# Phase transitions of McKean-Vlasov SDEs in Multi-well Landscapes

Alexander Alecio

###### Abstract.

Phase transitions and critical behaviour of a class of MV-SDEs, whose
concomitant non-local Fokker-Planck equation includes the Granular Media
equation with quadratic interaction potential as a special case, is studied.
By careful analysis of an implicit auxiliary integral equation, it is shown
for a wide class of potentials that below a certain ‘critical threshold’ there
are exactly as many stationary measures as extrema of the potential, while
above another the stationary measure is unique, and consequently phase
transition(s) between. For symmetric bistable potentials, these critical
thresholds are proven to be equal and a strictly increasing function of the
aggregation parameter. Additionally, a simple condition is provided for
symmetric multi-well potentials with an arbitrary number of extrema to
demonstrate analogous behaviour. This answers, with considerably more
generality, a conjecture of Tugaut [Stochastics, 86:2, 257-284].

To the best of our knowledge many of these results are novel. Others simplify
the proofs of known results whilst greatly increasing their applicability.

###### Key words and phrases:

McKean Vlasov diffusions; phase transitions; invariant probabilities;
interacting particles systems

###### 1991 Mathematics Subject Classification:

60H10, 60G10, 60J60, 82C22, 35Q82, 35Q83

The general form of McKean-Vlasov SDEs (MV-SDEs) [16] in one dimension is

$dX_{t}=b(X_{t},\mu_{t})dt+\sigma(X_{t},\mu_{t})dW_{t},\qquad\mu_{t}=\text{Law}(X_{t}),\,t>0$

In this work we focus on one-dimensional MV-SDEs with separable drifts that
depend on $\mu_{t}$ via an integral functional,
$b_{\theta}(x,\mu)=f_{1}(x)+\theta\mathbb{E}_{\mu}(f_{2}(x))$ and diffusion
$\sigma(x)=\sigma k(x)$, where aggregation and diffusion strength parameters
($\theta$, $\sigma$ respectively) are strictly positive constants. Concretely,
this is

(1)
$dX_{t}=\big{(}-V^{{}^{\prime}}(X_{t})-\theta(P^{{}^{\prime}}(X_{t})-\mathbb{E}_{\rho_{t}}[P^{{}^{\prime}}(x)]\big{)}dt+\sigma
k(X_{t})dB_{t},\qquad\rho_{t}=\mathrm{Law}(X_{t}),\,t>0$

where $k>\epsilon$. MV-SDEs of this form have found numerous applications, of
which we mention systemic risk [11] and global optimisation [17].

The concomitant Fokker-Planck Equation of (1) is of non-linear, non-local
type,

(2) $\frac{\partial}{\partial t}\rho=\frac{\partial}{\partial
x}\Big{(}(-V^{{}^{\prime}}(x)-\theta(P^{{}^{\prime}}(x)-\mathbb{E}_{\rho}[P^{{}^{\prime}}(x)]))\rho+\frac{\sigma}{2}\frac{\partial}{\partial
x}k^{2}(x)\rho\Big{)}$

The particular form of the drift with respect to aggregation parameter is such
that, when $P^{{}^{\prime}}=x$ and $k=1$, the granular media equation is
recovered

(3) $\frac{\partial}{\partial t}\rho=\frac{\partial}{\partial
x}(\rho\frac{\partial}{\partial x}(\log(\rho)+V+F*\rho))$

for $F=\frac{1}{2}x^{2}$, [6]. Although unused in this work, it noteworthy
that (3) is a gradient flow, with respect to the Wasserstein metric, of the
free energy functional

(4) $\mathcal{F}[\rho]=\sigma^{2}\int\rho\ln\rho\,dx+\int V\rho\,dx+\iint
F(x-y)\rho(x)\rho(y)\,dxdy$

for constant $k$; [22, 23].

Customarily, solutions of MV-SDE (1) are derived as the hydrodynamic limit of
a system of $N$-SDEs driven by independent Wiener processes, with the same
diffusion and drift as (1) but with their empirical measure replacing $\mu$:

$dX^{i,N}_{t}=b(X^{i,N}_{t},\tilde{\mu}_{t})dt+\sigma(X^{i,N}_{t},\tilde{\mu}_{t})dW_{t}^{i},\qquad\tilde{\mu}_{t}=\sum_{j}^{N}\delta_{X_{t}^{j,N}},\,t>0$

This is the ‘Propogation of Chaos’, which is described by a number of papers
with differing conditions, for instance [15, 21].

Existence and uniqueness of solutions to the Fokker-Planck equation (2) is
more straightforward to demonstrate, where positivity of solutions yield a
priori estimates on functional $\mathbb{E}(P^{{}^{\prime}})$. At stationarity,
direct integration of (2) combined with no-flux boundary conditions yields an
equivalent integral formulation, provided
$V^{{}^{\prime}},\,P^{{}^{\prime}},\,k\in C(\mathbb{R})$, entirely analogously
with the linear stationary Fokker-Planck equation [20]

In turn, this can be solved for the general form of the stationary measure

(5)
$\rho(\sigma,x,m)=\exp\Big{(}-\frac{2}{\sigma^{2}}\int^{x}\frac{V^{{}^{\prime}}+\sigma^{2}kk^{{}^{\prime}}+\theta(P^{{}^{\prime}}(x)-m)}{k^{2}}\Big{)}$

(assuming such $V^{{}^{\prime}},\,P^{{}^{\prime}},\,k^{{}^{\prime}}$ such that
$\rho\in L^{1}(\mathbb{R})$) with the important caveat that $\rho(m)$ is an
admissible stationary measure if and only if $m$ is a solution to the
auxiliary equation $m=\int-V^{{}^{\prime}}\rho(m)dx$, better known as the
self-consistency function.

There are well known examples of MV-SDE (1) ([7, 19] for instance), whose
self-consistency equation can have a single or multiple solutions depending on
parameters choice, thus multiple stationary measures and a richness of long
time behaviour. Casting $\sigma$ as the control parameter, admissible
stationary solutions can be viewed as phases, whose characteristic property
$\mathbb{E}_{\rho_{0}(m)}(P^{{}^{\prime}}(x))$ (solutions of the self-
consistency function) plays the rôle of order parameter. Bifurcations of the
order parameter as a function of the control parameter are then continuous
phase transitions. For a comprehensive tutorial introduction, with discussion
of compatiblity with Landau theory and connexions to self-organisation and
synergetics, consult [10].

The purpose of this work is to better understand this critical behaviour
which, given much of the attraction of MV-SDE models is in this critical
behaviour and multitude of long time dynamics, would seem a timely
contribution. Particularly, we study the number of stationary solutions and
their location, along with phase transition points (or critical transitions)
and their dependence on aggregation parameter.

###  0.1. Set-up

While it is clear that these questions can be approached via a study of the
self-consistency equation, our results take advantage of a useful equivalence
to the first moment equation, which becomes the central object of study in
this paper.

The genesis of this approach arose from numerical studies simulating SDE (1)
without a free energy functional (4). Earlier simulation approaches would
expand $P^{{}^{\prime}},\,V^{{}^{\prime}}$ as a truncated power series
$a^{(n)},\,b^{(n)}$ and recast the SDE as a denumerable system of ODEs,

$\dot{m}_{i}=\mathbb{E}(dX^{i}_{t})=i\mathbb{E}[a^{(n)}(X_{t})X_{t}^{i-1}dt]+\sigma\frac{i(i-1)}{2}\mathbb{E}[b^{(n)}(X_{t})X_{t}^{i-2}dt]$

allied with a method of closure; [3, 4].

Even the simplest of these methods were surprisingly effective in representing
critical behaviour with only a few retained moments. This was surprising, as
it is not immediate in what manner the self-consistency equation manifested in
the moment evolution equations. That the number of retained moments could be
taken so low suggested it was the lowest order moment equation that
encapsulated the self-consistency equation.

This motivated further work, eventually showing the first moment evolution
equation and self-consistency equation were equivalent. This has interesting
ramifications for the applicability of moment truncation schemes to SDE (1)
and, by extension cumulant truncations; [2]

In order to proceed, we formalise these remarks, and those made in the
introduction, and introduce definitions needed throughout the rest of this
work.

######  Definition 0.1.

(Self-Consistency Function) The self-consistency function is

(6) $G_{(\sigma,\theta)}(m)=\int(P^{{}^{\prime}}(x)-m)\rho(\sigma,x,m)dx$

Solutions of the equation $F_{\sigma}(m)=0$ correspond to admissible
stationary measures of (1)

The following term has become established in related literature, highlighting
parallels between (7) and a stationary measure of a Smoluchowski SDE:

######  Definition 0.2.

(The Effective Potential)
$\bar{V}_{\theta}(x,m)=\int\frac{V^{{}^{\prime}}+\theta(P^{{}^{\prime}}-m)}{k^{2}}$,
$\theta\in\mathbb{R}^{+}$

We will also refer to the potential $V$, although in this work we actually
take its derivative $V^{{}^{\prime}}$ rather than the potential as the
fundamental object of study. Assumptions on $V$ will be given in the relevant
sections.

It is useful, in Section 1 in particular, to distinguish the exponential
‘Gibbs’ measure’ part of the stationary measure

(7)
$\rho(\sigma,x,m)=\frac{1}{k^{2}}\exp\Big{(}-\frac{2}{\sigma^{2}}\bar{V}_{\theta}(x,m)\Big{)}:=\frac{1}{k^{2}}\rho_{0}(\sigma,x,m)$

######  Proposition 0.3.

The self-consistency equation is equivalent to to the first moment evolution
equation

(8)
$F_{(\sigma,\theta)}(m)=\frac{1}{\theta}\int-\frac{V^{{}^{\prime}}}{k^{2}}\rho_{0}(\sigma,x,m)dx$

###### Proof.

Working from Definition 6

(9) $\displaystyle G_{\sigma}(m)=\int(P^{{}^{\prime}}(x)-m)\rho(\sigma,x,m)$
$\displaystyle
dx=\int(P^{{}^{\prime}}(x)-m)\frac{1}{k^{2}}\exp(\bar{V}_{\theta}(x,m))dx$
(10)
$\displaystyle=\int\partial_{x}\big{(}\exp(-\frac{2}{\sigma^{2}}\int\frac{P^{{}^{\prime}}-m}{k^{2}})\big{)}k^{2}\exp(-\frac{2}{\sigma^{2}}\int\frac{V^{{}^{\prime}}}{k^{2}})dx$

Assuming sufficient regularity to ignore the boundary terms, the result
follows from an integration by parts. ∎

Working with the first moment evolution equation leads to robust results with
intuitive proofs without much technical obfuscation. Further they can yield
quantitative estimates of parameters if required. Motivated by applications,
we refrain from overly prescriptive assumptions on $V^{{}^{\prime}}$ to
maintain flexibility.

Throughout this work $J$ always denotes a connected compact set. Conditions
introduced in each section will always ensure $\rho$ is normalisable and
$F_{\sigma}$ exists for all $m,\sigma,\theta$ in their respective domains. We
take $\rho$ normalised in Section 1 and unnormalised in Section 2. We will
frequently suppress $\theta$ dependence when context allows.

In terms of $F_{\sigma}$, a critical transition is any value of $\sigma$ where
the number of roots changes. Assuming they exist, the upper (lower) critical
transition is the largest (smallest) such value.

###  0.2. Outline and relation to other works

This paper naturally divides into two halves, the first deals in problems with
fairly general conditions, while the second involves results that require some
symmetry.

The main results are:

  * •

For suitably smooth multi-well potentials with unimodal $\rho$ (specifically
that $\bar{V}^{{}^{\prime}}$ is a diffeomorphism, subsequently weakened to a
homeomorphism).

    * –

Below the lower critical threshold, $\sigma<\sigma_{c}^{l}$, there are exactly
as many stationary measures as simple roots of $\bar{V}^{{}^{\prime}}$ \-
Proposition 1.2 and Corollary 1.3

    * –

Above the upper critical threshold, $\sigma_{c}^{u}<\sigma$, there is only one
stationary measure - Proposition 1.8

    * –

Multimodal $\rho$ is considered as a special case and a counter-example given
where the number of stationary measures is less than the number of roots of
$V^{{}^{\prime}}$ \- Remark 1.6

  * •

For antisymmetric $V^{{}^{\prime}},\,P^{{}^{\prime}},k$, but otherwise looser
assumptions,

    * –

With a symmetric bistable potential $\sigma_{c}^{l}=\sigma_{c}^{u}$ and
$\sigma_{c}^{u}(\theta)$ is an increasing function - Proposition 2.2 and 2.9.

    * –

We define a class of symmetric multi-well potentials that behave similarly to
the bistable potential, particularly that $\sigma_{c}^{u}(\theta)$ remains an
increasing function - Proposition 2.16 and Corollary 2.17

There has been much sustained activity with regard to the convergence to
stationary measures of MV-SDEs, [5, 9]. Recently, there has been a growing
literature based on modern variational methods for the granular media equation
such as [24], the progenitor of which appears to be [19], which variationally
studies convergence for the symmetric polynomial bistable problem.

Exact studies of the properties of these stationary solutions has been
scarcer. For the symmetric polynomial bistable problem, a result substantially
like Proposition 2.2 is given, through a study of the self-consistency
equation by [19]. The proof relies heavily on the the GHS inequality, which is
resistant to generalisation. For a similar, entirely non-variational study,
see [7]

In a comprehensive series of papers, author Tugaut has studied many problems
related to those considered here. The closest in scope and applicability to
this paper is [25], which employs a self-consistency equation centered method
to study polynomial bistable potentials and positive interactions of the form
$x^{2n}*\rho,\,n\geq 2$. The first half of [25] is dedicated to quadratic
interactions, corresponding to the Granular Media equation with interaction
kernel $F=\frac{x^{2}}{2}$ which, as already discussed, our interactions
incorporate. In this overlap our results strengthen his with more general
assumptions, in the process simplifying the proofs. For instance, Proposition
1.2/ Corollary 1.3 of [25] demonstrates that, for sufficiently small $\sigma$,
there are at least as many stationary measures as roots $V^{{}^{\prime}}$
which is improved to equality by our Proposition 1.2. Theorem 2.1 of [25]
establishes there are exactly three solutions for bistable potentials of the
form $\sum^{N}_{j=2}|a_{j}|x^{2j}-x^{2}$. Our Proposition 2.2 simplifies the
argumentation and extends the result to a large class of bistable potential.
Further, for such polynomial bistable potentials, on p.270 it is conjectured
$\sigma^{u}_{c}(\theta)$ is increasing, which we answer in Proposition 2.9 and
extend to symmetric multi-well potentials in Proposition 2.16.

##  1\. Generic Multi-well Potentials

This section is devoted to MV-SDE (1) with a large class of polynomial bounded
multi-well potentials and strictly convex (unimodal)effective potentials,
subsequently loosened to convex. The non-convex (or multimodal) problem is
considered as a series of convex ones.

####  1.0.1. Definitions and Assumptions

We introduce the following definitions for this section. First is the inverse
of a function that maps $m$ to the mode of (7)

######  Definition 1.1.

(Modal dependence) $x^{*-1}=\frac{1}{\theta}(V^{{}^{\prime}}+\theta
P^{{}^{\prime}})$.

The second is the convenient short-hand $a=\int_{0}^{x}\frac{1}{k^{2}}$.

Next, we introduce a series of conditions for the results of this section.
They are phrased in terms of $V^{{}^{\prime}}$ and $\bar{V}$, which was seen
to be their most natural form given the proof of Proposition 1.2 and others.

  1. (1)

$V^{{}^{\prime}}(x,0),\,P^{{}^{\prime}}\in C^{2}(\mathbb{R})$, $k\in
C^{1}(\mathbb{R})$.

  2. (2)

$\lVert
a^{{}^{\prime}}\rVert_{\infty}=\lVert\frac{1}{k^{2}}\rVert_{\infty}<\infty$

  3. (3)

$\lim\limits_{|x|\rightarrow\infty}\frac{\bar{V}(x,0)}{x^{2}}>0$

  4. (4)

$\lim\limits_{x\rightarrow\pm\infty}\bar{V}^{{}^{\prime}}(x,0)=\lim\limits_{x\rightarrow\pm\infty}V^{{}^{\prime}}=\pm\infty$

  5. (5)

$|V^{{}^{\prime}}|<K(1+x^{2N})$

  6. (6)

$(x^{*-1})^{{}^{\prime}}>0$

Assumptions 1, 3, 4 & 5 provide useful bounds ensuring, amongst other things
that $F_{\sigma}(m)$ exists for all $m$, there is than sufficient regularity
for the integration by parts and the boundary terms are null. Assumption 4
ensures $F_{\sigma}(m)$ is bounded away from zero far away from the origin. It
is possible to weaken the last condition, which we consider as special cases
of the above. See Corollary 1.3 and the Remark 1.6.

######  Proposition 1.2 (Convergence of $F_{\sigma}(m)$ (6)).

Under Assumptions (1)-(6), $x^{*}$ exists and, as $\sigma\downarrow 0$,
$F_{\sigma}(m)$ converges to $-\frac{V^{{}^{\prime}}}{k^{2}}\circ x^{*}(m)$
uniformly on compact sets.

Moreover, let $J$ denote some connected compact set containing $N<\infty$
zeros of $V^{{}^{\prime}}$ strictly in its interior. If all these zeros are
simple, then there exists a critical threshold $\sigma_{c}$ such that for
$\sigma<\sigma_{c}$, $F_{\sigma}(m)|_{x^{*-1}(J)}$ has precisely $N$ zeros.

###### Proof.

That $x^{*}(m)$ exists and its range is $\mathbb{R}$ is an immediate
consequence of Condition 4 and 6.

By Laplace’s Method,

(11) $\lim\limits_{\sigma\downarrow
0}\int_{\mathbb{R}}f(x)\frac{\exp(-\frac{2}{\sigma^{2}}(\bar{V}+ma))}{\sqrt{\frac{\pi}{\bar{V}^{{}^{\prime\prime}}(x^{*})}}\exp(-\frac{2}{\sigma^{2}}\bar{V}(x^{*}))}=f(x^{*})$

where we have suppressed $m$ dependence in $x^{*}$.

On compact sets (of $m$), it follows from our assumptions that
$V^{{}^{\prime}},\,P^{{}^{\prime}},\,\lVert\frac{V^{{}^{\prime}}}{k^{2}}\exp(-(\int\frac{\bar{V}-\theta
ma}{k^{2}})\rVert_{1}$ are all bounded and
$\bar{V}^{{}^{\prime\prime}}(x^{*})=\frac{(x^{*-1})^{{}^{\prime}}}{k^{2}}(x^{*})>\epsilon>0$.
The standard proof of the Laplace method shows that convergence is uniform on
compact sets when $f=-\frac{V^{{}^{\prime}}(x)}{k^{2}}$111Strictly speaking, a
constant may need to be added to $-V^{{}^{\prime}}$ to ensure it is of one
sign on the chosen compact set before applying the Laplace method. This is
then subtracted off the limit to get the result or a constant.

As limit (11) with $f=1$ is bounded away from 0, the limit of the reciprocal
exists and must be uniformly convergent also. $\lim\limits_{\sigma\downarrow
0}F_{\sigma}(m)$ is the product of this reciprocal and (11) with
$-\frac{V^{{}^{\prime}}(x)}{k^{2}}$, it is also uniformly converging too with
the limit $-\frac{V^{{}^{\prime}}}{k^{2}}\circ x^{*}$, given the denominators
from (11) cancel each other.

By the uniform convergence of $F_{\sigma}$ and the intermediate value theorem,
there must be at least one root in a neighbourhood of $x_{0}$, a simple root
of $-V^{{}^{\prime}}$222To be precise, in the neighbourhood of $m_{0}$ where
$x^{*}(m_{0})=x_{0}$ which can be made arbitrarily small with $\sigma$. To
establish the second part of the Proposition, it’s sufficient to show
$\frac{\partial F_{\sigma}}{\partial m}$ has the sign of
$-V^{{}^{\prime\prime}}(x_{0})$ in a fixed interval centered on $x_{0}$, for
all $\sigma$ arbitrarily small. Given there must be at least one root in the
neighbourhood by the intermediate value theorem, and that $F_{\sigma}$ is
there strictly increasing (respectively decreasing), we can conclude there is
one root for sufficiently small $\sigma$.

Concretely, we will establish there is a some compact interval $A$ on which
$\frac{\partial F_{\sigma}}{\partial m}$, has the sign of
$-V^{{}^{\prime\prime}}(x_{0})$, for all $x^{*}\in A$. The crux of the method
is to identify and exploit the covariance structure of
$\frac{\partial}{\partial m}F_{\sigma}$:

$\frac{\partial}{\partial
m}F_{\sigma}=\frac{2}{\sigma^{2}}\Big{(}\int_{\mathbb{R}}-aV^{{}^{\prime}}\rho
dx-\int_{\mathbb{R}}a\rho dx\int_{\mathbb{R}}-V^{{}^{\prime}}\rho
dx\Big{)}=\frac{2}{\sigma^{2}}\mathrm{Cov}_{x^{*}}(a,-V^{{}^{\prime}})$

with $a=\int^{x}\frac{1}{k^{2}}dx$. Particularly useful is the alternate form
$\mathrm{Cov}_{m}(a,-V)$,

(12)
$\iint_{\mathbb{R}^{2}}\big{(}a(x)-a(y)\big{)}\big{(}-V^{{}^{\prime}}(x)-(-V^{{}^{\prime}}(y))\big{)}\frac{\rho_{0}(x,x^{*})}{k^{2}(x)}\frac{\rho_{0}(y,x^{*})}{k^{2}(y)}dxdy$

where we have used (7). Assuming $x,y$ are sufficiently close to $x_{0}$ that
$V^{{}^{\prime}}(x)$ is suitably well approximated by its Taylor expansion,
and recalling $a(x)$ is strictly increasing, it can be seen that the integrand
$(a(x)-a(y))(-V^{{}^{\prime}}(x)-(-V^{{}^{\prime}}(y)))$ is positive or
negative depending on the sign of $-V^{{}^{\prime\prime}}(x_{0})$, by taking
the cases $x<y$, $y<x$. By bounding the integrand, we will demonstrate
rigorously that $\rho$ weights this region sufficiently that the integral has
the same sign.

Without loss of generality, we assume $-V^{{}^{\prime\prime}}(x_{0})\geq 0$.
As before, we replace $\mathcal{Z}$ with

$\mathcal{\bar{Z}}=\sqrt{\pi}\sigma\exp(-\frac{2}{\sigma^{2}}\bar{V}(x^{*}))$

incorporating $\bar{V}(x^{*})$ from $\mathcal{\bar{Z}}$ into the exponent part
of $\rho$. We proceed by splitting the domain of integration into two parts,
$R_{1}(x^{*})=B_{R}{(x^{*},\,x^{*})}$ for $x^{*}$ in some set $A$, and its
complement $R_{2}$.

To define these regions, we consider the following factors. By Assumption 1
there exists an interval around root $x_{0}$,

$I=[x_{0}-2\delta,x_{0}+2\delta]$

on which $-V^{{}^{\prime\prime}}>-\underline{V}^{{}^{\prime\prime}}>0$.
Further on $I$ we can bound $a^{{}^{\prime}}=\frac{1}{k^{2}}$ below by
$\frac{1}{\underline{k}^{2}}>0$ by Assumption (1) and the Extremal Value
theorem applied to $k$. Then with the mean value theorem,

$(a(x)-a(y))(-V^{{}^{\prime}}(x)-(-V^{{}^{\prime}}(y)))\geq-\underline{V}^{{}^{\prime\prime}}\frac{1}{\underline{k}^{2}}(x-y)^{2}>0$

for $x,y\in I\times I$.

We bound $\bar{V}(x)-\bar{V}(x^{*})$ above by $\alpha(x-x^{*})^{2},\,\alpha>0$
for $[x,x^{*}]\subset I$ using the Taylor remainder theorem and Assumption 1.

Consequently we identify

$\begin{split}&A=[x_{0}-\delta,x_{0}+\delta]\\\
&R(x^{*})=\bar{B}_{\delta}(x^{*},x^{*}),\qquad R=\delta,\quad x^{*}\in
A\end{split}$

By construction, $R(x^{*})\subset I\times I$ when $x^{*}\in A$, so the
previously established bounds are still applicable.

We bound (12) below on $R_{1}$ by

$\frac{2}{\sigma^{2}}\iint_{R_{1}}\frac{-\underline{V}^{{}^{\prime\prime}}}{\underline{k}^{2}}\big{(}(x-x^{*})-(y-y^{*})\big{)}^{2}\frac{\exp\Big{(}-\frac{2\alpha}{\sigma^{2}}((x-x^{*})^{2}+(y-x^{*})^{2})\Big{)}}{\underline{k}^{4}2\pi\sigma^{2}}dxdy$

where we have dropped the dependency of $R$ on $x^{*}$, as our bounds are
uniform in $x^{*}$. Transforming to polar coordinates -
$r\cos(\theta)=(x-x^{*}),\,r\sin(\theta)=(y-x^{*})$ \- we have

$\frac{2}{2\pi\sigma^{4}}\int_{0}^{2\pi}(1-\sin(2\theta))d\theta\,\frac{-\underline{V}^{{}^{\prime\prime}}}{\underline{k}^{6}}\int_{R_{1}}r^{2+1}\exp(-\frac{2}{\sigma^{2}}\alpha
r^{2})dr\quad\propto\quad\frac{1}{\sigma^{4}}(\sigma^{4}-\exp(-\frac{2}{\sigma^{2}})(\dots))$

where (…) is polynomial in $\frac{1}{\sigma^{2}}$. Consequently the second,
negative, term can be made arbitrarily small.

On $R_{2}$, our argument is similar. We can bound the integrand
$|(V^{{}^{\prime}}(x)-V^{{}^{\prime}}(y))(a(x)-a(y))|<2K(2+x^{2N}+y^{2N})|x|$
by Assumption 2 and 5. Moreover, independently of bounded $x^{*}$, as
$\bar{V}-\bar{V}(x^{*})$ is (super-) quadratic outside some finite radius
(Assumption 3) and $x^{*}$ is the sole minimum, we can bound
$\bar{V}(x)-\bar{V}(x^{*})>\beta(x-x^{*})^{2},\,\beta>0,\,x^{*}\in A$.

Putting these bounds together and transferring to polar coordinates once
again, we have the following lower bound on $R_{2}$

$-K\int\frac{r^{2N+2}}{\bar{k}^{6}}\exp(-\frac{2\beta}{\sigma^{2}}r^{2})dr$

which decays $\exp(-\frac{2}{\sigma^{2}})$. Adding these two bounds we see the
integral has the sign of $-V^{{}^{\prime\prime}}(x_{0})$ for all $x^{*}\in A$,
and the result follows. ∎

Assumption 6 is by far the most onerous restriction needed for Proposition
1.2, requiring $\bar{V}^{{}^{\prime}}$ to be a diffeomorphism. This can be
loosened to a homeomorphism with the following assumptions.

  1. (7)

$\bar{V}^{{}^{\prime\prime}}(x,0)\geq 0$, where the lower bound is attained at
a finite number of isolated points $\\{\tilde{x}_{i}\\}_{i}^{n}$ which are
still global maxima of $\rho$ when $x^{*}=\tilde{x}$
and

  2. (8)

$V^{{}^{\prime\prime}}(\tilde{x},0)\neq 0$,
$\forall\tilde{x}\in\\{\tilde{x}_{i}\\}^{n}_{i}$

######  Corollary 1.3.

With Assumption 6 replaced by 7 & 8, Proposition 1.2 still holds in its
entirety.

###### Proof.

If Assumption 7 holds, Proposition 1.2 can be applied to all but a finite
number of open intervals containing a $\tilde{x}$ which can be made
arbitrarily small with $\sigma$. It remains to prove the result in these
intervals which as it happens, save for establishing uniform convergence, can
be handled with an argument substantially similar to Proposition 1.2. We
outline this, carefully noting any divergences, returning to the question of
uniform convergence at the end.

As $\tilde{x}$ is the unique maxima of $\rho(x,\tilde{x})$ , it is still
possible to define $x^{*}$, which is continuous. Moreover, the Laplace method
can be adapted to the lowest order non-null derivative which must be of even
order and positive. Applying the method to both the integral and $\mathcal{Z}$
in the same way as the first part of the above formulation, we see the limit
is as described.

The second part concerning bounding the derivative at roots of
$V^{{}^{\prime}}$ away from 0 is almost entirely applicable, except we caveat
that the bound $\bar{V}-\bar{V}(x^{*})>\beta(x-x^{*})^{2}$ only holds outside
an arbitrarily small interval around $x^{*}$. As this bound need only hold on
$R_{2}$, this is not an obstacle.333The derivative at $\tilde{x}$ may blow up
as product of $\sigma^{-2}$ and an integral with a non-zero
$\sigma^{2-\delta}$ term. As $F_{\sigma}$ is totally bounded this does not
affect the proof

To show convergence is uniform, we invoke Assumption 8. With this, we know
$V^{{}^{\prime\prime}}(x)$ is of one sign in some closed interval centered on
$\tilde{x}$. To that interval, we apply the method of the proof of the second
part of Proposition 1.2 to demonstrate the derivative cannot be null (with the
same caveats as described on the last paragraph), and conclude $F_{\sigma}$ is
the limit of strictly/increasing decreasing functions on this region. As
$F_{\sigma}$ is bounded on such an interval, convergence is uniform by
corollary of the Helly Selection theorem, Exercise 7.13 (b) of [18]. As there
are at most a finite number of such intervals, we conclude convergence is
uniform on $J$. ∎

We furnish these results with a few remarks and examples.

######  Remark 1.4.

[Examples] Consider the simplest polynomial bistable potential
$V^{{}^{\prime}}=x^{3}-x$ with quadratic interaction $P^{{}^{\prime}}=x$. Then
$\bar{V}^{{}^{\prime}}=x^{3}+(\theta-1)x$. There are three cases
$\theta<1,\theta=1,\theta>1$.

  * $\centerdot$

$\theta>1$, $\bar{V}^{{}^{\prime}}$ is strictly increasing, so Proposition 1.2
can be applied.

  * $\centerdot$

$\theta<1$, $\bar{V}^{{}^{\prime}}$ is not strictly increasing and cannot be
tackled with neither Proposition 1.2 of Corollary 1.3, though it can be with
the results of section 2.1

  * $\centerdot$

$\theta=1$ is the transition between the two regimes, which manifests as a
point of inflexion at 0. However $-V^{{}^{\prime\prime}}(0)=1$ so Corollary
1.3 can be applied.

[1] considers the related problem of
$\bar{V}^{{}^{\prime}}=\frac{x^{3}+(\theta-1)x}{1+x^{2}}$, where the same
parameter regime holds.

######  Remark 1.5.

(Simple zeros) To understand the restriction to simple zeros, consider
$V^{{}^{\prime}}=x^{3}$, $P^{{}^{\prime}}=x^{2}$. $V^{{}^{\prime}}$ has a
double root at 0, but for sufficiently small $\sigma$, $F_{\sigma}>0$ near
zero (see Remark 2.6). Only at $\sigma=0$ can there be a zero of $F_{\sigma}$
at $0=\bar{x}^{*-1}(0)$.

######  Remark 1.6.

(Multimodal $\rho$) In requiring $\bar{V}^{{}^{\prime\prime}}>0$, Proposition
1.2 is limited to unimodal $\rho$. This can be generalised to $\rho$
possessing a finite number of modes, where Laplace’s method applies to the
largest of them.

The general form of $\rho$ in (7) means $\rho$ can still parameterised by $m$
as $x^{*-1}(m):=\arg\max_{x}\rho(m,x)$ is an increasing, piecewise continuous
function. Discontinuities correspond to multiple modes of equal height, where
it is still possible to define the limit $F_{\sigma}$. For example, the
bistable potential, $P^{{}^{\prime}}=x$ with $\theta<1$, $\bar{x}^{*-1}(m)$
will be discontinuous at $m=0$ but $F_{\sigma}(0)$ is the average of the left
and right limit, which is itself zero.

Discontinuities may restrict the zeros of $F_{\sigma}$. Consider
$V^{{}^{\prime}}=x(x^{2}-1)(x^{2}-4)$, $P^{{}^{\prime}}=x$ and $\theta=1$.
$V^{{}^{\prime}}$ has a root at 1, but $x^{*}(m)$ cannot be unity because of
discontinuity.

For the sequel, we note that if $P^{{}^{\prime}}$ is increasing, any $\bar{V}$
will eventually be unimodal for sufficiently large $\theta$ as
$|V^{{}^{\prime}}|$ is eventually unbounded, Assumption 4

######  Lemma 1.7.

Suppose $P^{{}^{\prime\prime}}$ has a positive lower bound. Then there exists
$\theta^{*}$, such that for all $\theta>\theta^{*}$
$\bar{V}_{\theta}^{{}^{\prime}}(x,0)$ is strictly increasing.

It is considerably easier to demonstrate the existence of $\sigma_{c}^{u}$
than $\sigma_{c}^{l}$. The following Proposition uses a very straightforward
bounding argument with no additional assumptions to Proposition 1.2

######  Proposition 1.8.

There exists an Upper Critical threshold $\sigma_{c}^{u}$, such that for
$\sigma>\sigma_{c}^{u}$, $F_{\sigma}(m)$ has only one root.

###### Proof.

If it could be shown that above some $\sigma_{c}^{u}$ $\frac{\partial
F}{\partial m}(m)<-\epsilon<0$, then it is clear that $F(m)$ could be bounded
above anywhere by a decreasing linear function. Combined with the facts that
$F(m)$ exists for all finite $m$, and
$\lim_{m\rightarrow\pm\infty}F(m)=\mp\infty$, we could conclude there exists a
root, which must be unique.

It remains to establish the inequality on $\frac{\partial F}{\partial m}(m)$.
As an inequality on the sign, we work with the unnormalised stationary
measure,

$\frac{\partial F}{\partial
m}=\int_{\mathbb{R}}a(-V^{{}^{\prime}})\rho(\sigma,x,m)dx$

By Assumption 2 and 4 there exists an $R$ such that $aV^{{}^{\prime}}>c$, on
$\mathbb{R}\backslash B_{0}(R)$. To control the integral is split over an
inner region, $B_{0}(R)$ and outer domain, $\mathbb{R}\backslash B_{0}(R)$.
Setting $\mathcal{Z}=\exp\big{(}-\frac{2}{\sigma^{2}}\bar{V}(x^{*}(0))\big{)}$
and repeatedly using $a$ is increasing with $a(-R)<0<a(R)$:

(13)
$\displaystyle\int_{-R}^{R}a(-V^{{}^{\prime}})\frac{\mathcal{Z}}{\mathcal{Z}}\rho(\sigma,x,m)dx$
$\displaystyle<C\mathcal{Z}\int_{-R}^{R}\frac{\rho(\sigma,x,0)}{\mathcal{Z}}\exp(\frac{2}{\sigma^{2}}am)dx$
(14) $\displaystyle<C_{1}\mathcal{Z}\big{(}\mathbb{I}_{m\geq
0}\exp(\frac{2a(R)}{\sigma^{2}}m)+\mathbb{I}_{m<0}\exp(\frac{2a(-R)}{\sigma^{2}}m)\big{)}$

with $C=\sup_{[-R,R]}\frac{|a(-V^{{}^{\prime}})|}{k^{2}}$ and $C_{1}=2RC$.
Similarly,

(15) $\displaystyle\mathcal{Z}\int_{\mathbb{R}\backslash
B_{0}(R)}aV^{{}^{\prime}}$
$\displaystyle\frac{\rho_{0}(\sigma,x,m)}{\mathcal{Z}}dx>C_{2}\mathcal{Z}\int_{R}^{\infty}\frac{\rho(\sigma,x,0)}{\mathcal{Z}}\exp(\frac{2}{\sigma^{2}}am)dx$
(16) $\displaystyle>2C_{2}\mathcal{Z}I_{\sigma}\Big{(}\mathbb{I}_{m\geq
0}\exp(\frac{2a(R)}{\sigma^{2}}m)+\mathbb{I}_{m<0}\exp(\frac{2a(-R)}{\sigma^{2}}m)\Big{)}$

where $C_{2}=c\cdot\mathrm{min}(-a(-R),a(R))$ and
$I_{\sigma}=2\int_{R}^{\infty}\frac{\rho(\sigma,x,0)}{\mathcal{Z}}dx$

With the monotone convergence theorem, $I_{\sigma}$ is an increasing unbounded
function. Therefore it is possible to find $\sigma_{c}^{u}$ such that, for
$\sigma>\sigma_{c}^{u}$, $I_{\sigma}>C_{1}+\epsilon$. Then, subtracting (16)
from (14), we have $\frac{\partial F}{\partial m}(m)<-\epsilon<0$. The claim
is therefore proven and the desired result established.

∎

######  Remark 1.9.

While this argument can be extended to multimodal $\rho$, we advance a
different argument for the analogous Corollary 2.12 which is idiomatic to
symmetric potentials.

As $F_{\sigma}$ is bounded away from zero outside some finite interval by
Assumption 4, we can count roots on all $\mathbb{R}$. We combine this with a
restatement of Proposition 1.2 and 1.8, with the same assumptions, in terms of
stationary measures of MV-SDE(1).

######  Theorem 1.10.

Stationary Measures of MV-SDE (1) Suppose $V^{{}^{\prime}}$ has $N<\infty$
zeros, $\\{x_{i}\\}_{i}^{N}$, all of which are simple. Then there exists
critical transition thresholds $\sigma_{c}^{l},\,\sigma_{c}^{u}$ such that
$\sigma>\sigma_{c}^{u}$ MV-SDE (1) has one stationary measure while
$\sigma<\sigma_{c}^{l}$, it possesses exactly $N$ stationary measures.

###### Proof.

Proposition 1.8 can be applied verbatim. By Assumption 4, there exists a
finite interval $J$ such that $\\{x_{i}\\}^{N}_{i}\in J^{o}$ and
$|\frac{V^{{}^{\prime}}}{k^{2}}|>\delta,\,x\in J^{c}$. Applying Proposition
1.2 to $J$, noting $F_{\sigma}$ must be bounded away from 0 for sufficiently
small $\sigma$, and the bijection between zeros and stationary measures, we
can conclude the result. ∎

######  Remark 1.11.

We are now in a position to compare MV-SDE (1) with $\theta>0$ to $\theta=0$,
with assumptions 1-8. Below $\sigma_{c}^{l}$, every stationary measure is a
unimodal distribution whose mean (and mode) can be made arbitrarily close to
any extremal point (maxima and minima) of $V$. When $\theta=0$, there is one
stationary measure whose modes coincide with the minima of $V$. Above
$\sigma_{c}^{u}$, MV-SDE (1) becomes ergodic but is still unimodal.

######  Remark 1.12 (More Examples).

Section 4 of [12] introduces simple polynomial uni-stable
($V^{{}^{\prime}}=x$) and bistable potential perturbed by separable and non-
separable fluctuations, with quadratic interaction.

[12] approaches the problem by defining the potential
$V_{0}=\int_{x}\frac{V^{{}^{\prime}}}{k^{2}}$ to which a perturbation is
added, rather than working with its derivative as we have favoured here. In
the case of separable perturbation, $\delta\cos(\frac{x}{\epsilon})$,
Proposition 1.2 easily applies with $\theta>\frac{\delta}{\epsilon^{2}}$.

The non-separable perturbation
$\delta\mathbb{I}_{[-a,a]}x^{2}\cos(\frac{x}{\epsilon})$ requires a little
more work as it is non-differentiable. Reading the remarks in [12], the
indicator function is producted into the definition to control the
fluctuations as $|x|\rightarrow\infty$ to later apply the homogenisation
theorem, so $a$ should be read as ‘large’. In light of this, it is reasonable
to only consider $a$ very close to a zero of $\cos(\frac{x}{\epsilon})$, and
interpolate between the two with quadratic interpolants. This is in the
unbounded region, so again Proposition 1.2 can be applied for suitably large
$\theta$

##  2\. Symmetric Effective Potentials

This section is concerned with SDE (1) with a symmetric effective potential
(antisymmetric $P^{{}^{\prime}}$ $-V^{{}^{\prime}}$, symmetric $k$). Whilst a
few results could be deduced from Section 1, they would have narrower
applicability and most, such as those on the Critical Transition and its
dependence on $\theta$, cannot.

This is because, in return for this symmetry, some of the most onerous
restrictions of Section 1 can be relaxed or even lifted, such as the
prohibition on double roots in $-V^{{}^{\prime}}$ and non-increasing $\bar{V}$
can be lifted. Amongst other interesting possibilities, this allows repulsive-
attractive $P^{{}^{\prime}}$.

###  2.1. Bistable Potential

This section is concerned with SDE (1) with a antisymmetric $P^{{}^{\prime}}$
and $-V^{{}^{\prime}}$, which is bistable, i.e possessing three roots at
$-x^{*},0,x^{*}$. The polynomial bistable potential is well studied, [7, 8,
14, 19].

The approach here is still rooted in a study of the self-consistency equation
$F_{\sigma}$, or more precisely the equivalent first moment equation. After
establishing some key propositions about the number of roots of $F_{\sigma}$,
critical transition points are studied in detail.

The most basic assumptions are

  1. (1)

$-V^{{}^{\prime}}$, $P^{{}^{\prime}}$ and $k^{{}^{\prime}}$ are antisymmetric
and $C(\mathbb{R})$.

  2. (2)

$-V^{{}^{\prime}}$ has three roots at $\\{0,\pm|x^{*}|\\}$, and
$-V^{{}^{\prime}}(x)<0\,x>x^{*}$

  3. (3)

$\lim\limits_{|x|\rightarrow\infty}\frac{\bar{V}(x,0)}{x^{2}}>0$

  4. (4)

$|V^{{}^{\prime}}|<K(1+x^{2N})$

Assumption 1 and 3 ensures enough regularity for the integration by parts in
section 0.1 and that the boundary terms can be discarded. The polynomial
bounds in the fourth ensure integrability and the second simply outlines a
bistable potential. Further assumptions (5-8) will be introduced when the
critical transition point is studied.

In much of this section, $\sigma$ dependence of $\rho$ is supressed. Parity
restrictions (Assumption 1) are manifested in $F_{\sigma}$ according to the
following lemma

######  Lemma 2.1.

F(m) is antisymmetric

###### Proof.

Assumption 1 implies $\rho(x,m)=\rho(-x,-m)$, where
$\rho(x,m)=\exp(-\frac{2}{\sigma^{2}}(\bar{V}(x,m)))$. Then

$\begin{split}F(-m)=\int_{-\infty}^{\infty}-V^{{}^{\prime}}(x)\rho(-m,x)dx\stackrel{{\scriptstyle
y=-x}}{{=}}-\int^{-\infty}_{\infty}(-V^{{}^{\prime}}(y))\rho(-m,-y)dy=\\\
\int_{-\infty}^{\infty}V^{{}^{\prime}}(y)\rho(m,y)dy=-F(m)\end{split}$

holding regardless of $\sigma$ or $\theta$ ∎

The following proposition characterises the roots of $F_{\sigma}$. For
constant diffusion, compare with (3.21) of [19], Theorem 3.31 of [7] and,
particularly, Theorem 2.1 of [25]

######  Proposition 2.2 (Properties of $F(m)$ ).

  1. (i)

F has a root at 0.

  2. (ii)

There is at most one strictly positive (negative) root of $F_{\sigma}$.

  3. (iii)

These additional roots exist iff $\left.\frac{\partial F}{\partial
m}\right|_{m=0}>0$

###### Proof.

Part (i) follows immediately from the antisymmetry of $F_{\sigma}$, Lemma 2.1.

For Parts (ii) & (iii), we derive the series representation of $F_{\sigma}$ by
substituting $\exp(\frac{2}{\sigma^{2}}mx)$ with its series representation,

$F(m)=\int-V^{{}^{\prime}}(x)\rho(x,0)\sum_{n}\frac{(\frac{2}{\sigma^{2}}ma)^{n}}{n!}dx$

Given the antisymmetry of $F_{\sigma}$, we focus on $m\in\mathbb{R}^{+}$ The
strategy for the proof is to show the coefficients of the series are at most
change sign once, from positive to negative. It is then straightforward to
demonstrate that $F(m)$ must also have at most one root in $\mathbb{R}^{+}$

With the polynomial growth conditions 3 & 4, $F(m)$ can be bounded above as
moments of a Gaussian distribution. Exchange of the summation and integral

(17)
$F(m)=\sum_{n}\frac{(\frac{2}{\sigma^{2}}m)^{n}}{n!}\int-V^{{}^{\prime}}a^{n}\rho(x,0)dx:=\sum_{n}\frac{(\frac{2}{\sigma^{2}}m)^{n}}{n!}I(n)$

can then be justified with an application of the Tonelli-Fubini Theorem.

From the antisymmetry of the integrand, $I(2n)=0$. Reciprocally, by the
symmetry of the integrand of $I(2n-1)$

(18) $I(2n-1)=2\int_{0}^{\infty}-V^{{}^{\prime}}a^{2n-1}\rho(x,0)dx$

By assumption on $k$, $a$ must be strictly increasing. Consequently,

(19)
$\displaystyle\begin{split}\big{(}\frac{a}{a(x^{*})}\big{)}^{2n-1}<1,\,x<x^{*}\\\
\big{(}\frac{a}{a(x^{*})}\big{)}^{2n-1}\geq 1,\,x\geq x^{*}\end{split}$

With these facts, and introducing the rescaled coefficient

(20)
$\tilde{I}(2n-1,[0,\infty))=2\int_{0}^{\infty}-V^{{}^{\prime}}\Big{(}\frac{a}{a(x^{*})}\Big{)}^{2n-1}\rho(x,0)dx$

with

$\frac{a}{a(x^{*})}=\tilde{a}$

it can be seen both

(21)
$\displaystyle\begin{split}0<\,&\tilde{I}(2n-3,[0,x^{*}])<\tilde{I}(2n-1,[0,x^{*}])\\\
&\tilde{I}(2n-3,[x^{*},\infty))<\tilde{I}(2n-1,[x^{*},\infty))<0\end{split}$

by the monotonicity of the integral. Adding these two monotonically decreasing
inequalities, we see $\tilde{I}(2n-1,[0,\infty))$ is also strictly decreasing,

(22) $\tilde{I}(2n-1)<\tilde{I}(2n-3)$

Applying the Dominated convergence theorem,
$\lim\limits_{n\rightarrow\infty}\tilde{I}(2n-1,[0,x^{*}])\rightarrow 0$.
Therefore there must exist a threshold $n_{t}$ such that
$|\tilde{I}(2n_{t}-1,[0,x^{*}])|<|\tilde{I}(1,[x^{*},\infty))|$ so, combined
with monotonicity (22), we conclude $\tilde{I}(2n-1,[0,\infty))$ must
eventually become negative.

As a positive multiple of $\tilde{I}(2n-1,[0,\infty))$, $I(2n-1)$, while not
necessarily monotonic, inherits the crucial property that there exists a
threshold $n_{c}<\infty$ such that $I(2n-1)\geq 0$ iff $n\leq n_{c}$.

If all the $I(2n-1)$ are negative we set $n_{c}=0$, and clearly $F(m)<0,m>0$

For $n_{c}\neq 0$, splitting the series by sign at $n_{c}$ and factorising

(23)
$F(m)=m^{2n_{c}-1}\Big{(}\sum_{1}^{n_{c}}\frac{(\frac{2}{\sigma^{2}})^{2n-1}}{2n-1!}I_{2n-1}\frac{1}{m^{2(n_{c}-n)}}+\sum_{n_{c}+1}^{\infty}\frac{(\frac{2}{\sigma^{2}})^{2n-1}}{2n-1!}I_{2n-1}m^{2(n-n_{c})}\Big{)}$

Inside the parenthesis is the difference of a strictly decreasing and
increasing functions, where the limit as $m\downarrow 0$ is $\infty$ and
$m\uparrow\infty$ is $-\infty$. We conclude $F_{\sigma}$ has at most one root
in $\mathbb{R}^{+}$, where the sign must change from positive to negative.

As has just been outlined, a root can only occur if $n_{c}\neq 0$. By the
monotonicity of $\tilde{I}(2n-1)$, $n_{c}\neq 0$ iff $I(1)>0$. Then
$I(1)=\int-V^{{}^{\prime}}a\rho=\frac{\partial F}{\partial m}$ which is part
(iii) ∎

The rest of this section is devoted to studying critical transitions in
$\sigma$: whether and when the one solution regime turns to the three, and/or
vice versa.

This can be restated in terms of the number of roots of $F_{\sigma}(m)$,
section 0.1 and in the bistable case, this reduces by Proposition 2.2 (iii) to
studying the sign of $\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma)$, i.e

######  Lemma 2.3.

The critical transition(s) $\sigma_{c}$ corresponds to roots of
$\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)$.

The forthcoming Proposition entirely characterises the possible number of
roots and their existence. For this the following assumptions are needed, and
additionally for all remaining results of this section.

  1. (5)

$\sup\limits_{0\leq x\leq x^{*}}\bar{V}(x,0)=\bar{V}(x^{*},0)>0$

  2. (6)

$\inf\limits_{x\geq x^{*}}\bar{V}(x,0)=\bar{V}(x^{*},0)$

######  Remark 2.4.

These assumptions generalise $\bar{V}(x,0)$ being strictly increasing

######  Proposition 2.5.

(The Roots of $\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)$)

$\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)$ can have at most
one root. The exact number of roots can be characterised as follows:

  * $\centerdot$

There are no roots iff $\lim\limits_{\sigma\downarrow 0}\left.\frac{\partial
F}{\partial m}\right|_{m=0}(\sigma)<0$.

  * $\centerdot$

There is exactly one root iff $\lim\limits_{\sigma\downarrow
0}\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)>0$

###### Proof.

Suppose there is a root of $\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma)$ at $\sigma_{c}\in(0,\infty)$. Differentiating,

$\left.\frac{\partial^{2}F}{\partial\sigma\partial
m}\right|_{m=0}=-\left.\frac{2}{\sigma}\frac{\partial F}{\partial
m}\right|_{m=0}+\frac{8}{\theta\sigma^{5}}\int-V^{{}^{\prime}}a\bar{V}(x,0)\rho(x,0)dx$

At $\sigma_{c}$, the first term is 0 by assumption. For the second, set
$\tilde{V}(x,0)=\frac{1}{k_{\bar{V}}}\bar{V}(x,0)$ with
$k_{\bar{V}}=\bar{V}(x^{*},0)>0$. Then as $0\leq\tilde{V}(x,0)\leq 1$,
$-V^{{}^{\prime}}a\tilde{V}(x,0)<-V^{{}^{\prime}}a$ so at $\sigma_{c}$

(24)
$\int\left.-V^{{}^{\prime}}a\tilde{V}(x,0)\rho(\sigma_{c},x,0)dx<\int-V^{{}^{\prime}}a\rho(\sigma_{c},x,0)=k\frac{\partial
F}{\partial m}\right|_{m=0}(\sigma_{c})=0$

where the strict inequality follows from $\rho>0$. By inequality (24), the
gradient at any root $\sigma_{c}$ must be negative, so if a root exists it
must be unique. Reciprocally, $\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma)\leq 0$ cannot have any roots and must be of constant
sign.

If $\lim\limits_{\sigma\downarrow 0}\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma)>0$ there must be a root as
$\lim\limits_{\sigma\uparrow\infty}\left.\frac{\partial F}{\partial
m}\right|_{m=0}<0$. This can be seen by factoring the integral

(25)
$\exp(-\frac{2}{\sigma^{2}}\mathcal{Z})\int\tilde{a}(-V^{{}^{\prime}})\exp(-\frac{2}{\sigma^{2}}(\bar{V}(x,0)-\mathcal{Z}))$

where $\mathcal{Z}=\min_{x}V(x,0)$ must exist by Assumption 1.

In $[0,x^{*}]$ the integrand is positive and bounded, so can be bounded above
by $\int_{0}^{x^{*}}a(-V^{{}^{\prime}})dx$. Outside, it is negative and
unbounded, so there exists some compact set $K\in[x^{*},\infty)$ such that
$\int_{0}^{x^{*}}a(-V^{{}^{\prime}})dx<\int_{K}|a(-V^{{}^{\prime}})|dx$.

Using
$\lim\limits_{\sigma\uparrow\infty}\exp(-\frac{2}{\sigma^{2}}(\bar{V}-\mathcal{Z})\equiv
1$ and applying the dominated convergence theorem, integral (25) must
eventually be negative. It follows that $\lim\limits_{\sigma\downarrow
0}\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)>0$ there must be a
root, and it is unique by the first claim in this Proposition. ∎

In other words if $\lim\limits_{\sigma\downarrow 0}\left.\frac{\partial
F}{\partial m}\right|_{m=0}(\sigma)<0$ the only stationary measure possible is
the symmetric one. Otherwise there are three stationary measures below
$\sigma_{c}$, and only the one (symmetric) above. The uniqueness of this point
implies $\sigma_{c}^{u}=\sigma_{c}^{u}:=\sigma_{c}$. This rigidity in
bifurcation pattern is striking and further investigated in [1].

######  Remark 2.6.

If the limit $\lim\limits_{\sigma\downarrow 0}\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma)=0$ the behaviour will depend on whether the limit is
approached from above or below. A very important example of this is if the
maxima is at 0, as it coincides with a root of $a(-V^{{}^{\prime}})$, the
limit will be 0. However as $a(-V^{{}^{\prime}})\geq 0$ in a neighbourhood of
that root the limit will be approached from above, due of the assumed
polynomial bounds on $-V^{{}^{\prime}}$ and more careful estimates in the
style of Theorem 1.2. The result of Proposition 2.5 can then be applied, so we
conclude there must be a root strictly greater than 0.

######  Remark 2.7.

An upcoming assumption, 7, implies $\rho$ must have a maximum in $[0,x^{*})$
and, with Remark 2.6, that $\lim\limits_{\sigma\downarrow
0}\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)>0$ by the Laplace
theorem, guaranteeing $\sigma_{c}$’s existence.

The second part of the outlined programme - a study of the dependence of
$\sigma_{c}$ on $\theta$ \- is made possible by Proposition 2.5. If
$\sigma_{c}$ exists, it is unique, so the mapping between $\theta$ and
$\sigma_{c}$ is a well defined function.

######  Definition 2.8 (The Critical Transition function).

$\sigma^{*}:\theta\rightarrow\sigma_{c}$

(26) $\sigma^{*}(\theta)=\begin{cases}\sigma_{c}&\sigma_{c}\mathrm{exists}\\\
0&\mathrm{otherwise}\end{cases}$

It was conjectured and numerically demonstrated in [25] that $\sigma_{c}$
exists $\theta>0$ and that $\sigma^{*}(\theta)$ is an increasing function, for
$P^{{}^{\prime}}=x$ and the simple polynomial bistable potential (Remark 1.4).
Using the results above, this conjecture can be proven under considerably
weaker conditions.

This final result requires

  1. (7)

$\sup\limits_{0\leq x\leq
x^{*}}\int^{x}\frac{P^{{}^{\prime}}}{k^{2}}=\int^{x^{*}}\frac{P^{{}^{\prime}}}{k^{2}}>0$

  2. (8)

$\inf\limits_{x\geq
x^{*}}\int^{x}\frac{P^{{}^{\prime}}}{k^{2}}=\int^{x^{*}}\frac{P^{{}^{\prime}}}{k^{2}}>0$

Additionally it is assumed that Assumptions 1-8 hold on $\theta\in
J\subseteq\mathbb{R}^{+}$. Then

######  Proposition 2.9.

$\sigma^{*}(\theta)$ is an increasing function on $J$.

###### Proof.

Assumption 5 and 7 imply $P^{{}^{\prime}}(x^{*})>0$ which further implies
$-\bar{V}(x,0)$ has a maxima in $[0,x^{*})$. As explained in Remark 2.6, with
the Laplace theorem, $\lim\limits_{\sigma\downarrow 0}\left.\frac{\partial
F}{\partial m}\right|_{m=0}(\sigma)>0$ (or 0 but approaching from above) and
so $\sigma^{*}(\theta)>0$ (i.e a root of $\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma)$) by Proposition 2.5.

Differentiating with respect to $\theta$,

$\left.\frac{\partial^{2}F}{\partial\theta\partial
m}\right|_{m=0}=-\left.\frac{1}{\theta}\frac{\partial F}{\partial
m}\right|_{m=0}-\frac{4}{\theta\sigma^{4}}\int-V^{{}^{\prime}}a\int^{x}\frac{P^{{}^{\prime}}}{k^{2}}\rho(x,0)dx$

With Assumptions 7 and 8, at $\sigma^{*}(\theta)=\sigma_{c}$

(27)
$-\left.\int-V^{{}^{\prime}}a\tilde{P}\rho(\sigma^{*},x,0)dx>-\int-V^{{}^{\prime}}a\rho(\sigma^{*},x,0)=c\frac{\partial
F}{\partial m}\right|_{m=0}(\sigma_{c})=0$

with $\tilde{P}=\int\frac{P^{{}^{\prime}}}{k_{P}k^{2}}$ with
$k_{P}=\int^{1}\frac{P^{{}^{\prime}}}{k^{2}}dx$. With identical reasoning to
Proposition 2.5 it can be deduced that the derivative with respect to $\theta$
at any root $\sigma_{c}$ must be positive.

By the chain rule,

(28)
$\frac{d\sigma^{*}}{\partial\theta}(\theta)=-\frac{\frac{\partial^{2}F}{\partial\theta\partial
m}|_{m=0}}{\frac{\partial^{2}F}{\partial\sigma\partial m}|_{m=0}}>0$

where inequality comes from inequalities (24) and (27). So
$\sigma_{c}(\theta)$ is an increasing function, and the claim proven. ∎

Returning to the very particular case considered by the conjecture of [25],
the maximum of $V_{\theta}$ lies in $[0,x^{*}]$ for $\theta>0$ and so with the
reasoning of Remark 2.6, $J=(0,\infty)$.

###  2.2. Multi-Well Potential

This section is concerned with the natural generalisation of the symmetric
double well potential: a symmetric potential which possesses multiple extrema
in some finite interval. With more degrees of freedom, a greater multiplicity
of behaviours can be exhibited. Rather than document this myriad of
possibilities here, the primary interest is in finding criterion such that the
symmetric multi-well case of self-consistency equation (6) behaves like the
symmetric bistable case of section 2.1.

Heuristically, it is reasonable to suppose that, for a multi-well potential
$-V^{{}^{\prime}}$ that is ‘minimally’ negative in $[0,x^{*}]$, the self-
consistency equation would exhibit ‘close to bistable’ behaviour. The non-
contiguousness of the positive regions of $-V^{{}^{\prime}}$ present an
immediate barrier to applying the results of Section 2.1 complicating this
intuitive picture. This section will overcome these issues, developing
attractively conjunct criteria on $-V^{{}^{\prime}}$ to satisfy this
programme, avoiding clumsy, overly prescriptive assumptions on the location of
roots or extrema.

####  2.2.1. Definitions and Assumptions

In light of this philosophy, the assumptions needed for this section are
Assumptions 1-8 from Section 2.1, lightly modified. As a corollary of
Assumption 1, specifically the anti-symmetry of $-V^{{}^{\prime}}$, there
necessarily a root at 0. No further assumptions are demanded on the location
or number of the roots.

Having been in essence already presented, the modifications are listed here

  1. (2*)

$x^{*}$ is the root of $-V^{{}^{\prime}}$ farthest from the origin, and
$-V^{{}^{\prime}}(x)<0$, $x>x^{*}$

  2. (5*)

$\sup\limits_{0\leq x\leq x^{*}}\bar{V}_{D}(x,0)=\bar{V}_{D}(x^{*},0)>0$

  3. (6*)

$\inf\limits_{x\geq x^{*}}\bar{V}_{D}(x,0)=\bar{V}_{D}(x^{*},0)$

where
$\bar{V}_{D}(s,0)=\int^{s}_{0}\frac{V^{{}^{\prime}}_{D}+P^{{}^{\prime}}}{k^{2}}$
and
$-V^{{}^{\prime}}_{D}:=(-V^{{}^{\prime}})_{+}-\mathbb{I}_{[x^{*},\infty)}(-V^{{}^{\prime}})_{-}$444$(f)_{-}=\max(0,-f)$

Assumption 2 has been loosened to admit multi-well potentials. Assumptions 5-6
have have been modified to apply to a dominating bistable potential
$-V^{{}^{\prime}}_{D}\geq-V^{{}^{\prime}}$ for Corollary 2.12. When Assumption
2/5/6 is referred to in the following, it should be read as
$2^{*}$/$5^{*}$/$6^{*}$. All the results in this section require Assumptions
1-8, unless otherwise stated.

Intuitively, as $\sigma$ increases, the contribution to $F_{\sigma}$ of the
integrand over $[x^{*},\infty)$ will dominate that over $[0,x^{*}]$, just as
in the bistable case. This first result established a threshold $\sigma_{r}$
above which $[x^{*},\infty)$ will dominate the negative parts of the integrand
in $[0,x^{*}]$, which is crucial to the sequel

######  Proposition 2.10.

$\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx$ has exactly one
root, $\sigma_{r}$. Moreover

(29) $\int_{0}^{\infty}(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho
dx>0$

for $\sigma>\sigma_{r}$

###### Proof.

To highlight the similarities with Proposition 2.5, define
$H(\sigma):=-\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx$. Now
the proof runs identically: bound $\frac{dH}{d\sigma}$ above by $G$ using
Assumptions 5 & 6 to show the gradient at any root must be negative. Conclude
there can only be at most one root, that exist iff
$\lim\limits_{\sigma\downarrow 0}H(\sigma)>0$

There are then two cases:

  * $\centerdot$

There are no roots iff $\lim\limits_{\sigma\downarrow
0}\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx>0$,

  * $\centerdot$

There is exactly one root iff $\lim\limits_{\sigma\downarrow
0}\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx<0$

However there must be a maxima for $\rho$ in $[0,x^{*}]$ by Assumptions 7-8,
so $\lim\limits_{\sigma\downarrow
0}\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx\leq 0$ by the
Laplace method. If strictly negative, the claim follows immediately. If the
limit is 0, it will be approached from below by the same reasoning as Remark
2.6 and the defintion of $(-V^{{}^{\prime}})_{-}$. Therefore $\sigma_{r}$ must
exist.

That $\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx>0$ for
$\sigma>\sigma_{r}$ follows from establishing the limit $\sigma\uparrow\infty$
is negative in the same manner as Proposition 2.5. ∎

The importance of $\sigma_{r}$ becomes apparent in this next Proposition, as
the point where the series expansion for $F_{\sigma}$ behaves much like
Proposition 2.2, with monotonically decreasing (scaled) coefficients.

######  Proposition 2.11.

For $\sigma>\sigma_{r}$, $F_{\sigma}(m)$ has at most one strictly positive
(respectively negative) root of $F_{\sigma}$, which exist iff
$\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)>0$

###### Proof.

We will demonstrate that if
$\int_{0}^{\infty}(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho dx>0$
then $\\{\tilde{I}_{\sigma}(2n-1)\\}_{n}$ is decreasing. The full result
follows identically to Proposition 2.2.

As the positive and negative regions of $(-V^{{}^{\prime}})$ in $[0,x^{*}]$
are not contiguous, to easily represent them the following are introduced:

(30)
$\displaystyle\begin{split}a_{n}=\int_{0}^{x^{*}}\tilde{a}^{2n-1}(-V^{{}^{\prime}})_{+}\rho
dx\\\ b_{n}=\int_{0}^{x^{*}}\tilde{a}^{2n-1}(-V^{{}^{\prime}})_{-}\rho dx\\\
c_{n}=\int^{\infty}_{x^{*}}\tilde{a}^{2n-1}(-V^{{}^{\prime}})_{-}\rho
dx\end{split}$

all of which are bounded below by 0.

In the terminology established in Proposition 2.2,
$-c_{n}=\tilde{I}_{\sigma}(2n-1,[x^{*},\infty))$ and
$a_{n}-b_{n}=\tilde{I}_{\sigma}(2n-1,[0,x^{*}])$

Then the condition $\int(\tilde{a}^{3}-\tilde{a})(-V^{{}^{\prime}})_{-}\rho
dx>0$ is equivalent to

(31) $(b_{2}-b_{1})+(c_{2}-c_{1})>0$

the terms in the first bracket being negative increasing, the second positive
increasing which can be seen by directly considering the integrands.
Therefore, $\forall n$

(32) $(b_{n}-b_{n-1})+(c_{n}-c_{n-1})>(b_{2}-b_{1})+(c_{2}-c_{1})>0$

By identical reasoning to Proposition 2.2 we can represent self consistency
function $F_{\sigma}$ by its power series (17). In terms of (30), the
difference between any two rescaled coefficients is

(33)
$\tilde{I}_{\sigma}(2n+1)-\tilde{I}_{\sigma}(2n-1)=a_{n+1}-a_{n}-\Big{(}(b_{n+1}-b_{n})+(c_{n+1}-c_{n})\Big{)}$

To complete the proof of the claim, it suffices to show (33) must be negative.
That $a_{n+1}-a_{n}$ is negative in was demonstrated in inequality (21). It
remains to be shown that the terms inside the parenthesis must be positive,
but this was the content of inequality (32). ∎

The next corollary contains the same result as Proposition 1.8, but with an
entirely different proof to cover symmetric multi-well potentials

######  Corollary 2.12.

There exists an upper critical threshold $\sigma_{c}^{u}$ such that,
$\sigma>\sigma_{c}^{u}$, $F_{\sigma}(m)<0,\,m>0$.

###### Proof.

By identical reasoning to Proposition 2.2, $F_{\sigma}(m)$ can be represented
by power series 17. Then, for $\sigma>\sigma_{r}$ the coefficients
$I_{\sigma}(n)$ are negative iff $I_{\sigma}(1)<0$.

Recalling the related dominating bistable potential

$-V^{{}^{\prime}}_{D}:=\mathbb{I}_{[0,x^{*}]}(-V^{{}^{\prime}})_{+}-\mathbb{I}_{[x^{*},\infty)}(-V^{{}^{\prime}})_{-}$

it can be seen that

(34)
$\tilde{I}_{\sigma}(1)=a_{1}-b_{1}-c_{1}<a_{1}+0-c_{1}=\int-V^{{}^{\prime}}_{D}\tilde{a}\rho
dx:=\tilde{I}^{D}_{\sigma}(1)$

where the notation established in (30) has been used.

$-V^{{}^{\prime}}_{D}$ is a bistable potential satisfying all the salient
original assumptions, so Proposition 2.5 is applicable, whence there exists
some $\sigma_{c}^{D}$ such that
$\tilde{I}^{D}_{\sigma}(1)<0,\,\sigma>\sigma_{c}^{D}$ Inequality (34) implies
$\tilde{I}_{\sigma}(1)$ must be negative above the same threshold, so
$\sigma_{c}^{u}$ must exist and

$\sigma_{c}^{u}\leq\max(\sigma_{c}^{D},\sigma_{r})$

∎

Given its existence, we set $\sigma_{c}^{u}$ to be the smallest $\sigma$ above
which $F_{\sigma}(m)$ has no positive (negative respectively) roots

######  Definition 2.13.

The Critical Transition function
$\sigma_{c}(\theta)=\mathrm{inf}\\{\sigma:F_{(s,\theta)}(m)<0,\,\forall
m>0,\,\forall s>\sigma\\}$ is well-defined.

###### Proof.

This set is non-empty by Corollary 2.5 ($\sigma_{c}<\sigma_{c}^{u}$), and
bounded below above 0.∎

Again, in terms of stationary measures of MV-SDE (1):

######  Theorem 2.14 (Stationary Measures of MV-SDE (1)).

There exists an upper critical threshold $\sigma_{c}^{u}$ such that,
$\sigma>\sigma_{c}^{u}$, MV-SDE (1) has only one stationary measure.

In the multi-well case the upper and lower critical thresholds will not
necessarily be equal, and it is the upper threshold that is the true analogue
of the critical threshold of Section 2.1. The rest of this section is
dedicated to a study of the dependence of the upper critical transition
dependence on $\theta$.

######  Remark 2.15.

Applying the results of the foregoing section to the dominating bistable
process at most would show that the critical transition of the multi-well
process is bounded above by an increasing function.

In section 2.1 this relied upon the bijection between critical points and
roots of $\left.\frac{\partial F}{\partial m}\right|_{m=0}(\sigma)$, which in
turn relies on ordered, decreasing coefficients $I(2n-1)$ of the power series
expansion of $F_{\sigma}(m)$. This is no longer globally true, however if
$I_{\sigma_{r}}(1)>0$ then $\sigma^{u}_{c}>\sigma_{r}$, where the coefficients
are ordered and much of the machinery developed for the bistable potential can
be repurposed.

The next Proposition translates the result of 2.1 as directly as possible. As
with Proposition 2.9, let Assumptions 1-8 hold for $\theta\in J$.

######  Proposition 2.16.

Suppose,

(35) $\displaystyle\left.\frac{\partial F}{\partial
m}\right|_{m=0}(\sigma_{r})=\tilde{I}_{\sigma_{r}}(1)>0$

and

(36)
$\displaystyle\int_{0}^{\infty}\tilde{a}(-V^{{}^{\prime}})(1-\tilde{V})\rho
dx>0$ (37)
$\displaystyle\int_{0}^{\infty}\tilde{a}(-V^{{}^{\prime}})(1-\tilde{P})\rho
dx>0$

for $\sigma\in\mathbb{R}^{+}$ and $\theta\in J\subseteq\mathbb{R}^{+}$

Then $\sigma^{u}_{c}>\sigma_{r}$ and $\sigma^{u}_{c}(\theta)$ is an increasing
function on $J$.

###### Proof.

With (35), $\sigma_{c}^{u}$ must necessarily be greater than $\sigma_{r}$
Inequality (36) implies (24) so by the same process as Proposition 2.5, we
know $\tilde{I}_{\sigma}(1)$ has one root, $\sigma>\sigma_{r}$. By the
monotonicity of the $\tilde{I}(2n-1)$ from Proposition 2.11, $\sigma_{c}$ will
coincide with the root of $\tilde{I}_{\sigma}(1)$.

Inequality (37) implies (27), so we conclude the result with the chain rule
(28), identically to Proposition 2.9. ∎

By maintaining the utmost generality, the proposition is cumbersome in use. It
requires the calculation of $\sigma_{r}$ rather than simply relying on its
existence, while global inequalities (36) and (37) depend on
$(\sigma,\,\theta)$ through $\rho$. Contrastingly, in the bistable case,
inequalities (36) and (37) were implied by conditions directly on
$\bar{V}^{{}^{\prime}}$ and $P^{{}^{\prime}}$, unmoderated by $\rho$. In the
‘close to bistable’ regime with which we are concerned there is good reason to
suppose conditions on $-V^{{}^{\prime}},\,P^{{}^{\prime}}$ alone would be
sufficient to imply (36) and (37), see the remarks that head this section.

As we will demonstrate, if $\bar{V}^{{}^{\prime}}(x,0)$ and $P^{{}^{\prime}}$
are strictly increasing, it is possible to find intuitive integral
inequalities on $(-V^{{}^{\prime}})_{+}$ versus $(-V^{{}^{\prime}})_{-}$ at a
small cost, see Remark 2.18. Further, a similar inequality can be derived on
the potential to replace the need for the precise location of $\sigma_{r}$,
with no further penalty. The following corollary exemplifies the above
discussion. Although dealing specifically with the quadratic interaction it
can be generalised to $P^{{}^{\prime}}$ strictly increasing and $k$ such that
$\lim_{x\uparrow\infty}\tilde{a}(x)>\sqrt{2}$ for $\theta\in J$ such that
$\bar{V}^{{}^{\prime\prime}}(x,0)>0$ with no change in argumentation.

######  Corollary 2.17 (Multi-Well Potential with Quadratic Interaction).

Consider SDE (1) with $P^{{}^{\prime}}=x$, $k=1$ and $-V^{{}^{\prime}}$ a
multi-well potential satisfying Assumptions 1-4. Suppose $-V^{{}^{\prime}}$
additionally satisfies Inequalities

(38)
$\displaystyle\int_{0}^{t}x(1-x)(-V^{{}^{\prime}})_{+}-x(-V^{{}^{\prime}})_{-}dx>0,$
$\displaystyle\quad\forall t<x^{*}$ (39)
$\displaystyle\int_{0}^{t}x\big{(}(-V^{{}^{\prime}})_{+}-2(-V^{{}^{\prime}})_{-}\big{)}dx>0,$
$\displaystyle\quad\forall t<\sqrt{2}x^{*}$

Then there exists $\theta^{*}$ such that the upper critical temperature
function is decreasing on $[\theta^{*},\infty)$

###### Proof.

As $P^{{}^{\prime\prime}}$ is bounded below, by Lemma 1.7, above some
$\theta^{*}$, $\bar{V}^{{}^{\prime\prime}}(x,0)>0$ . This implies Assumptions
5-6 and further implies $P^{{}^{\prime}}$ strictly increasing, which in turn
implies Assumptions 7-8 on $[x^{*},\infty)$.

To reduce the need for tildes, we rescale the above inequalities in order to
be able to take $x^{*}=1$. For inequality (36) it is sufficient to require

(40) $\int_{0}^{1}x(-V^{{}^{\prime}})(1-\tilde{V})\rho dx>0,\,\forall\sigma>0$

as the contribution to the integral over $[1,\infty)$ is positive by
Assumption 2.555If needed, a less restrictive inequality can be derived by
bounding below the integral in this outer region using inequality (31). For
this inequality to hold $(-V^{{}^{\prime}})_{+}$ must ‘dominate’
$(-V^{{}^{\prime}})_{-}$ in $[0,1]$, which is intuituvely the case if $V$ is
‘close to bistable’.

We can further simplify this inequality further using the convexity of
$\tilde{V}$, guaranteed by the second derivative test. Given $\tilde{V}(0)=0$
and $\tilde{V}(1)=1$ by definition, $1-x<1-\tilde{V}<1$. Consequently (40) can
be bounded below with

$\int_{0}^{1}x\big{(}(-V^{{}^{\prime}})_{+}-(-V^{{}^{\prime}})_{-}\big{)}(1-\tilde{V})\rho
dx>\int_{0}^{1}x(1-x)(-V^{{}^{\prime}})_{+}\rho
dx-\int_{0}^{1}x(-V^{{}^{\prime}})_{-}\rho dx$

so it is suffices that

(41) $\int_{0}^{1}x(1-x)(-V^{{}^{\prime}})_{+}\rho
dx>\int_{0}^{1}x(-V^{{}^{\prime}})_{-}\rho dx$

for all $\sigma\in\mathbb{R}^{+}$.

In general an argument like that just performed would be required to simplify
(27). However with this choice of $P^{{}^{\prime}}$ and $k$, (27) such a
manipulation yields (29), which is of course true for $\sigma>\sigma_{r}$ by
Proposition 2.10.

Finally, to ensure $\sigma_{r}<\sigma_{c}$, it suffices to find conditions
that imply $I_{\sigma}^{*}(1)>0$. Contrastingly, an inequality on the sign
$I_{\sigma}(1)$ cannot be global in $\sigma$, because it must be unbounded
below as $\sigma\uparrow\infty$, as expatiated in Corollary 2.12.

For $\sigma\leq\sigma_{r}$,

(42) $\int_{0}^{1}(x-x^{3})(-V^{{}^{\prime}})_{-}\rho
dx\geq\int_{1}^{\infty}(x^{3}-x)(-V^{{}^{\prime}})_{-}\rho dx$

by Proposition 2.10. Then

$\int_{1}^{\infty}x(-V^{{}^{\prime}})_{-}\rho
dx<\int_{1}^{\sqrt{2}}x(-V^{{}^{\prime}})_{-}\rho
dx+\int_{0}^{1}(x-x^{3})(-V^{{}^{\prime}})_{-}\rho
dx<\int_{0}^{\sqrt{2}}x(-V^{{}^{\prime}})_{-}\rho dx$

where the first inequality comes from simple bounding of the polynomial terms,
while the second from inequality (42).

Inserting this into $\tilde{I}_{\sigma}(1)$,

(43)
$\displaystyle\tilde{I}_{\sigma}(1)=\int_{0}^{1}x(-V^{{}^{\prime}})_{+}\rho
dx-\int_{0}^{1}x(-V^{{}^{\prime}})_{-}\rho
dx-\int_{1}^{\infty}x(-V^{{}^{\prime}})_{-}\rho dx$ (44)
$\displaystyle>\int_{0}^{1}x(-V^{{}^{\prime}})_{+}\rho
dx-2\int_{0}^{\sqrt{2}}x(-V^{{}^{\prime}})_{-}\rho dx$

So, if $\forall\sigma\in\mathbb{R}^{+}$

(45) $\int_{0}^{1}x(-V^{{}^{\prime}})_{+}\rho
dx>2\int_{0}^{\sqrt{2}}x(-V^{{}^{\prime}})_{-}\rho dx$

then $\tilde{I}_{\sigma}(1)>0$ when $\sigma\leq\sigma^{*}$

As $\rho$ is strictly decreasing, inequality conditions (41) and (45) are
implied by

(46)
$\displaystyle\int_{0}^{t}x(1-x)(-V^{{}^{\prime}})_{+}-x(-V^{{}^{\prime}})_{-}dx>0,$
$\displaystyle\quad\forall t<1$ (47)
$\displaystyle\int_{0}^{t}x\big{(}(-V^{{}^{\prime}})_{+}-2(-V^{{}^{\prime}})_{-}\big{)}dx>0,$
$\displaystyle\quad\forall t<\sqrt{2}$

by the second mean value theorem theorem for integrals.

Therefore the assumptions of this corollary imply those of Proposition 2.16,
which proves the result. ∎

######  Remark 2.18.

With strictly increasing $\bar{V}(x,0)$, $\rho$ must have global maxima at 0.
It is possible to chose $\epsilon$ such that $x(-V^{{}^{\prime}})\geq 0$ or
$\leq 0$ on $[-\epsilon,\epsilon]$. The limit of integrals (36) and (37) as
$\sigma\downarrow 0$ is, applying the Laplace method, 0 and the inequalities
imply the limit is approached from above. So, by Remark 2.6,
$-xV^{{}^{\prime}}\geq 0$ for arbitrarily small $x$ implying
$-V^{{}^{\prime}}(x)>0$ necessarily, for some small interval $0<x<\epsilon$

This can be remedied by finding some new lower bound for $\sigma$ that those
inequalities hold. However that would prevent further simplification by
application of the second mean value theorem for integrals.

These assumptions are flexible enough to admit a potential with any number of
extrema. Indeed, we demonstrate this by outlining a procedure to construct an
admissible $-V^{{}^{\prime}}$ with an arbitrary number of roots. Perhaps the
most direct method is to start with a multi-well potential with the required
number of roots that is positive in $[0,x_{1}]$ and $[x_{2},x^{*}]$ and
satisfies Assumptions 1 to 4. It is possible to find coefficients
$\alpha_{i}$666Assumptions 1-4 are not affected by such a scaling, although
$\theta^{*}$ will increase such that
$\alpha_{1}\int_{0}^{x_{1}}x(1-x)(-V^{{}^{\prime}})\rho
dx>\int_{x_{1}}^{x_{2}}x(-V^{{}^{\prime}})_{-}\rho dx$ and
$\alpha_{2}\int_{x_{2}}^{x^{*}}x(-V^{{}^{\prime}})\rho
dx>2\int_{x^{*}}^{\sqrt{2}}x(-V^{{}^{\prime}})_{-}\rho dx$. Then for
$\alpha_{1}\mathbb{I}_{[0,x_{1}]}(-V^{{}^{\prime}})+\mathbb{I}_{[x_{1},x_{2}]\cup[1,\infty)}(-V^{{}^{\prime}})+\alpha_{2}\mathbb{I}_{[0,x_{1}]}(-V^{{}^{\prime}})$
satisfies inequalities (38 & 39).

##  3\. Conclusion

In this work, we have studied the possible phases and their transition points
for MV-SDE (1) We have shown for sufficiently small $\sigma$ there are exactly
as many stationary measures as roots of $V^{{}^{\prime}}$ and sufficiently
large there is only one. In the case of symmetrical potentials we have gone
further and additionally demonstrated the (upper) critical transition is a
strictly increasing function of the aggregation parameter.

The approach utilised is direct, relying upon the first MEE, and robust enough
to generate quantitative estimates. In addition to entirely novel results,
where similar results have presented before, their proofs have been simplified
and their applicability greatly increased. A choice was made to keep
assumptions as general as possible, and the results can yield more when more
is known of $V^{{}^{\prime}},\,P^{{}^{\prime}}$

For related future work, an idea that immediately presents itself is whether
this machinery can be brought to bear on MV-SDE (1) with coloured noise. [13]
studied phase transitions using a small parameter expansion approach, to which
our methodology can be employed to fully understand the individual correction
terms.

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