)

###### Proof.

By (8) and (9) we have for all $f\in C(\mathbb{M})$

$(\mu_{e^{t+s}}-\Phi(s,\mu_{e^{t}}))(f)=-\int_{0}^{s}\left(\mu_{e^{t+r}}-\Phi(r,\mu_{e^{t}})\right)(f)dr+\int_{0}^{s}\left(\varepsilon^{1}_{t+r}+\varepsilon^{2}_{t+r}\right)(f)dr,$

so

$(\mu_{e^{t+s}}-\Phi(s,\mu_{e^{t}}))(f)=\int_{t}^{t+s}e^{-(t+s-r)}\left(\varepsilon^{1}_{r}+\varepsilon^{2}_{r}\right)(f)dr.$
(40)

Since integration by parts yields

$\int_{t}^{t+s}e^{-(t+s-r)}\varepsilon^{1}_{r}(f)dr=\int_{t}^{t+s}\varepsilon^{1}_{r}(f)dr-\int_{t}^{t+s}e^{-(t+s-r)}\left(\int_{t}^{r}\varepsilon^{1}_{u}(f)du\right)dr,$

we get from (40) that

$\left\|\mu_{e^{t+s}}-\Phi(s,\mu_{e^{t}})\right\|\leq\left\|\int_{t}^{t+s}\varepsilon^{1}_{r}dr\right\|+\int_{t}^{t+s}e^{-(t+s-r)}\left\|\int_{t}^{r}\varepsilon^{1}_{u}du\right\|dr+\int_{t}^{t+s}\left\|\varepsilon^{2}_{r}\right\|dr.$

The claim now follows from Propositions 2.2 and 2.8. ∎

Intuitively, Proposition 2.9 says that $\mu_{e^{t}}$ is exponentially close to
the behavior of a solution of (9). Since any solution of (9) converges
exponentially fast to $\mathcal{U}$, Theorem 1.4 now follows from a general
result about pseudotrajectories.

###### Proof of Theorem 1.4.

By (38),

$\limsup_{t\to\infty}\frac{1}{t}\log\left\|\Phi(t,\nu_{0})-\mathcal{U}\right\|=-1\quad\text{for
all $\nu_{0}\in\mathcal{M}(\mathbb{M})$.}$

Thanks to this and Proposition 2.9, all of the conditions of part (i) of [1,
Lemma 8.7] (with $B=\mathcal{M}(\mathbb{M})$, $K=\\{\mathcal{U}\\}$,
$X(t)=\mu_{e^{t}}$, $Y(t)=\mathcal{U}$, $\alpha=-\eta>-1=\lambda$) are
fulfilled, and hence we get

$\limsup_{t\to\infty}\frac{1}{t}\log\left\|\mu_{e^{t}}-\mathcal{U}\right\|\leq-\eta\quad\text{almost
surely,}$

which is equivalent to (7). ∎

######  Remark 2.10.

Note that the final step of the proof of Proposition 2.9 also works with the
supremum in (39) being taken over all $s\geq 0$ instead of only $s\in[0,T]$.
However, this stronger variant of the pseudotrajectory property is not needed
for the argument in the proof of Theorem 1.4 to work.

##  3 A closer investigation of the case $\mathbb{M}=\mathbb{S}^{n}$ and
$v(x)=x$

For this entire section, let

$\mathbb{M}=\mathbb{S}^{n}\quad\text{and}\quad v(x)=x.$

In the case of weak self-interaction, i.e. $\beta\equiv b$, Theorem 4.5 of [4]
then implies that444Note that our notation differs slightly from that in [4]:
the parameter $a$ there corresponds to what is $b/2$ in our notation, and
there is another factor 2 in [4, (11)] that is not in our definition of $\Pi$
(compare Remark 1.7).

$\mu_{t}\xrightarrow{w}\mathcal{U}\quad\text{almost
surely}\quad\Leftrightarrow\quad b\geq-(n+1),$ (41)

while Theorem 1.4 implies

$b>-\frac{1}{\Lambda}\quad\Rightarrow\quad\left\|\mu_{t}-\mathcal{U}\right\|\to
0\quad\text{almost surely with polynomial speed}$ (42)

(compare Example 1.5). Because of (41) and (42), we already know that
$\Lambda\geq(n+1)^{-1}$. In this section, we provide a way to calculate
$\Lambda$, show numerically that it is in fact strictly larger than
$(n+1)^{-1}$, but then prove that the conclusion of (42) nevertheless actually
holds for all $b>-(n+1)$.

###  3.1 Calculating $\Lambda$

In the following, for any $m\in\mathbb{R}^{n+1}$ we use the notation

$\bar{m}=\begin{cases}\frac{m}{|m|},&\text{if $m\neq 0$,}\\\ 0,&\text{if
$m=0$}.\end{cases}$

######  Lemma 3.1.

Let

$\displaystyle\varrho(r)=-\frac{\int_{0}^{\pi}\cos xe^{-r\cos x}(\sin
x)^{n-1}dx}{\int_{0}^{\pi}e^{-r\cos x}(\sin x)^{n-1}dx}\quad\text{for all
$r\geq 0$}.$

  1. 1.

For all $m\in\mathbb{R}^{n+1}$ we have

$\Pi(m)(v)=-\varrho(|m|)\bar{m}.$ (43)

  2. 2.

We have

$\lim_{r\to 0}\frac{\varrho(r)}{r}=\frac{1}{n+1},$

and we therefore interpret $\frac{\varrho(0)}{0}$ as $\frac{1}{n+1}$ in the
following.

  3. 3.

For all $m\in\mathbb{R}^{n+1}$ we have

$\operatorname{Cov}_{\Pi(m)}(v)=\left(\varrho^{\prime}(|m|)-\frac{\varrho(|m|)}{|m|}\right)\bar{m}\bar{m}^{T}+\frac{\varrho(|m|)}{|m|}1_{(n+1)\times(n+1)}$

and its largest eigenvalue is given by $\lambda(|m|)$, where

$\lambda(r)=\frac{2\varrho(r)}{r}-\varrho^{\prime}(r)\quad\text{for all $r\geq
0$}.$ (44)

  4. 4.

We have

$\Lambda=\max_{r\geq 0}\lambda(r)\in\left[\frac{1}{n+1},\frac{2}{n+1}\right).$
(45)

Figure 1: Plots of $\varrho$ and $\lambda$ for $n\in\\{1,2,3,4\\}$.

###### Proof.

The first part of this lemma is just a reformulation of [4, Lemma 4.7]. Note
that for all $r\geq 0$

$\displaystyle\varrho(r)=\frac{H^{\prime}(r)}{H(r)}\quad\text{with}\quad
H(r)=\int_{0}^{\pi}e^{-r\cos x}(\sin x)^{n-1}dx$

and hence

$\varrho^{\prime}(r)=\frac{H^{\prime\prime}(r)}{H(r)}-\left(\frac{H^{\prime}(r)}{H(r)}\right)^{2}>0$
(46)

where the inequality follows from Cauchy-Schwarz. As shown in the proof of [4,
Lemma 4.8],

$\frac{d}{dr}\frac{H^{\prime\prime}(r)}{H(r)}>0,\quad\frac{H^{\prime\prime}(0)}{H(0)}=\frac{1}{n+1},$

and since an integration by parts yields

$H^{\prime}(r)=\frac{r}{n}(H(r)-H^{\prime\prime}(r))\quad\text{for all $r\geq
0$,}$ (47)

we get

$\frac{d}{dr}\frac{\varrho(r)}{r}=\frac{d}{dr}\left(\frac{1}{n}\left(1-\frac{H^{\prime\prime}(r)}{H(r)}\right)\right)<0,\quad\frac{\varrho(r)}{r}\xrightarrow{r\to
0}\frac{1}{n+1}.$ (48)

In particular, we have

$0>\frac{d}{dr}\frac{\varrho(r)}{r}=\frac{\varrho^{\prime}(r)}{r}-\frac{\varrho(r)}{r^{2}}\quad\text{for
all $r>0$}$

so, together with (46),

$0<\varrho^{\prime}(r)<\frac{\varrho(r)}{r}\quad\text{for all $r>0$}.$ (49)

Combining (43) with the second part of Lemma 2.3, we get

$\displaystyle\operatorname{Cov}_{\Pi(m)}(v)$
$\displaystyle=\left(\partial_{m_{i}}\left(\frac{\varrho(|m|)}{|m|}m_{j}\right)\right)_{i,j=1,\ldots,n+1}$
$\displaystyle=\left(\varrho^{\prime}(|m|)-\frac{\varrho(|m|)}{|m|}\right)\bar{m}\bar{m}^{T}+\frac{\varrho(|m|)}{|m|}1_{(n+1)\times(n+1)}$

and hence its largest eigenvalue is

$\displaystyle\sup_{x\in\mathbb{S}^{n}}x^{T}\operatorname{Cov}_{\Pi(m)}(v)x=\left|\varrho^{\prime}(|m|)-\frac{\varrho(|m|)}{|m|}\right|+\frac{\varrho(|m|)}{|m|}=\lambda(|m|),$

where the last equality uses (49). Plugging (49) into (44) yields

$\frac{\varrho(r)}{r}<\lambda(r)<\frac{2\varrho(r)}{r}\quad\text{for all
$r>0$,}$

which, in combination with (48), implies (45). ∎

######  Remark 3.2.

With the help of Lemma 3.1, one can easily show that if
$(m_{k})_{k\in\mathbb{N}}\subset\mathbb{R}^{n+1}$ satisfies $|m_{k}|\to\infty$
and $\bar{m}_{k}\to m\in\mathbb{S}^{n}$ for $k\to\infty$, then
$\Pi(m_{k})\xrightarrow{w}\delta_{-m}$ for $k\to\infty$.

$n$ | $\Lambda=\max\lambda$ | $\Lambda\cdot(n+1)$ | $\operatorname{argmax}\lambda$
---|---|---|---
1 | 0.548 | 1.096 | 1.442
2 | 0.363 | 1.090 | 1.930
3 | 0.272 | 1.087 | 2.405
4 | 0.217 | 1.084 | 2.876
5 | 0.180 | 1.083 | 3.345
6 | 0.154 | 1.081 | 3.812
7 | 0.135 | 1.080 | 4.278
8 | 0.120 | 1.079 | 4.744
9 | 0.108 | 1.079 | 5.210
10 | 0.098 | 1.078 | 5.676
20 | 0.051 | 1.075 | 10.329
50 | 0.021 | 1.073 | 24.288
100 | 0.011 | 1.073 | 47.552
Table 2: Numerical approximations for $\Lambda$, $\Lambda\cdot(n+1)$, and
$\operatorname{argmax}\lambda$.

######  Remark 3.3.

It follows from (46) and (47) that $\varrho$ satisfies the differential
equation

$\varrho^{\prime}(r)=1-\varrho(r)\left(\frac{n}{r}+\varrho(r)\right),$

so thanks to (44), we can express both $\lambda(r)$ and $\lambda^{\prime}(r)$
as functions of $r$ and $\varrho(r)$. This makes it easy to calculate
$\lambda$ and $\lambda^{\prime}$ and thus approximate the maximum of $\lambda$
numerically. Simulations suggest that $\lambda(r)$ attains $\Lambda$ as its
unique local and global maximum in a position $r=\operatorname{argmax}\lambda$
that grows linearly in $n$. Furthermore, these simulations suggest that
$\Lambda\cdot(n+1)$ is indeed strictly greater than 1, even though it is
decreasing in $n$ and the upper bound $2$ from (45) is far from optimal. Note
however, that even (45) is a vast improvement over the trivial bound from (6).
See Table 2 for some approximate values and Figure 1 for a visualisation of
$\varrho$ and $\lambda$.

###  3.2 Improving Theorem 1.4 in the case of weak self-attraction

The following proposition means that we can improve [4, Theorem 4.5 (i)] for
all cases in which the uniform distribution is the limit, with the exception
of the critical value $b=-(n+1)$.

######  Proposition 3.4.

Let $\beta\equiv b>-(n+1)$. Then

$\limsup_{t\to\infty}\frac{\log\left\|\mu_{t}-\mathcal{U}\right\|}{\log
t}\leq-\min\left\\{\frac{1}{2},1+\frac{b}{n+1}\right\\}\quad\text{almost
surely.}$

###### Proof.

The case $b\geq 0$ is already covered entirely in Example 1.5, so let

$0>b>-(n+1).$

Since Proposition 2.2 implies

$\limsup_{t\to\infty}\frac{1}{t}\log\left(\sup_{s\geq
0}\left\|\int_{t}^{t+s}\varepsilon^{1}_{r}dr\right\|\right)\leq-\frac{1}{2}\quad\text{almost
surely},$

it only remains to show that

$\limsup_{t\to\infty}\frac{1}{t}\log\left\|\varepsilon^{2}_{t}\right\|\leq-\min\left\\{\frac{1}{2},1-\frac{|b|}{n+1}\right\\}\quad\text{almost
surely},$ (50)

since then the proof can be completed in the exact same way as in Section 2.3.
In order to prove (50), we will need to revisit and refine the arguments from
Lemmas 2.4 and 2.6 (the notation of which we adapt in the sequel) and make
explicit use of [4, Theorem 4.5 (i)].

Since the second and third parts of Lemma 3.1 imply

$\lim_{r\to 0}\lambda(r)=\lambda(0)=\frac{1}{n+1}<|b|^{-1},$

we can choose $\vartheta>0$ such that

$\zeta:=\sup_{m\in\mathbb{R}^{n+1},|m|\leq\vartheta}\left(\sup_{x\in\mathbb{S}^{n}}x^{T}\operatorname{Cov}_{\Pi(bm)}(v)x\right)=\sup_{r\in[0,\vartheta|b|]}\lambda(r)<|b|^{-1}.$

Then with the same argument as in Lemma 2.4 we get

$m\cdot\nabla J_{t}(m)\geq(1-|b|\zeta)\left|m\right|^{2}\quad\text{for all
$m\in\mathbb{R}^{n+1}$ with $|m|\leq\vartheta$.}$ (51)

We already know from [4, Theorem 4.5 (i)] that almost surely
$\mu_{t}\xrightarrow{w}\mathcal{U}$ (compare (41)), so $m_{t}\to 0$ and hence
also $\tilde{m}_{t}\to 0$ (compare (22) and Lemma 2.1). Therefore, there is an
almost surely finite random time $\tau=\tau(\vartheta)$ such that
$|\tilde{m}_{t}|\leq\vartheta$ for all $t>\tau$, so with (51) we get

$-\tilde{m}_{t}\cdot\nabla
J_{t}(\tilde{m}_{t})\leq-(1-|b|\zeta)\left|\tilde{m}_{t}\right|^{2}\quad\text{for
all $t\geq\tau$.}$ (52)

Also note that

$-\tilde{m}_{t}\cdot\nabla J_{t}(\tilde{m}_{t})\leq C\quad\text{for all
$t\in[t_{0},\tau)$.}$ (53)

With

$0<\alpha<\min\left\\{1,2-2|b|\zeta\right\\},$

we can show as in the proof of Lemma 2.6 that

$d|\tilde{m}_{t}|^{2}\leq\left(-2\tilde{m}_{t}\cdot\nabla
J_{t}(\tilde{m}_{t})+Ce^{-\alpha t}\right)dt+2\tilde{m}_{t}\cdot
dM^{v}_{t}\quad\text{for all $t\geq t_{0}$.}$ (54)

If we set $\xi:=2-2|b|\zeta$, plug (54) into

$d\left(e^{\xi t}|\tilde{m}_{t}|^{2}\right)=\xi e^{\xi
t}|\tilde{m}_{t}|^{2}dt+e^{\xi t}d|\tilde{m}_{t}|^{2},$

and then apply (52) and (53), we can estimate

$\displaystyle\begin{split}|\tilde{m}_{t}|^{2}&\leq
e^{-\xi(t-t_{0})}|\tilde{m}_{t_{0}}|^{2}+C\int_{t_{0}}^{t}e^{-\xi(t-r)-\alpha
r}dr+C\int_{t_{0}}^{\tau}e^{-\xi(t-r)}dr+2\int_{t_{0}}^{t}e^{-\xi(t-r)}\tilde{m}_{r}\cdot
dM^{v}_{r}\\\ &\leq Ke^{-\alpha t}+2e^{-\xi t}\int_{t_{0}}^{t}e^{\xi
r}\tilde{m}_{r}\cdot dM^{v}_{r},\end{split}$

where $K$ is a positive, almost surely finite random number. Now, (50) follows
from the same line of reasoning as in the last step of the proof of Lemma 2.6
and in the proof of Proposition 2.8. ∎

######  Remark 3.5.

The general idea behind the proof of Proposition 3.4 is not restricted to the
specific case treated in this section: there is potential for the Assumption
1.3 on the lower bound for $\beta$ to be weakened, whenever we already know
that $\mu_{t}\xrightarrow{w}\mathcal{U}$ and thus only need to bound the
eigenvalues of $\beta(e^{t})\operatorname{Cov}_{\mathcal{U}}(v)$ in order to
make the arguments in the proofs of Lemmas 2.4 and 2.6 work.

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