# Symmetric Cooperative motion in one dimension

Louigi Addario-Berry  Department of Mathematics and Statistics, McGill
University<EMAIL_ADDRESS>,  Erin Beckman  Department of
Mathematics and Statistics, McGill University<EMAIL_ADDRESS>and
Jessica Lin  Department of Mathematics and Statistics, McGill University
<EMAIL_ADDRESS>
###### Abstract.

We explore the relationship between recursive distributional equations and
convergence results for finite difference schemes of parabolic partial
differential equations (PDEs). We focus on a family of random processes called
symmetric cooperative motions, which generalize the symmetric simple random
walk and the symmetric hipster random walk introduced in [2]. We obtain a
distributional convergence result for symmetric cooperative motions and, along
the way, obtain a novel proof of the Bernoulli central limit theorem. In
addition, we prove a PDE result relating distributional solutions and
viscosity solutions of the porous medium equation and the parabolic
$p$-Laplace equation, respectively, in one dimension.

###### Key words and phrases:

recursive distributional equations, monotone finite difference schemes,
monotone couplings

###### 2010 Mathematics Subject Classification:

Primary: 60F05, 60K35; Secondary: 65M12, 35K61, 35K92

##  1\. Introduction

###  1.1. Description of the model and the main result.

We consider a family of random processes called symmetric cooperative motions,
which generalize the symmetric simple random walk on $\mathbb{Z}$. Informally,
symmetric cooperative motion is a random walk where at each step, the walker
requires the assistance of $m$ other walkers (independent copies of the
process) in order to move. If all $m$ copies are at the same location as the
walker, the walker can take a step. Otherwise, it must stay put.

The model can be constructed in two ways. The first, which is the primary
construction we will work with, is as follows. Given an initial distribution
$\mu$ on $\mathbb{Z}\cup\\{-\infty,\infty\\}$, we define $\left(X_{n},n\geq
0\right)$ via the following recursive distributional equation (RDE):

$X_{n+1}:=\begin{cases}X_{n}+E_{n}&\mbox{ if
}X_{n}=X_{n,1}=\ldots=X_{n,m}\,,\\\ X_{n}&\mbox{ otherwise.}\end{cases}$ (1.1)

Here $(E_{n},n\geq 0)$ are IID with
${\mathbf{P}}\left(E_{0}=1\right)=1/2={\mathbf{P}}\left(E_{0}=-1\right)$, and
$X_{n,1},\ldots,X_{n,m}$ are IID copies of $X_{n}$. We set $-\infty+x=-\infty$
and $\infty+x=\infty$ for $x\in\mathbb{R}$; allowing the initial condition to
take values $\pm\infty$ will be useful in upcoming sections. We write
$\mathrm{SCM}(m,\mu)$ for the law of the process $(X_{n},n\geq 0)$ defined by
(1.1) when $X_{0}$ has distribution $\mu$.

The second construction interprets symmetric cooperative motion as a tree-
indexed random process. Let $\mathcal{T}$ be the complete rooted $(m+1)$-ary
tree, with root labeled by $\varnothing$ and node $v$ having children
$(vi,1\leq i\leq m+1)$. Let $\mathcal{T}_{n}$ denote the subtree of
$\mathcal{T}$ containing nodes at distance at most $n$ from the root, and let
$\mathcal{L}_{n}$ denote the leaves of $\mathcal{T}_{n}$.

In order not to cause confusion, we will use a different set of variables
(rather than $X_{n}$) to define the cooperative motion in this setting. Fix a
probability distribution $\mu$ on $\mathbb{Z}\cup\\{-\infty,\infty\\}$ and for
each $n\in\mathbb{N}$, let $\Sigma^{n}=(\Sigma^{n}_{v},v\in\mathcal{L}_{n})$
be a collection of IID, $\mu$-distributed random variables. Let
$E^{n}=(E_{v},v\in\mathcal{T}_{n}\setminus\mathcal{L}_{n})$ denote a separate
collection of IID Bernoulli($\tfrac{1}{2}$) random variables, independent of
$\Sigma^{n}$, and for $v\in\mathcal{T}_{n}\setminus\mathcal{L}_{n}$,
recursively define

$\Sigma^{n}_{v}:=\begin{cases}\Sigma^{n}_{v1}+E_{v}&\text{if
$\Sigma_{v1}=\Sigma_{v2}=\ldots=\Sigma_{v(m+1)}$},\\\
\Sigma^{n}_{v1}&\text{otherwise.}\end{cases}$

This recursion yields an “output value” $\Sigma^{n}_{\varnothing}$ at the
root, which has the same distribution as $X_{n}$ from the first construction.
(Note: the processes $(\Sigma_{\varnothing}^{n},n\geq 0)$ and $(X_{n},n\geq
0)$ have different laws, but their one-dimensional distributions are the
same.)

Returning to our notation $(X_{n},n\geq 0)$ as defined by (1.1), for
$p^{n}_{k}:={\mathbf{P}}\left(X_{n}=k\right)$, we have

$p^{n+1}_{k}=p^{n}_{k}(1-(p^{n}_{k})^{m})+\frac{1}{2}(p^{n}_{k+1})^{m+1}+\frac{1}{2}(p^{n}_{k-1})^{m+1}\,.$
(1.2)

The first term, $p^{n}_{k}(1-(p^{n}_{k})^{m})$, corresponds to the event that
$X_{n}=k$ and at least one of $X_{n,1},\ldots,X_{n,m}$ is different from $k$.
The second term corresponds to the event that $X_{n},X_{n,1},\ldots,X_{n,m}$
all equal $k+1$ and $E_{n}=-1$, and the third term corresponds to the event
that $X_{n},X_{n,1},\ldots,X_{n,m}$ all equal $k-1$ and $E_{n}=+1$.
Rearranging gives

$p^{n+1}_{k}-p^{n}_{k}-\frac{1}{2}\left((p^{n}_{k+1})^{m+1}-2(p^{n}_{k})^{m+1}+(p^{n}_{k-1})^{m+1}\right)=0.$

In the case when $m>0$ is non-integer, setting $p^{0}_{k}=\mu(\\{k\\})$ and
defining $p^{n}_{k}$ inductively using (1.2), it still holds that
$\sum_{k\in\mathbb{Z}}p^{n}_{k}=1$ for all $n$; we may therefore define a
sequence of random variables $(X_{n},n\geq 0)$ with
${\mathbf{P}}\left(X_{n}=k\right)=p^{n}_{k}$, and study the asymptotic
behaviour of $X_{n}$ even when $m$ is non-integer. However, in this case the
interpretations in terms of the recurrence (1.1) and in terms of the tree-
indexed process are unavailable.

Our main result describes the asymptotic distributional behaviour of $X_{n}$.

######  Theorem 1.1.

Fix $m>0$, and any probability distribution $\mu$ on $\mathbb{Z}$, and let
$(X_{n},n\geq 0)$ be $\mathrm{SCM}(m,\mu)$-distributed. Let

$\rho:=\frac{m}{2(m+2)}\quad\mbox{ and }\quad
D:=\rho^{-1/2}{\mathbf{B}}\left(\frac{1}{2},\frac{m+1}{m}\right)\,,$

where ${\mathbf{B}}(\cdot\,,\cdot)$ denotes the Beta Function. Then

$\frac{X_{n}}{n^{1/(m+2)}}\xrightarrow{d}\frac{2^{\frac{m+1}{m+2}}(m+1)^{1/(m+2)}}{D^{\frac{m}{m+2}}\rho^{1/2}}\left(B-\frac{1}{2}\right),$
(1.3)

where $B$ is
$\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)$-distributed.

As explained in [2], in the case $m=1$ there is a compelling heuristic
connection between the asymptotic behaviour of symmetric cooperative motion
and that of the critical random hierarchical lattice, a model of random
electrical network studied by Hambly and Jordan [12]. This connection yields
new predictions (but thus far, no proofs) for the scaling behaviour of the
effective resistance in such random networks.

### An aside: the $m\to 0$ limit of symmetric cooperative motion

This short section relates the asymptotic distribution of $X_{n}$ to that seen
for a symmetric simple random walk (SSRW); it is not necessary for the rest of
the paper.

Taking $m=0$ in the cooperative motion dynamics, the condition that
$X_{n}=X_{n,1}=\ldots=X_{n,m}$ becomes vacuous. Thus, the process makes a
random move at every step, and we recover the SSRW dynamics. In this sense,
one may say that at the level of processes, the $m\to 0$ limit of the
cooperative motion is just the symmetric simple random walk.

This $m\to 0$ limit is also reflected in the behaviour of the limiting
distributions. To see this, rewrite

$\displaystyle{}{}\frac{1}{D^{\frac{m}{m+2}}\rho^{1/2}}$
$\displaystyle\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$
$\displaystyle=\frac{\rho^{\frac{m}{2(m+2)}}}{\textbf{B}\left(\frac{1}{2},\frac{m+1}{m}\right)^{\frac{m}{m+2}}}\frac{1}{\rho^{1/2}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$
$\displaystyle=\frac{1}{\rho^{\frac{1}{m+2}}\textbf{B}\left(\frac{1}{2},\frac{m+1}{m}\right)^{\frac{m}{m+2}}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$

By Stirling’s formula, since $\frac{m+1}{m}\xrightarrow[m\to 0]{}\infty$, we
have

$\textbf{B}\left(\frac{1}{2},\frac{m+1}{m}\right)=\left[1+o(1)\right]\Gamma\left(\frac{1}{2}\right)\sqrt{\frac{m}{m+1}}.$

Continuing from the prior computation, and using the definition of $\rho$, we
then have

$\displaystyle\frac{1}{D^{\frac{m}{m+2}}\rho^{1/2}}$
$\displaystyle\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$
$\displaystyle=\frac{1+o(1)}{\left[\frac{m}{2(m+2)}\right]^{\frac{1}{m+2}}\left[\Gamma\left(\frac{1}{2}\right)\sqrt{\frac{m}{m+1}}\right]^{\frac{m}{m+2}}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right].$

We thus obtain the distributional limit

$\displaystyle\frac{2^{\frac{m+1}{m+2}}(m+1)^{1/(m+2)}}{D^{\frac{m}{m+2}}\rho^{1/2}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$
$\displaystyle=\left[1+o(1)\right]\frac{2^{\frac{m+1}{m+2}}(m+1)^{1/(m+2)}}{\left[\frac{m}{2(m+2)}\right]^{\frac{1}{m+2}}\left[\Gamma\left(\frac{1}{2}\right)\sqrt{\frac{m}{m+1}}\right]^{\frac{m}{m+2}}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$
$\displaystyle=\left[1+o(1)\right]2^{\frac{m+1}{m+2}}\left[2(m+2)\right]^{\frac{1}{m+2}}\sqrt{\frac{m+1}{m}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right]$
$\displaystyle\stackrel{{\scriptstyle d}}{{\longrightarrow}}\mathcal{W}_{1}$

as $m\to 0$, where $\mathcal{W}_{1}$ is a standard Gaussian; here we have used
the fact that

$\sqrt{8n}\left[\operatorname{Beta}(n,n)-\frac{1}{2}\right]\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{W}_{1}$

as $n\to\infty$. (We use the notation $\mathcal{W}_{1}$ as later
$\mathcal{W}_{t}$ will denote a centered Gaussian with variance $t$.)

###  1.2. The main ideas, related work, and an overview of the paper.

We begin with a brief introduction of the main ideas of the proof of Theorem
1.1. Our proof technique is based on the observation that the recurrence (1.2)
looks like a discrete approximation of a PDE: the porous medium equation

$u_{t}-\frac{1}{2}(u^{m+1})_{xx}=0\,.$ (1.4)

Letting $F^{n}_{k}:={\mathbf{P}}\left(X_{n}\leq
k\right)=\sum_{j=-\infty}^{k}p^{n}_{j}$, (1.2) then also yields a recurrence
relation for $F^{n}_{k}$. Indeed, we have

$\displaystyle F^{n+1}_{k}$ $\displaystyle=\sum_{j=-\infty}^{k}p^{n+1}_{j}$
$\displaystyle=\sum_{j=-\infty}^{k}p^{n}_{j}(1-(p^{n}_{j})^{m})+\frac{1}{2}(p^{n}_{j+1})^{m+1}+\frac{1}{2}(p^{n}_{j-1})^{m+1}$
$\displaystyle=F^{n}_{k}+\sum_{j=-\infty}^{k}-(p^{n}_{j})^{m+1}+\frac{1}{2}(p^{n}_{j+1})^{m+1}+\frac{1}{2}(p^{n}_{j-1})^{m+1}$
$\displaystyle=F^{n}_{k}+\sum_{j=-\infty}^{k}\left(\frac{1}{2}\left[(p^{n}_{j+1})^{m+1}-(p^{n}_{j})^{m+1}\right]-\frac{1}{2}\left[(p^{n}_{j})^{m+1}-(p^{n}_{j-1})^{m+1}\right]\right)$
$\displaystyle=F^{n}_{k}+\frac{1}{2}(p^{n}_{k+1})^{m+1}-\frac{1}{2}(p^{n}_{k})^{m+1}.$

Rearranging and using that $p^{n}_{k}=F^{n}_{k}-F^{n}_{k-1}$, we obtain

$F^{n+1}_{k}-F^{n}_{k}-\frac{1}{2}\left[(F^{n}_{k+1}-F^{n}_{k})^{m+1}-(F^{n}_{k}-F^{n}_{k-1})^{m+1}\right]=0.$
(1.5)

We can think of this as a discretization of the PDE

$v_{t}-\frac{1}{2}\left((v_{x})^{m+1}\right)_{x}=0\,,$ (1.6)

or equivalently $v_{t}-\tfrac{m+1}{2}v_{x}^{m}v_{xx}=0$. When $v_{x}\geq 0$,
this PDE can be rewritten in the form

$v_{t}-\frac{1}{2}\left(|v_{x}|^{m}v_{x}\right)_{x}=0\,,$ (1.7)

or $v_{t}-\tfrac{m+1}{2}|v_{x}|^{m}v_{xx}=0$, which is called the parabolic
$p$-Laplace equation. We can think of (1.6) as an “integrated version” of the
porous medium equation, since if $u$ solves (1.4) in the classical sense, then
the antiderivative

$v(x,t):=\int_{-\infty}^{x}u(y,t)\,dy$ (1.8)

formally solves (1.6). We will focus our analysis primarily on (1.5) since the
integration in (1.8) leads us to expect that $v$ should have an additional
(higher) order of regularity in space compared to $u$.

As in our previous paper [1], our main approach to establish distributional
convergence will be to use convergence results for finite difference
approximation schemes of nonlinear PDEs in order to demonstrate convergence of
the rescaled CDFs to a continuous function. In the setting of this paper, the
CDFs of the rescaled process

$\left(\frac{X_{t}}{n^{1/(m+2)}},0\leq t\leq n\right)\,,$ (1.9)

converge to the solution of an initial value problem corresponding to the PDEs
(1.6) and/or (1.7) in the viscosity sense.

At this point, a word about initial conditions is in order. If the random
variable $X_{0}$ is $\mu$-distributed then the “initial condition” of the
sequence of CDFs is given by $F^{0}_{k}=\mathbf{P}(X_{0}\leq
k)=\mu(-\infty,k]$. However, if we rescale as in (1.9), then the initial
condition of the rescaled process is $X_{0}/n^{1/(m+2)}$, and the CDF of this
random variable approaches a Heaviside function as $n\to\infty$. For this
reason, to connect the probabilistic evolution with that of PDE (1.7), a
Heaviside initial condition for the PDE is required.

In [1], we considered a variant of the above dynamics, where the random
variables $(E_{n},n\geq 0)$ are not symmetric, but instead have a bias to the
right or left. In the non-symmetric case, the recurrence relation for
$F^{n}_{k}$ looks like a discretization of a different PDE known as a
Hamilton-Jacobi equation, $v_{t}+\sigma|v_{x}|^{m+1}=0$, for an appropriate
value $\sigma$ depending on the parameters in the distribution of the random
walk steps. In the symmetric case which we address here, several challenges
arise compared to the setting of [1], of which the following two are notable:

  * •

The theory of lower semicontinuous viscosity solutions, which was required
throughout [1] to handle the Heaviside initial condition appearing in the
limit, does not exist for degenerate parabolic equations. While there have
been some attempts to develop a theory of discontinuous viscosity solutions
for parabolic equations such as (1.7) (see for example [14]), these theories
generally do not produce unique solutions.

  * •

In [1], we frequently used the Hopf-Lax formula to understand properties of
solutions to the relevant equations. However, in this setting, there is no
natural, physically motivated, explicit representation formula for a
“candidate viscosity solution” of (1.7) with Heaviside initial conditions.

To overcome these challenges, we use a combination of techniques from
nonlinear PDEs to approximate, via continuous viscosity solutions, what we
believe to be the “physically correct” viscosity solution of the parabolic
$p$-Laplace equation with Heaviside initial conditions. Our approach is
essentially a classical “vanishing viscosity” argument, which was the original
motivation for the definitions appearing in the modern theory of viscosity
solutions. It turns out that this is enough to identify the limiting behaviour
of the rescaled symmetric cooperative motions, as we are able to obtain an
explicit formula for the continuous viscosity solutions which approximate our
desired solution. A key ingredient of our argument is to rigorously relate
solutions of the porous medium equation (1.4) and those of the parabolic
$p$-Laplace equation (1.7). In essence, we prove that $u$ is a distributional
solution of (1.4) if and only if $v$ is a viscosity solution of (1.7) (see
Theorem 3.1 for a more precise statement). To our knowledge, this is a novel
result for degenerate parabolic equations in one spatial dimension.

This paper is part of a broader attempt to understand the relationship between
RDEs and convergence results for finite difference schemes of PDEs. We focus
on illustrating our method in the proof of Theorem 1.1, and as an expository
example,we use our techniques to provide a novel proof of the Bernoulli
central limit theorem in Section 2.2. Our approach builds upon that used in
our prior work [1], but we believe we now have a clearer perspective on the
method and its robustness. In Section 7.1 we present a framework and summary
of how to use this method to obtain distributional convergence results for
other discrete-time, $\mathbb{R}$-valued random processes.

The structure of the remainder of the paper is as follows. In Section 2, we
present a convergence result of Barles and Souganidis [3] for finite
difference schemes of degenerate parabolic PDEs; this will be the main
technical input from the theory of numerical approximation of PDEs used
throughout our paper. In order to illustrate the main ideas we explain how our
approach applies in the setting of simple random walk in Section 2.2; this
yields a novel proof of the Bernoulli central limit theorem. Section 2.3 then
contains a discussion of the challenges in adapting this method to symmetric
cooperative motion. In Section 3, we establish the ingredients from PDEs which
will be needed for our analysis. These include a discussion of the Zakharov-
Kuznetsov-Burgers (ZKB) solution of the porous medium equation, as well as the
statement and proof of Theorem 3.1, which relates distributional solutions and
viscosity solutions of certain degenerate parabolic PDEs. This section relies
on Appendix A, in which we review the notions of PDE solutions we work with
throughout the paper. Sections 4, 5 and 6 then contain the key technical
ingredients, culminating in the proof of Theorem 1.1 in Section 6.2. We
conclude our paper in Section 7 with a “recipe” for how to apply our method in
other settings. This section also presents several additional results whose
proofs are straightforward adaptations of those of Theorem 1.1 and of
analogous results from [1].

###  1.3. Definitions and Notation

Given a Euclidean space $\mathcal{X}$, we use $L^{p}(\mathcal{X})$ to denote
the classical $L^{p}$ space equipped with the norm
$\left|\left|f\right|\right|_{L^{p}(\mathcal{X})}=\left(\int_{\mathcal{X}}|f|^{p}\right)^{1/p}.$
We say that $f\in C^{k}(\mathcal{X})$ for $k\in\mathbb{N}$ if $f$ is $k$-th
order continuously differentiable, and $f\in C^{k}(\mathcal{X})$ for
$k\in(0,\infty)$ if $f\in C^{\lfloor k\rfloor}(\mathcal{X})$ and the $\lfloor
k\rfloor$-th order derivative belongs to the Hölder space $C^{k-\lfloor
k\rfloor}(\mathcal{X})$. We reserve the notation $f\in C^{k,j}(\mathcal{X})$,
for $k,j\in(0,\infty)$, whenever $f$ depends on space and time (so, e.g.,
$\mathcal{X}=\mathbb{R}\times[0,T]$), with $f\in C^{k}$ in the spatial
variables and $f\in C^{j}$ in the time variable. We denote $f\in
C^{\infty}_{c}(\mathcal{X})$ if $f$ is an infinitely differentiable function
with compact support. We say that
$u:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{R}$ belongs to
$C([0,T];L^{p}(\mathbb{R}))$ if $u(\cdot,t)\in L^{p}(\mathbb{R})$ for every
$t\in[0,T]$ and $u(\cdot,t)$ is continuous in $t$ with respect to the
$L^{p}(\mathbb{R})$-norm.

##  2\. The Barles-Souganidis approximation framework

###  2.1. Introducing the framework

The main external input to our proof is a result of Barles and Souganidis [3]
on the convergence of approximation schemes of viscosity solutions for second-
order PDEs, and the goal of this section is to present that result (Theorem
2.1, below) in the restricted setting we use here. The definition of viscosity
solutions, and facts about viscosity solutions relevant to this paper and in
particular to Theorem 2.1, appear in Appendix A.

We consider initial value problems (IVPs) of the following form

$\begin{cases}v_{t}+G(v_{x},v_{xx})=0&\text{in
$\mathbb{R}\times(0,\infty)$,}\\\ v(x,0)=v_{0}(x)&\text{in
$\mathbb{R}$}\,,\end{cases}$ (2.1)

where $G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ and
$v_{0}:\mathbb{R}\to\mathbb{R}$.We refer to $v_{0}$ as the initial condition
of (2.1). We say that the initial value problem (2.1) is good if the following
conditions hold:

  1. (1)

$G$ is continuous.

  2. (2)

$G$ is degenerate elliptic, i.e., $G(p,a)\leq G(p,b)$ if $a\geq b$, for all
$p\in\mathbb{R}$.

  3. (3)

$v_{0}$ is bounded and uniformly continuous.

Under these conditions, it is known (see [8], Section 5 and the discussion on
page 50) that the initial value problem satisfies strong uniqueness: there
exists a unique continuous viscosity solution $v$ of (2.1), and moreover, if
$v^{-}$ is any upper semicontinuous subsolution of (2.1) and $v^{+}$ is any
lower semicontinuous supersolution of (2.1), then $v^{-}\leq v^{+}$.

We next consider approximation schemes for (2.1). For each $N\in(0,\infty)$,
fix time and space mesh sizes $\Delta_{t}^{N},\Delta_{x}^{N}>0$ which, in our
applications, will always go to $0$ as $N\to\infty$, and fix a function
$\mathcal{G}^{N}:\mathbb{R}^{3}\to\mathbb{R}$. Introducing the abbreviation
$\langle
f(x,t)\rangle_{N}:=(f(x-\Delta_{x}^{N},t),f(x,t),f(x+\Delta_{x}^{N},t))$, we
define a finite difference scheme for (2.1) by

$\begin{cases}\frac{v^{N}(x,t+\Delta_{t}^{N})-v^{N}(x,t)}{\Delta_{t}^{N}}+\mathcal{G}^{N}\langle
v^{N}(x,t)\rangle_{N}=0,&\text{in
$\mathbb{R}\times[\Delta_{t}^{N},\infty)$}\,,\\\
v^{N}(x,0)=v_{0}\big{(}\big{\lfloor}\frac{x}{\Delta_{x}^{N}}\big{\rfloor}\Delta_{x}^{N}\big{)},&\text{in
$\mathbb{R}\times[0,\Delta_{t}^{N})$}\,.\end{cases}$ (2.2)

We also refer to $v_{0}$ as the initial condition of (2.2). For each $N>0$,
once we choose the initial condition $v_{0}$, the approximation scheme (2.2)
uniquely determines the function
$v^{N}:\mathbb{R}\times[0,\infty)\to\mathbb{R}$. However, in what follows we
will often need to compare the functions defined by (2.2) for various initial
conditions. As such, we write $u^{N},v^{N}$ and $w^{N}$ to indicate the
functions defined by (2.2) when started from initial conditions
$u_{0},v_{0},w_{0}:\mathbb{R}\to\mathbb{R}$, respectively.

Now fix a family $\mathcal{C}\subset L^{\infty}(\mathbb{R})$ with
$v_{0}\in\mathcal{C}$. We say that (2.2) is a good approximation scheme for
(2.1) on $\mathcal{C}$ if the following conditions hold.

  1. (1)

Monotonicity: For all $N$ sufficiently large and all
$u_{0},w_{0}\in\mathcal{C}$,

$u_{0}\leq w_{0}\mbox{ on }\mathbb{R}\Rightarrow u^{N}\leq w^{N}\mbox{ on
}\mathbb{R}\times[0,\infty)\,.$ (2.3)

  2. (2)

Stability:
$\limsup_{N\to\infty}\sup_{(x,t)\in\mathbb{R}\times(0,\infty)}|v^{N}(x,t)|<\infty$.

  3. (3)

Consistency: If $\varphi:\mathbb{R}\times(0,\infty)\to\mathbb{R}$ is any
bounded smooth function, then for all $(x,t)\in\mathbb{R}\times(0,\infty)$,

$\lim_{(N,y,s,\varepsilon)\to(\infty,x,t,0)}\mathcal{G}^{N}\langle\varphi(y,s)+\varepsilon\rangle_{N}=G(\varphi_{x}(x,t),\varphi_{xx}(x,t))\,.$

In the last point, we use the notation
$\lim_{(N,y,s,\varepsilon)\to(\infty,x,t,0)}$ to indicate that the limit may
be taken in any order.

The first point, the monotonicity condition, plays a crucial role in our
analysis. We note that since $u^{N}$ and $w^{N}$ are piecewise constant in
time, to establish monotonicity it is enough to verify that
$u^{N}(\cdot,n\Delta_{t}^{N})\leq w^{N}(\cdot,n\Delta_{t}^{N})$ for all
$n\in\mathbb{N}$. For the case $n=1$, note that by (2.2) applied to $w^{N}$,
we have

$w^{N}(x,\Delta_{t}^{N})=w^{N}(x,0)-\Delta_{t}^{N}\mathcal{G}^{N}(w^{N}(x-\Delta_{x}^{N},0),w^{N}(x,0),w^{N}(x+\Delta_{x}^{N},0))\,.$

So to prove that $u^{N}(\cdot,\Delta_{t}^{N})\leq
w^{N}(\cdot,\Delta_{t}^{N})$, it suffices to show that the map

$(a,b,c)\mapsto b-\Delta_{t}^{N}\mathcal{G}^{N}(a,b,c)$ (2.4)

is nondecreasing in all of its arguments, for all arguments which arise from
the family $\mathcal{C}$ (with
$a=w^{N}(x-\Delta_{x}^{N},0),b=w^{N}(x,t),c=w^{N}(x+\Delta_{x}^{N},0)$). To
extend beyond $n=1$, if we can further show that $u^{N}(\cdot,\Delta_{t}^{N})$
and $w^{N}(\cdot,\Delta_{t}^{N})$ both belong to $\mathcal{C}$, then by
induction $u^{N}(\cdot,n\Delta_{t}^{N})\leq w^{N}(\cdot,n\Delta_{t}^{N})$ for
all $n\in\mathbb{N}$, and thus $u^{N}\leq w^{N}$ pointwise, so (2.3) holds.
This is the general strategy by which we shall verify (2.3) throughout the
rest of the paper.

The result of Barles and Souganidis says that good approximation schemes for
good initial value problems converge to the unique viscosity solutions of
those initial value problems. In fact, their result applies to a broader
family of PDEs and approximation schemes than the ones considered here; we
have specialized their result, in order to make it easier to explain the
application to our setting.

######  Theorem 2.1.

[Theorem 1, [3]] Fix a family $\mathcal{C}\subset L^{\infty}(\mathbb{R})$, and
a function $v_{0}\in\mathcal{C}$. Consider a good initial value problem of the
form (2.1) started from initial condition $v_{0}$, and let $v$ be its unique
viscosity solution. Next, fix an approximation scheme (2.2) which is a good
approximation scheme for (2.1) on $\mathcal{C}$, and for $N>0$ let $v^{N}$ be
the solution of (2.2) with initial condition $v_{0}$. Then $v^{N}\to v$
locally uniformly as $N\to\infty$.

###  2.2. A case-study: the central limit theorem for simple random walk.

In order to understand how Theorem 2.1 can be used to prove distributional
limit theorems, we start with a simple example: in this section, we show how
Theorem 2.1 can be applied to prove the Bernoulli central limit theorem.

Let $(S_{n},n\geq 0)$ be a symmetric simple random walk on $\mathbb{Z}$, so
$S_{n+1}=S_{n}\pm 1$, each with equal probability. Defining
$p^{n}_{k}:=\mathbf{P}(S_{n}=k)$ and $F^{n}_{k}:=\mathbf{P}(S_{n}\leq k)$, we
then have

$p^{n+1}_{k}=\frac{1}{2}p^{n}_{k-1}+\frac{1}{2}p^{n}_{k+1}.$

Summing over $j\leq k$ and rearranging gives

$F^{n+1}_{k}-F^{n}_{k}=\frac{1}{2}(F^{n}_{k-1}-2F^{n}_{k}+F^{n}_{k+1})\,.$
(2.5)

This recursion suggests that the appropriate PDE to describe the limiting
behavior of the CDF is the heat equation $u_{t}=\tfrac{1}{2}u_{xx}$. With this
in mind, we fix the family of functions
$\mathcal{C}=\\{f:\mathbb{R}\to[0,1]:f\mbox{ is a cumulative distribution
function}\\}$, fix $\varepsilon>0$ small, and let

$v_{0}(x)=\frac{1}{(2\pi\varepsilon)^{1/2}}\int_{-\infty}^{x}e^{-y^{2}/(2\varepsilon)}\,dy.$
(2.6)

Consider the initial value problem

$\begin{cases}v_{t}-\frac{1}{2}v_{xx}=0&\text{in
$\mathbb{R}\times(0,\infty)$,}\\\ v(x,0)=v_{0}(x)&\text{in
$\mathbb{R}$}\,.\end{cases}$ (2.7)

Using the notation of (2.1), we have here $G(x,y)=-(1/2)y$. It is clear that
$G$ is both continuous and degenerate elliptic. The initial condition,
$v_{0}$, is an antiderivative of a Gaussian density, so it is both bounded and
uniformly continuous and belongs to the class $\mathcal{C}$. Therefore, (2.7)
is a good initial value problem.

Next we define the approximation scheme which arises in this setting. We let
$\Delta_{x}^{N}=1/N$ and $\Delta_{t}^{N}=1/N^{2}$, and set

$\mathcal{G}^{N}(a,b,c)=-\frac{1}{\Delta_{t}^{N}}\frac{1}{2}(a-2b+c).$ (2.8)

Then define $v^{N}$ via the scheme (2.2), which we restate here for
convenience:

$\begin{cases}\frac{v^{N}(x,t+\Delta_{t}^{N})-v^{N}(x,t)}{\Delta_{t}^{N}}+\mathcal{G}^{N}\langle
v^{N}(x,t)\rangle_{N}=0,&\text{in
$\mathbb{R}\times[\Delta_{t}^{N},\infty)$}\,,\\\
v^{N}(x,0)=v_{0}\big{(}\big{\lfloor}\frac{x}{\Delta_{x}^{N}}\big{\rfloor}\Delta_{x}^{N}\big{)},&\text{in
$\mathbb{R}\times[0,\Delta_{t}^{N})$}\,.\end{cases}$ (2.9)

For $t\geq\Delta_{t}^{N}$, this yields

$\displaystyle\frac{v^{N}(x,t+N^{-2})-v^{N}(x,t)}{N^{-2}}$
$\displaystyle=\frac{v^{N}(x,t+\Delta_{t}^{N})-v^{N}(x,t)}{\Delta_{t}^{N}}$
$\displaystyle=\frac{1}{\Delta_{t}^{N}}\frac{1}{2}(v^{N}(x-\Delta_{x}^{N},t)-2v^{N}(x,t)+v^{N}(x+\Delta_{x}^{N},t))$
$\displaystyle=\frac{1}{N^{-2}}\frac{1}{2}(v^{N}(x-1/N,t)-2v^{N}(x,t)+v^{N}(x+1/N,t))\,.$
(2.10)

This is simply a rescaling of the recurrence (2.5). Thus, writing
$\mathbf{P}_{N}$ for the measure under which $(S_{n},n\geq 0)$ is a symmetric
simple random walk with initial distribution given by

$\mathbf{P}_{N}(S_{0}\leq
k)=v^{N}(k\Delta_{x}^{N},0)=\frac{1}{\sqrt{2\pi\varepsilon}}\int_{-\infty}^{\frac{k}{N}}e^{-y^{2}/(2\varepsilon)}dy\,,$
(2.11)

then for all $n\in\mathbb{N}$ and $k\in\mathbb{Z}$ we have

$\mathbf{P}_{N}(S_{n}\leq
k)=F^{n}_{k}=v^{N}(k/N,n/N^{2})=v^{N}(k\Delta_{x}^{N},n\Delta_{t}^{N})\,.$
(2.12)

We now verify that with $\mathcal{G}^{N}$ as in (2.8), the scheme (2.9) is a
good approximation scheme of the PDE (2.7). Fix initial conditions
$u_{0},w_{0}\in\mathcal{C}$ and let $u^{N},w^{N}$ be defined by (2.9) with the
initial condition $v_{0}$ replaced by $u_{0},w_{0}$, respectively. Using that
$\Delta_{t}^{N}=N^{-2}$, we have that for any $a,b,c\in\mathbb{R}$,

$b-\Delta_{t}^{N}\mathcal{G}^{N}(a,b,c)=b+\frac{1}{2}(a-2b+c)=\frac{1}{2}(a+c)\,,$

so the map $(a,b,c)\mapsto b-\Delta_{t}^{N}\mathcal{G}^{N}(a,b,c)$ is
nondecreasing in all its arguments. It follows that
$u^{N}(\cdot,\Delta_{t}^{N})\leq w^{N}(\cdot,\Delta_{t}^{N})$. Moreover, by
the analogues of (2.12) for $u^{N}$ and $w^{N}$, the functions
$u^{N}(\cdot,\Delta_{t})$ and $w^{N}(\cdot,\Delta_{t})$ are again cumulative
distribution functions, which lie in $\mathcal{C}$. It follows by induction
that $u^{N}\leq w^{N}$ everywhere; this establishes the monotonicity of the
scheme. (Note that, in view of (2.12), the monotonicity of the scheme is in
fact equivalent to the statement that for simple random walk, if $S_{0}\preceq
S_{0}^{\prime}$, then also $S_{1}\preceq S_{1}^{\prime}$. We could have used
this to give a “purely probabilistic” proof of monotonicity, but we preferred
to spell out the inductive approach to provide an example of its use in a
simple setting. However, the connection between monotonicity and preservation
of stochastic ordering under the random process dynamics will reoccur later in
the paper.)

Next, the relation (2.12) implies in particular that the codomain of $v^{N}$
is contained within $[0,1]$, from which stability is immediate.

Finally, if $\varphi:\mathbb{R}\times(0,\infty)\to\mathbb{R}$ is a bounded
smooth function, then using that $\Delta_{x}^{N}=1/N$, we have

$\displaystyle\mathcal{G}^{N}\langle\varphi(y,s)+\varepsilon\rangle_{N}$
$\displaystyle=-\frac{1}{N^{-2}}\left(\frac{1}{2}(\varphi(y-1/N,s)+\varepsilon-2(\varphi(y,s)+\varepsilon)+\varphi(y+1/N,s)+\varepsilon)\right)$
$\displaystyle=-\frac{1}{2}\frac{\varphi(y-1/N,s)-2\varphi(y,s)+\varphi(y+1/N,s)}{(1/N)^{2}}.$

This is a second centred difference approximation of
$-\tfrac{1}{2}\varphi_{xx}(y,s)$, which converges to
$-\tfrac{1}{2}\varphi_{xx}(x,t)$ as $(N,y,s)\to(\infty,x,t)$ in any order
since $\varphi$ is smooth. (The $\varepsilon\to 0$ limit is irrelevant due to
the linear role of $\varepsilon$ in the approximation scheme.)

It follows that with $\mathcal{G}^{N}$ defined as in (2.8), the approximation
scheme defined by (2.9) is indeed a good approximation scheme for the
corresponding initial value problem (2.7). The (unique, continuous) viscosity
solution of the heat equation (2.7) is

$v(x,t)=\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi(\varepsilon+t)}}e^{-y^{2}/(2(\varepsilon+t))}dy\,.$

Since $v^{N}(k/N,n/N^{2})=\mathbf{P}_{N}(S_{n}\leq k)$, taking $n=N^{2}$ and
$k=xN$, and ignoring inconsequential rounding issues, we can now apply Theorem
2.1 to see that

$\mathbf{P}_{N}(S_{N^{2}}\leq
xN)\to\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi(\varepsilon+1)}}e^{-y^{2}/(2(\varepsilon+1))}dy=\mathbf{P}(\mathcal{W}_{\varepsilon+1}\leq
x)\,,$ (2.13)

where $\mathcal{W}_{\varepsilon+1}$ is a centred Gaussian with variance
$\varepsilon+1$.

The proof above required the use of a smooth initial condition $v_{0}$ in
order to apply the result of Barles and Souganidis. However,
probabilistically, we are interested in a simple random walk $S^{\prime}_{n}$
which is started from 0. Because the Heaviside function cannot be thought of
as the discretization of any Lipschitz initial condition, the result of
Theorem 2.1 is not applicable directly. We use the following coupling argument
to connect a simple random walk $S_{n}$ started from a discretization of
$v_{0}$, as defined above, and the simple random walk $S^{\prime}_{n}$ started
from 0.

Fix $\delta>0$, and let $(S_{n}^{\prime},n\geq 0)$ be a simple random walk
with initial condition $S_{0}^{\prime}=0$. The random walks
$(S_{n}^{\prime},n\geq 0)$ and $(S_{n},n\geq 0)$ can be coupled so that both
the increments of both walks are equal (i.e. so that
$S_{n+1}^{\prime}-S_{n}^{\prime}=S_{n+1}-S_{n}$ for all $n\geq 0$). Under such
a coupling, we have $S_{n}-S_{n}^{\prime}=S_{0}-S_{0}^{\prime}=S_{0}$ for all
$n\geq 0$. It follows that if $S_{N^{2}}^{\prime}\leq xN$ then either
$S_{N^{2}}\leq(x+\delta)N$ or else $S_{0}>\delta N$, so

$\displaystyle\mathbf{P}(S_{N^{2}}^{\prime}\leq xN)$
$\displaystyle\leq\mathbf{P}_{N}(S_{N^{2}}\leq(x+\delta)N)+\mathbf{P}_{N}(S_{0}\geq\delta
N)\,.$

By (2.11) we have $\lim_{N\to\infty}\mathbf{P}_{N}(S_{0}\geq\delta
N)=\mathbf{P}(\mathcal{W}_{\varepsilon}>\delta)$. Together with (2.13) this
yields that

$\limsup_{N\to\infty}\mathbf{P}(S_{N^{2}}^{\prime}\leq
xN)\leq\mathbf{P}(\mathcal{W}_{\varepsilon+1}\leq
x+\delta)+\mathbf{P}(\mathcal{W}_{\varepsilon}>\delta)\,.$

The identity $S_{n}-S_{n}^{\prime}=S_{0}$ likewise implies that if
$S_{N^{2}}\leq(x-\delta)N$ then either $S_{0}\leq-\delta N$ or
$S^{\prime}_{N^{2}}\leq xN$, so

$\displaystyle\mathbf{P}(S_{N^{2}}^{\prime}\geq xN)$
$\displaystyle\leq\mathbf{P}_{N}(S_{N^{2}}\geq(x-\delta)N)-\mathbf{P}_{N}(S_{0}\leq-\delta
N)\,,$

from which it similarly follows that

$\liminf_{N\to\infty}\mathbf{P}(S_{N^{2}}^{\prime}\leq
xN)\geq\mathbf{P}(\mathcal{W}_{\varepsilon+1}\leq
x-\delta)-\mathbf{P}(\mathcal{W}_{\varepsilon}<-\delta)\,.$

These bounds hold for all $\varepsilon,\delta>0$. (We’ll later refer to this
sort of argument as a sandwiching argument.)

The first probabilities on the right in the preceding $\limsup/\liminf$
equations approximate $\mathbf{P}(\mathcal{W}_{1}\leq x)$ for
$\varepsilon,\delta$ small, and the second probabilities may be made as small
as we like by choosing $\varepsilon$ small as a function of $\delta$. We thus
conclude that

$\lim_{N\to\infty}\mathbf{P}(S_{N^{2}}^{\prime}\leq
xN)=\mathbf{P}(\mathcal{W}_{1}\leq x),$

where $\mathcal{W}_{1}$ is a centred Gaussian of variance 1. This is
equivalent to the Bernoulli central limit theorem.

###  2.3. The Barles-Souganidis framework and cooperative motion

Moving from the heat equation $u_{t}-\tfrac{1}{2}u_{xx}=0$ to the porous
medium equation $u_{t}-\tfrac{1}{2}(u^{m+1})_{xx}=0$, or its integrated
version, the parabolic $p$-Laplace equation (1.6), makes the application of
the Barles-Souganidis convergence framework more delicate. There are three
issues, two related to monotonicity and one to the relevant PDE theory, which
we now discuss.

First, the monotonicity required by Theorem 2.1 was obvious in the simple
random walk/heat equation setting, and indeed held for the broadest natural
class $\mathcal{C}$ of initial conditions: all CDFs. For cooperative motion,
monotonicity simply does not hold in such generality, and to apply Theorem 2.1
we instead restrict our attention to the class $\mathcal{C}$ of Lipschitz
CDFs. This only allows us to prove an approximation result (and hence
convergence in distribution) for sequences of cooperative motion processes
whose initial CDFs arise as discretizations of Lipschitz functions.

The second issue relates to the use of (stochastic) monotonicity in the
sandwiching argument. In Section 2.2 we used that simple random walks may be
coupled so that if $S_{0}\preceq S_{0}^{\prime}$, then this stochastic
ordering is maintained in time (by simply using the same increments for both
walks). However, the dynamics of cooperative motion are not stochastically
monotone for all initial distributions. That is, there exist initial
conditions $X_{0}\preceq X_{0}^{\prime}$ for the cooperative motion process so
that $X_{1}\not\preceq X_{1}^{\prime}$.

This may seem like a death knell for the approach, given that stochastic
monotonicity is required for the sandwiching argument. However, there is hope.
The fact is that while stochastic monotonicity may not hold for all initial
conditions of cooperative motion, it does hold for a large subclass of initial
conditions. We will show that if, for example, the initial distributions
$X_{0}\preceq X_{0}^{\prime}$ additionally satisfy that
$\sup_{k}\mathbf{P}(X_{0}=k)<1/e$ and
$\sup_{k}\mathbf{P}(X^{\prime}_{0}=k)<1/e$, then indeed $X_{1}\preceq
X_{1}^{\prime}$; the stochastic ordering is maintained. Moreover, under this
condition, it turns out that $\sup_{k}\mathbf{P}(X_{1}=k)<1/e$ and
$\sup_{k}\mathbf{P}(X_{1}=k)<1/e$, so the argument may be iterated in order to
show that $X_{n}\preceq X_{n}^{\prime}$ for all $n$. (Proving all this takes
some work and requires a more technical coupling technique; this is
accomplished in Section 5, below.)

To extend the result to arbitrary initial distributions, we prove that from
any initial condition, after a bounded number of steps, cooperative motion
reaches a state where all single-site probabilities are small enough that
stochastic monotonicity is obtained. (This fact feels unsurprising, and even
seems like it should be “obvious” – but the only proof we found is rather
involved.) These first steps introduce a bounded error, which is washed away
by the rescaling when we take limits.

The final issue, which poses a significant challenge in comparison to the
setting of the simple random walk, is to identify an explicit representation
for the viscosity solution of the parabolic $p$-Laplace equation with suitable
initial conditions (it is from this representation that we identify the
limiting random variable in the distributional convergence). This was
completely straightforward in the setting of the heat equation thanks to the
theory of fundamental solutions for linear parabolic PDEs. In the case of the
parabolic $p$-Laplace equation, no such theory exists. We overcome this
obstacle using PDE techniques and analysis, which may be of independent
interest. We introduce the main PDE ideas, and the explicit characterization
of the relevant solution of the limiting PDE in Section 3.

##  3\. The ZKB solution and its approximation.

This section presents the relevant PDE results needed to characterize the
behaviour of the limiting dynamics. As briefly explained in the introduction,
we seek an appropriate “viscosity solution” of the parabolic $p$-Laplace
equation (1.7) started from a Heaviside initial condition. Our approach
consists of building an approximation by continuous viscosity solutions in
order to characterize this limit. This approximation is achieved by
identifying the distributional solution of the porous medium equation (1.4)
with a suitable initial condition, known as the Zakharov-Kuznetsov-Burgers or
ZKB solution, and considering its antiderivative.

###  3.1. The ZKB solution.

The evolution of the probability mass functions
$(p^{n}_{k},k\in\mathbb{Z},n\geq 0)$ of the random variables $(X_{n},n\geq 0)$
is a discrete approximation of the porous medium equation (1.4). If we assume
that $u(x,0)=\delta_{0}(x)$ is a Dirac delta at zero, it turns out that the
“correct” solution (in a sense to be specified in Section 3.2) of (1.4)
corresponding to these dynamics is the ZKB solution (also called a source-type
solution, source solution, Barenblatt solution, or Barenblatt-Pattle
solution), which we now describe. Our presentation is based on that of [18,
Chapter 17]. That reference presents the ZKB solution of
$u_{t}-\Delta(u^{m+1})=0$ in $\mathbb{R}^{d}\times(0,\infty)$ for any
dimension $d\geq 1$, but we only present the solution for
$u_{t}-\frac{1}{2}(u^{m+1})_{xx}=0$ in $\mathbb{R}\times(0,\infty)$, as this
is all that is relevant for us.

For every $\theta>0$, we define
$\hat{U}(\cdot\,,\cdot\,;\theta):\mathbb{R}\times(0,\infty)\rightarrow\mathbb{R}$
by

$\hat{U}(x,t;\theta)=\frac{1}{t^{\frac{1}{m+2}}}\left[\left(\frac{\sqrt{2}\theta}{D\sqrt{m+1}}\right)^{\frac{2m}{m+2}}-\frac{2\rho|x|^{2}}{(m+1)t^{\frac{2}{m+2}}}\right]^{\frac{1}{m}}_{+}\,,$
(3.1)

with $\rho:=m/(2(m+2))$ as defined in Theorem 1.1 and
$D:=2\int_{0}^{\infty}(1-\rho y^{2})_{+}^{\frac{1}{m}}\,dy$; we will see
shortly this definition of $D$ also agrees with the one given in Theorem 1.1.
As explained in [18, Chapter 17, (17.31)], the choice of constants yields that
for every $t>0$,

$\int_{-\infty}^{\infty}\hat{U}(x,t;\theta)\,dx=\theta.$ (3.2)

In particular, the function $\hat{U}(\cdot\,,\cdot\,;1)$ has the explicit form

$\hat{U}(x,t;1)=\frac{1}{t^{\frac{1}{m+2}}}\left[\left(\frac{\sqrt{2}}{D\sqrt{m+1}}\right)^{\frac{2m}{m+2}}-\frac{2\rho|x|^{2}}{(m+1)t^{\frac{2}{m+2}}}\right]^{\frac{1}{m}}_{+}\,,$
(3.3)

which is supported on

$\left\\{|x|\leq\frac{(m+1)^{\frac{1}{m+2}}}{2^{\frac{1}{m+2}}\rho^{\frac{1}{2}}D^{\frac{m}{m+2}}}t^{\frac{1}{m+2}}\right\\}\,.$

Evaluating at $t=1$ yields

$\hat{U}(x,1;1)=\left[\left(\frac{\sqrt{2}}{D\sqrt{m+1}}\right)^{\frac{2m}{m+2}}-\frac{2\rho|x|^{2}}{(m+1)}\right]^{\frac{1}{m}}_{+}.$

Up to an affine change of variables, $\hat{U}(x,1;1)$ is a Beta density.
Indeed, defining

$\displaystyle f(x)$
$\displaystyle:=\hat{U}\left(x-(m+1)^{\frac{1}{m+2}}2^{-\frac{1}{m+2}}\rho^{-\frac{1}{2}}D^{-\frac{m}{m+2}},1;1\right)$
$\displaystyle=\left[\left(\frac{\sqrt{2}}{D\sqrt{m+1}}\right)^{\frac{2m}{m+2}}-\frac{2\rho}{(m+1)}\left|x-\frac{(m+1)^{\frac{1}{m+2}}}{2^{\frac{1}{m+2}}\rho^{\frac{1}{2}}D^{\frac{m}{m+2}}}\right|^{2}\right]^{\frac{1}{m}}_{+}$
$\displaystyle=\left(\frac{2\rho}{m+1}\right)^{\frac{1}{m}}\left(x\left(\frac{2\cdot(m+1)^{\frac{1}{m+2}}}{2^{\frac{1}{m+2}}\rho^{\frac{1}{2}}D^{\frac{m}{m+2}}}-x\right)\right)_{+}^{\frac{1}{m}},$

then $\int_{-\infty}^{\infty}f(x)dx=1$ by (3.2). Next, if $X$ is distributed
as

$\frac{2^{\frac{m+1}{m+2}}(m+1)^{\frac{1}{m+2}}}{D^{\frac{m}{m+2}}\rho^{\frac{1}{2}}}\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)\,,$

then the PDF of $X$ is given by

$\displaystyle\frac{1}{{\mathbf{B}}\left(\frac{m+1}{m},\frac{m+1}{m}\right)}\cdot\frac{1}{(2^{\frac{m+1}{m+2}}D^{-\frac{m}{m+2}}\rho^{-\frac{1}{2}}(m+1)^{\frac{1}{m+2}})^{\frac{m+2}{m}}}\left(x\left(\frac{2\cdot(m+1)^{\frac{1}{m+2}}}{2^{\frac{1}{m+2}}\rho^{\frac{1}{2}}D^{\frac{m}{m+2}}}-x\right)\right)_{+}^{\frac{1}{m}}\,.$
(3.4)

Since this expression integrates to $1$, as does $f(x)$, and the two
expressions are the same up to a multiplicative constant, they must in fact be
equal. (The equality of the two expressions may also be verified directly,
using that $D$ can be re-expressed as

$\displaystyle
D=2\rho^{-\frac{1}{2}}\int_{0}^{1}(1-y^{2})^{\frac{1}{m}}\,dy=2\rho^{-\frac{1}{2}}\int_{0}^{1}(1-y)^{\frac{1}{m}}\frac{1}{2y^{\frac{1}{2}}}\,dy$
$\displaystyle=\rho^{-\frac{1}{2}}{\mathbf{B}}\left(\frac{1}{2},\frac{m+1}{m}\right),$

along with the identity

$\Gamma(2z)=(2\pi)^{-1/2}2^{2z-1/2}\Gamma(z)\Gamma\left(z+\frac{1}{2}\right)\,$

and the fact that
${\mathbf{B}}(x,y)=\tfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$.) The fact that
we fixed time $t=1$ above is arbitrary; a similar argument shows that
$\hat{U}(x,t;1)$ is a (scaled, shifted) Beta density for any $t>0$.

###  3.2. The integrated ZKB solution, and the relation between distributional
solutions and viscosity solutions

For readers who are less familiar with the notions of distributional and
viscosity solutions, we provide an overview in Appendix A, which will be
helpful to review before reading this section.

The parabolic $p$-Laplace equation (1.7) is an integrated version of the
porous medium equation (1.4), and in our setting, the “correct” viscosity
solution of (1.6) and/or (1.7) is given by the antiderivative of the ZKB
solution for the porous medium equation. This follows from a general result we
prove relating distributional solutions and viscosity solutions in
1-dimension.

Before beginning our analysis, we remark that we will always impose hypotheses
which guarantee existence and uniqueness for viscosity/distributional
solutions. For the parabolic $p$-Laplace equation, as discussed near (2.1),
the existence and uniqueness of viscosity solutions is guaranteed for any good
IVP. For PDEs of the form

$\begin{cases}u_{t}-(\Psi(u))_{xx}=0&\text{in $\mathbb{R}\times(0,T]$},\\\
u(x,0)=u_{0}(x)&\text{in $\mathbb{R}$},\end{cases}$ (3.5)

where $\Psi:\mathbb{R}\rightarrow\mathbb{R}$ is nondecreasing, continuous, and
$\Psi(0)=0$, it follows from the main result of [5] that if $u_{0}\in
L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ is nonnegative, then such IVPs
have a unique distributional solution $u\in C([0,T];L^{1}(\mathbb{R}))\cap
L^{\infty}(\mathbb{R}\times[0,T])$. (See also [4] for related results.)
Throughout the rest of the paper, by “the unique distributional solution” of a
PDE of the form (3.5), we always mean the one which belongs to
$C([0,T];L^{1}(\mathbb{R}))\cap L^{\infty}(\mathbb{R}\times[0,T])$. (In the
special case of the porous medium equation with initial data $u_{0}\in
L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, the existence and uniqueness of
a distributional solution is also a consequence of the existence and
uniqueness of strong $L^{1}$ solutions [18, Theorem 9.3].)

We are now ready to present our main result concerning the connection between
distributional and viscosity solutions.

######  Theorem 3.1.

Fix $\beta\geq 1$ and any nondecreasing function $\Psi\in
C^{\beta}(\mathbb{R})$ with $\Psi(0)=0$. Fix $u_{0}\in L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$ and nonnegative, and for any $T>0$, let $u\in
C([0,T];L^{1}(\mathbb{R}))\cap L^{\infty}(\mathbb{R}\times[0,T])$ be the
unique distributional solution of

$\begin{cases}u_{t}-(\Psi(u))_{xx}=0&\text{in $\mathbb{R}\times(0,T]$,}\\\
u(x,0)=u_{0}(x)&\text{in $\mathbb{R}$.}\end{cases}$ (3.6)

Let $v(x,t):=\int_{-\infty}^{x}u(y,t)\,dy$. Then $v$ is the unique viscosity
solution of

$\begin{cases}v_{t}-\Psi^{\prime}(v_{x})v_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ v(x,0)=\int_{-\infty}^{x}u_{0}(y)\,dy&\text{in
$\mathbb{R}$.}\end{cases}$ (3.7)

A version of the correspondence described in Theorem 3.1 was heuristically
presented in [18] in the special case of the porous medium equation and the
parabolic $p$-Laplace equation. The proof of Theorem 3.1 appears in Section
3.3, below.

######  Remark 3.2.

We note that the uniqueness theory of both distributional and viscosity
solutions yields that the converse of Theorem 3.1 essentially also holds.
Indeed, if under the same hypotheses on $\Psi$, we work from the premise that
$v$ is the unique viscosity solution of

$\begin{cases}v_{t}-\Psi^{\prime}(v_{x})v_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ v(x,0)=v_{0}(x)&\text{in
$\mathbb{R}$,}\end{cases}$

with $v_{0}$ Lipschitz continuous, nonnegative, and bounded, then we may
always express $v_{0}(x)=\int_{-\infty}^{x}v_{0}^{\prime}(y)\,dy$, where
$v_{0}^{\prime}$ is defined pointwise a.e. by Rademacher’s Theorem. Taking
$u_{0}(x)=v_{0}^{\prime}(x)$, then $u_{0}\in L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$, and letting $u$ denote the unique distributional
solution of (3.6) with this initial condition, Theorem 3.1 and the uniqueness
of viscosity solutions tells us a posteriori that
$v(x,t)=\int_{-\infty}^{x}u(y,t)\,dy$.

######  Remark 3.3.

Our result is similar in flavor to the well-known connection between entropy
solutions of scalar conservation laws and viscosity solutions of Hamilton-
Jacobi equations in 1-dimension (see for example [13]). In fact, given our
hypotheses on $\Psi$, it turns out that the unique distributional solution is
in fact also the (unique) entropy solution of (3.7). This is consequence of
[6, Section 4, Corollary 9], which establishes (in a more general setting than
ours) that distributional solutions are also entropy solutions. Therefore,
Theorem 3.1 can be viewed as a generalization of the classical result
connecting entropy and viscosity solutions of scalar conservation
laws/Hamilton-Jacobi equations.

As a special case of Theorem 3.1, we identify the unique viscosity solutions
of the parabolic $p$-Laplace equation with bounded and Lipschitz continuous
initial conditions as integrals of unique distributional solutions of the
porous medium equation:

######  Corollary 3.4.

Fix nonnegative $u_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, and
for any $T>0$, let $u\in C([0,T];L^{1}(\mathbb{R}))\cap
L^{\infty}(\mathbb{R}\times[0,T])$ denote the unique distributional solution
of

$\begin{cases}u_{t}-\frac{1}{2}(u^{m+1})_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ u(x,0)=u_{0}(x)&\text{in
$\mathbb{R}$.}\end{cases}$ (3.8)

Then $v(x,t):=\int_{-\infty}^{x}u(y,t)\,dy$ is the unique viscosity solution
of

$\begin{cases}v_{t}-\frac{m+1}{2}|v_{x}|^{m}v_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ v(x,0)=\int_{-\infty}^{x}u_{0}(y)\,dy&\text{in
$\mathbb{R}$.}\end{cases}$ (3.9)

###### Proof.

First, since $u_{0}$ is nonnegative, the comparison principle for the porous
medium equation [18, Theorem 9.2] guarantees that $u$ is nonnegative on all of
$\mathbb{R}\times[0,T]$, and hence we may rewrite (3.8) as

$\begin{cases}u_{t}-\frac{1}{2}(|u|^{m}u)_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ u(x,0)=u_{0}(x)&\text{in
$\mathbb{R}$.}\end{cases}$

Letting $\Psi(r):=\frac{1}{2}|r|^{m}r$, then $\Psi$ satisfies the hypotheses
of Theorem 3.1. Hence, by Theorem 3.1, $v(x,t):=\int_{-\infty}^{x}u(y,t)\,dy$
is the unique viscosity solution of (3.9), as desired. ∎

We highlight that Corollary 3.4 has the requirement that $u_{0}\in
L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, and in the case of the ZKB
solution, this is simply not true (the ZKB solution at time 0 is a Dirac delta
at the origin). We instead consider a ZKB solution shifted by some time
$\varepsilon>0$, which does indeed belong to $L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$. Applying Corollary 3.4 with initial condition
$\hat{U}(\cdot\,,\varepsilon;r_{\varepsilon})$ from (3.1) and with
$r_{\varepsilon}\rightarrow 1$ as $\varepsilon\to 0$ identifies a family of
viscosity solutions which approximate the desired solution of the parabolic
$p$-Laplace equation with Heaviside initial conditions.

This will be used as input to our sandwiching argument, comparing the CDF of
the original $\mathrm{SCM}(m,\mu)$ process to the CDFs of these viscosity
solution approximations. Picking suitable approximations and sending
$\varepsilon\to 0$, we will be able to conclude that the CDF is given
precisely by the shifted Beta density (3.4). The details of this argument
appear in the proof of Proposition 5.1, below.

###  3.3. The relationship between distributional solutions and viscosity
solutions

This section presents the proof of Theorem 3.1. We begin by recalling a result
of Benilan and Crandall [4], who considered equations of the form

$\begin{cases}u_{t}-(\Psi(u))_{xx}=0&\text{in $\mathbb{R}\times(0,T]$},\\\
u(x,0)=u_{0}(x)&\text{in $\mathbb{R}$},\end{cases}$ (3.10)

for $\Psi:\mathbb{R}\to\mathbb{R}$ continuous and nondecreasing, $u_{0}\in
L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, and for any $T>0$, with
solutions considered in the distributional sense (see Definition A.1). In
fact, [4] is more general than this; they present results for arbitrary
dimensions, which we do not include here. The main result of [4] is the
following stability property for solutions of equations of the form (3.10).

######  Theorem 3.5.

[4, Theorem, page 162.] Let $(\Psi_{n},1\leq n\leq\infty)$ be nondecreasing,
continuous functions $\Psi_{n}:\mathbb{R}\to\mathbb{R}$ with

$\lim_{n\rightarrow\infty}\Psi_{n}(r)=\Psi_{\infty}(r)\quad\text{for all
$r\in\mathbb{R}$},$

and let $(u_{0n},1\leq n\leq\infty)\in L^{1}(\mathbb{R})\cap
L^{\infty}(\mathbb{R})$ be such that

$\lim_{n\to\infty}\left|\left|u_{0n}-u_{0\infty}\right|\right|_{L^{1}(\mathbb{R})}=0.$
(3.11)

Fix $T>0$ and for $1\leq n\leq\infty$, let $u_{n}$ be the unique
distributional solution of

$\begin{cases}(u_{n})_{t}-\left(\Psi_{n}(u_{n})\right)_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$},\\\ u_{n}(x,0)=u_{0n}(x)&\text{in
$\mathbb{R}$},\end{cases}$

with $u_{n}\in C([0,T];L^{1}(\mathbb{R}))\cap
L^{\infty}(\mathbb{R}\times[0,T])$. Then

$u_{n}\rightarrow u_{\infty}\quad\text{in $C([0,T];L^{1}(\mathbb{R}))$.}$

In particular, this implies that

$\lim_{n\rightarrow\infty}\sup_{t\in[0,T]}\int_{\mathbb{R}}|u_{n}(x,t)-u_{\infty}(x,t)|\,dx=0.$

Equipped with Theorem 3.5, we proceed with the proof of Theorem 3.1. Our proof
is a vanishing viscosity argument, where we add an additional viscosity term
to regularize the PDE in order to obtain classical solutions, verify the
relationship (1.8) for the classical solutions, and use stability estimates to
send the viscosity parameter to 0.

###### Proof of Theorem 3.1.

Fix $u_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$ as in the
statement of Theorem 3.1, and let $u$ be the corresponding distributional
solution of (3.6). Since $u_{0}\in L^{1}(\mathbb{R})$, there exists a
collection of functions $\left(u^{\varepsilon}_{0},0<\varepsilon\leq
1\right)\subseteq C^{\infty}(\mathbb{R})$ so that

$\lim_{\varepsilon\to
0}\left|\left|u^{\varepsilon}_{0}-u_{0}\right|\right|_{L^{1}(\mathbb{R})}=0.$
(3.12)

Since $\Psi\in C^{\beta}$ for some $\beta\geq 1$, we may further consider a
sequence $(\tilde{\Psi}_{\varepsilon},0<\varepsilon<1)\subseteq
C^{\infty}(\mathbb{R})$ so that

$\displaystyle\tilde{\Psi}_{\varepsilon}\xrightarrow[\varepsilon\to
0]{}\Psi\quad\text{locally uniformly, and}$
$\displaystyle(\tilde{\Psi}_{\varepsilon})^{\prime}\xrightarrow[\varepsilon\to
0]{}\Psi^{\prime}\quad{\text{locally uniformly}.}$

The construction of $\tilde{\Psi}_{\varepsilon}$ can be done using a standard
mollifier [10, C.5], and this construction additionally yields that
$\tilde{\Psi}_{\varepsilon}$ can be chosen to be nondecreasing for every
$\varepsilon>0$.

Now, for each $\varepsilon>0$, we define $\Psi_{\varepsilon}(r):=\varepsilon
r+\tilde{\Psi}_{\varepsilon}(r)$. We now consider the uniformly parabolic
equation

$\begin{cases}u^{\varepsilon}_{t}-(\Psi_{\varepsilon}(u^{\varepsilon}))_{xx}=0&\text{in
$\mathbb{R}\times(0,T],$}\\\
u^{\varepsilon}(x,0)=u^{\varepsilon}_{0}(x)&\text{in
$\mathbb{R}$}.\end{cases}$ (3.13)

We may equivalently write the first line of (3.13) as
$u^{\varepsilon}_{t}-\varepsilon
u^{\varepsilon}_{xx}-\left(\tilde{\Psi}_{\varepsilon}(u^{\varepsilon})\right)_{xx}=0$.
By the introduction of the viscosity term $-\varepsilon u^{\varepsilon}_{xx}$,
the problem (3.13) becomes a uniformly parabolic quasilinear Cauchy problem,
with $\Psi_{\varepsilon}$ smooth, and with principal part in divergence form.
In particular, it is well known (see for example [15, V, Theorem 8.1]) that
under our hypotheses on $\Psi_{\varepsilon}$ and $u^{\varepsilon}_{0}$, for
every $\varepsilon>0$, $u^{\varepsilon}\in C^{2,1}(\mathbb{R}\times(0,T])$,
which implies (see Remark A.3) that $u^{\varepsilon}$ is a distributional
solution of (3.13). Moreover, by [15, V, Theorem 8.1] we have
$u^{\varepsilon}_{t},u^{\varepsilon}_{x}\in
L^{\infty}(\mathbb{R}\times[0,T])$.

Since $\Psi_{\varepsilon}(r)\rightarrow\Psi(r)$ as $\varepsilon\to 0$ for
every $r\in\mathbb{R}$, and (3.12) holds, it then follows by Theorem 3.5 that
for $u$ as in (3.6),

$\lim_{\varepsilon\rightarrow
0}\sup_{t\in[0,T]}\int_{\mathbb{R}}|u^{\varepsilon}(x,t)-u(x,t)|\,dx=0.$
(3.14)

We next define

$v^{\varepsilon}(x,t):=\int_{-\infty}^{x}u^{\varepsilon}(y,t)\,dy.$

This makes $v^{\varepsilon}\in C^{3,1}(\mathbb{R}\times(0,T])$ a classical
solution (and therefore a viscosity solution) of

$\begin{cases}v^{\varepsilon}_{t}-\left(\Psi_{\varepsilon}(v^{\varepsilon}_{x})\right)_{x}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\
v^{\varepsilon}(x,0)=\int_{-\infty}^{x}u^{\varepsilon}(y,0)\,dy&\text{in
$\mathbb{R}$,}\end{cases}$

which can be rewritten as

$\begin{cases}v^{\varepsilon}_{t}-\Psi_{\varepsilon}^{\prime}\left(v^{\varepsilon}_{x}\right)v^{\varepsilon}_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\
v^{\varepsilon}(x,0)=\int_{-\infty}^{x}u^{\varepsilon}(y,0)\,dy&\text{in
$\mathbb{R}$.}\end{cases}$

Since $\Psi_{\varepsilon}$ is nondecreasing, $\Psi^{\prime}_{\varepsilon}\geq
0$, and hence this problem is a good IVP, for which $v^{\varepsilon}$ is the
unique viscosity solution.

For $u$ the unique distributional solution of (3.6), we let

$v(x,t):=\int_{-\infty}^{x}u(y,t)\,dy.$

Equation (3.14) implies that for any $K\subseteq\mathbb{R}$ compact, we have

$\displaystyle\lim_{\varepsilon\to 0}\sup_{x\in
K}\sup_{t\in[0,T]}|v(x,t)-v^{\varepsilon}(x,t)|$
$\displaystyle=\lim_{\varepsilon\to 0}\sup_{x\in
K}\sup_{t\in[0,T]}\left|\int_{-\infty}^{x}u(y,t)-u^{\varepsilon}(y,t)\,dy\right|$
$\displaystyle\leq\lim_{\varepsilon\to 0}\sup_{x\in
K}\sup_{t\in[0,T]}\int_{-\infty}^{x}\left|u(y,t)-u^{\varepsilon}(y,t)\right|\,dy$
$\displaystyle\leq\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}\int_{\mathbb{R}}\left|u(y,t)-u^{\varepsilon}(y,t)\right|\,dy=0,$

which implies that

$v^{\varepsilon}\rightarrow v\quad\text{locally uniformly in $x,t$}\text{ on
}\mathbb{R}\times[0,T].$

Since $\Psi_{\varepsilon}^{\prime}\rightarrow\Psi^{\prime}$ locally uniformly,
the stability property of viscosity solutions (see Proposition A.5) now
guarantees that $v$ is both a viscosity subsolution and supersolution of

$v_{t}-\Psi^{\prime}(v_{x})v_{xx}=0\quad\text{in $\mathbb{R}\times(0,T]$.}$

Moreover, since $v\in C(\mathbb{R}\times[0,T])$, this yields that $v$ is the
unique viscosity solution of

$\begin{cases}v_{t}-\Psi^{\prime}(v_{x})v_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ v(x,0)=\int_{-\infty}^{x}u(y,0)\,dy&\text{in
$\mathbb{R}$},\end{cases}$

as desired. ∎

######  Remark 3.6.

While the proof of Theorem 3.1 is a standard vanishing viscosity argument, we
surprisingly were unable to find a version of Theorem 3.1 anywhere in the
literature, even for the porous medium equation. The idea to regularize PDEs
of the form (3.10) and then study their stability properties was pursued by
[16, 17], who regularized the equations by considering initial data
$u_{0}+\varepsilon$ for $\varepsilon>0,u_{0}\geq 0$.

##  4\. Finite Difference Schemes for Diffuse Initial Conditions

In this section, we show that the CDFs of the discrete random variables
$(X_{n},n\geq 0)$ yield a good approximation scheme for the integrated ZKB
solution, and that Theorem 2.1 applies in this setting, when the initial
condition (the CDF of $X_{0}$) is given by a fine-mesh discretization of a
Lipschitz continuous extended CDF.

Throughout the section, fix $m\in(0,\infty)$, and a probability distribution
$\mu$ supported on $\mathbb{Z}\cup\\{-\infty,\infty\\}$. Let $(X_{n},n\geq 0)$
be $\mathrm{SCM}(m,\mu)$-distributed, and for $k\in\mathbb{Z}$, write
$F^{n}_{k}=F^{n}_{k}(\mu)=\mathbf{P}(X_{n}\leq k)=\mu[-\infty,k]$. In this
case $F^{n}_{(\cdot)}$ is an extended CDF for each $n\in\mathbb{N}$. We
suppress the dependence on $m$ as it is fixed throughout, and also suppress
the dependence on $\mu$ whenever possible. As observed in Section 1,
$(F^{n}_{k})_{k\in\mathbb{Z},n\in\mathbb{N}}$ can be defined by the recurrence

$\begin{cases}F^{n+1}_{k}=F^{n}_{k}+\frac{1}{2}\left[(F^{n}_{k+1}-F^{n}_{k})^{m+1}-(F^{n}_{k}-F^{n}_{k-1})^{m+1}\right],\\\
F^{0}_{k}=\mu[-\infty,k].\end{cases}$ (4.1)

As before, for every $N>0$, we consider mesh sizes $\Delta_{x}^{N}$ and
$\Delta_{t}^{N}$ such that $\Delta_{x}^{N},\Delta_{t}^{N}\to 0$ as
$N\to\infty$. Due to the scaling properties of the parabolic $p$-Laplace
equation, the relation $\Delta_{t}^{N}=(\Delta_{x}^{N})^{m+2}$ is natural
(when $m=0$ this is Brownian scaling), and we enforce this relation on
$\Delta_{t}^{N}$ and $\Delta_{x}^{N}$ throughout, by defining

$\Delta_{t}^{N}=\frac{1}{N}\quad\mbox{ and
}\quad\Delta_{x}^{N}=\frac{1}{N^{1/(m+2)}}.$ (4.2)

For brevity, we suppress the dependence on $N$ wherever possible. Given
$v_{0}\in L^{\infty}(\mathbb{R})$, we define a function $v^{N}\in
L^{\infty}(\mathbb{R}\times[0,\infty))$ by

$\begin{cases}v^{N}(x,t+\Delta_{t})=v^{N}(x,t)+\frac{\Delta_{t}}{\Delta_{x}}\frac{1}{2}\left[\left(\frac{v^{N}(x+\Delta_{x},t)-v^{N}(x,t)}{\Delta_{x}}\right)^{m+1}\right.&\\\
\quad\qquad\qquad\qquad\qquad\qquad\qquad-\left.\left(\frac{v^{N}(x,t)-v^{N}(x-\Delta_{x},t)}{\Delta_{x}}\right)^{m+1}\right]&\text{in
$\mathbb{R}\times[\Delta_{t},\infty)$}\,,\\\
v^{N}(x,t)=v_{0}\big{(}\big{\lfloor}\frac{x}{\Delta_{x}}\big{\rfloor}\Delta_{x}\big{)}&\text{in
$\mathbb{R}\times[0,\Delta_{t})$}\,.\end{cases}$ (4.3)

######  Remark 4.1.

If $v_{0}$ is an extended CDF, then for $N\in\mathbb{N}$ we may define a
probability measure $\mu^{N}$ on $\mathbb{Z}\cup\\{-\infty,\infty\\}$ by
setting $\mu^{N}[-\infty,k]=v_{0}(k\Delta_{x}^{N})=v_{0}(k/N^{1/(m+2)})$. With
this definition, if $(X_{n},n\geq 0)$ is $\mathrm{SCM}(m,\mu^{N})$-distributed
then for $k\in\mathbb{Z}$,

$v^{N}(k\Delta_{x}^{N},0)=v_{0}(k\Delta_{x}^{N})=F^{0}_{k}\,,$

and inductively, $v^{N}(k\Delta_{x}^{N},n\Delta_{t}^{N})=F^{n}_{k}(\mu^{N})$
for all $n\in\mathbb{N}$ and $k\in\mathbb{Z}$. Since $v^{N}$ is piecewise
constant in each lattice rectangle of the mesh
$\Delta_{x}\mathbb{Z}\times\Delta_{t}\mathbb{N}=N^{-1/(m+2)}\mathbb{Z}\times
N^{-1}\mathbb{N}$, interpreted with closed lower and left boundaries and open
upper and right boundaries, for all $(x,t)\in\mathbb{R}\times[0,\infty)$, we
may then express $v^{N}(x,t)$ as

$v^{N}(x,t)=F^{\lfloor Nt\rfloor}_{\lfloor N^{1/(m+2)}x\rfloor}\,,$ (4.4)

where $(F^{n}_{k})_{k\in\mathbb{Z},n\in\mathbb{N}}$ is defined by (4.1). In
particular, this implies that for any $t>0$, $v^{N}(\cdot,t)$ is an extended
CDF which is piecewise constant on intervals of the form
$[kN^{-1/(m+2)},(k+1)N^{-1/(m+2)})$, for $k\in\mathbb{Z}$. This also yields
that in this case $v^{N}\in L^{\infty}(\mathbb{R}\times[0,\infty))$.

The main result of this section shows that, under suitable assumptions on the
initial conditions, $v^{N}\rightarrow v$ locally uniformly, where $v$ is the
unique viscosity solution of the parabolic $p$-Laplace equation.

######  Proposition 4.2.

Let $v_{0}$ be a Lipschitz continuous extended CDF with Lipschitz constant
$M$. Fix $N\in\mathbb{N}$ and define a probability distribution $\mu_{N}$ on
$\mathbb{Z}\cup\\{-\infty,\infty\\}$ by

$\mu_{N}[-\infty,k]:=v_{0}(kN^{-1/(m+2)})$

for $k\in\mathbb{Z}$. Let $(X_{n},n\geq 0)$ be
$\mathrm{SCM}(m,\mu_{N})$-distributed, and let $F^{n}_{k}=F^{n}_{k}(\mu_{N})$
be defined by (4.1). Finally, fix $T>0$, and $K\subseteq\mathbb{R}$ compact.
Then for every $\varepsilon>0$, there exists
$N_{0}=N_{0}(m,M,\varepsilon,K,T)$ such that if $N\geq N_{0}$, then for
$v^{N}$ defined by (4.3),

$\sup_{0\leq t\leq T}\sup_{x\in
K}\left|v^{N}(x,t)-v(x,t)\right|\leq\varepsilon,$ (4.5)

where $v$ is the viscosity solution of

$\begin{cases}v_{t}-\frac{m+1}{2}|v_{x}|^{m+1}v_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ v(x,0)=v_{0}(x)&\text{in
$\mathbb{R}$.}\end{cases}$ (4.6)

It follows that $v(x,1)$ is an extended CDF and that if $N\geq N_{0}$, then

$\sup_{x\in K}\left|\mathbf{P}\left\\{\frac{X_{N}}{N^{1/(m+2)}}\leq
x\right\\}-v(x,1)\right|\leq\varepsilon.$ (4.7)

In order to prove Proposition 4.2, we must verify that the conditions of
Theorem 2.1 are satisfied. We begin with a monotonicity lemma for certain
solutions of (4.1). Write $p^{*}:=\frac{1}{(m+1)^{1/m}}$. We say that a
probability distribution $\mu$ on $\mathbb{Z}$ is $p^{*}$-bounded if
$\mu(\\{z\\})\leq p^{*}$ for all $z\in\mathbb{Z}$.

######  Lemma 4.3.

Fix probability distributions $\mu,\nu$ on
$\mathbb{Z}\cup\left\\{-\infty,\infty\right\\}$. Let
$(G^{n}_{k})_{k\in\mathbb{Z},n\in\mathbb{N}}=(F^{n}_{k}(\mu))_{k\in\mathbb{Z},n\in\mathbb{N}}$
and
$(H^{n}_{k})_{k\in\mathbb{Z},n\in\mathbb{N}}=(F^{n}_{k}(\nu))_{k\in\mathbb{Z},n\in\mathbb{N}}$
be defined by the recurrence (4.1) with initial conditions given by $\mu$ and
$\nu$, respectively. If $\mu$ and $\nu$ are $p^{*}$-bounded and
$\mu[-\infty,k]\leq\nu[-\infty,k]$ for all $k\in\mathbb{Z}$, then
$G^{n}_{k}\leq H^{n}_{k}$ for all $k\in\mathbb{Z}$ and $n\in\mathbb{N}$, and
additionally $G^{n}_{k}-G^{n}_{k-1}\leq p^{*}$ and $H^{n}_{k}-H^{n}_{k-1}\leq
p^{*}$ for all $k\in\mathbb{Z}$ and $n\in\mathbb{N}$.

###### Proof.

Write

$\mathcal{S}(a,b,c)=b+\frac{1}{2}[(c-b)^{m+1}-(b-a)^{m+1}]\,,$

so that $F^{n+1}_{k}=\mathcal{S}(F^{n}_{k-1},F^{n}_{k},F^{n}_{k+1})$. Note
that $\mathcal{S}$ is nondecreasing in $a$ when $a\leq b$ and is nondecreasing
in $c$ when $b\leq c$. Moreover,

$\frac{\partial}{\partial
b}\mathcal{S}(a,b,c)=1-\frac{m+1}{2}[(c-b)^{m}+(b-a)^{m}]$

which is nonnegative provided that $0\leq b-a\leq p^{*}$ and $0\leq c-b\leq
p^{*}$. This in particular shows that the directional derivative of
$\mathcal{S}$ in directions $(1,0,0),(0,1,0)$ and $(0,0,1)$ is positive on the
set

$R=\big{\\{}(a,b,c):a\geq 0,b-a\in[0,p^{*}],c-b\in[0,p^{*}]\big{\\}}.$

It follows that if $(a,b,c),(a^{\prime},b^{\prime},c^{\prime})\in R$ and
$a\leq a^{\prime}$, $b\leq b^{\prime}$, $c\leq c^{\prime}$, then the
directional derivative of $\mathcal{S}$ is also nonnegative on the set $R$ in
direction $(x,y,z):=(a^{\prime}-a,b^{\prime}-b,c^{\prime}-c)$, since this
direction is a linear combination of the axis directions. Since $R$ is convex,
the line segment from $(a,b,c)$ to $(a^{\prime},b^{\prime},c^{\prime})$ lies
within $\mathcal{S}$, and we conclude that
$\mathcal{S}(a,b,c)\leq\mathcal{S}(a^{\prime},b^{\prime},c^{\prime})$.

Recalling that $G^{0}_{k}=\mu[-\infty,k]$ and $H^{0}_{k}=\nu[-\infty,k]$,
under the assumptions of the lemma it follows that for all $k\in\mathbb{Z}$,
we have the entry-wise inequality
$(G^{0}_{k-1},G^{0}_{k},G^{0}_{k+1})\leq(H^{0}_{k-1},H^{0}_{k},H^{0}_{k+1})$,
and both triples lie in $R$, so

$G^{1}_{k}=\mathcal{S}(G^{0}_{k-1},G^{0}_{k},G^{0}_{k+1})\leq\mathcal{S}(H^{0}_{k-1},H^{0}_{k},H^{0}_{k+1})=H^{1}_{k}\,.$

We now claim that $0\leq G^{1}_{k}-G^{1}_{k-1}\leq p^{*}$ and $0\leq
H^{1}_{k}-H^{1}_{k-1}\leq p^{*}$ for all $k\in\mathbb{Z}$, in which case it
follows by induction that $G^{n}_{k}\leq H^{n}_{k}$ and that
$G^{n}_{k}-G^{n}_{k-1}\leq p^{*}$ and $H^{n}_{k}-H^{n}_{k-1}\leq p^{*}$ for
all $n\in\mathbb{N}$ and $k\in\mathbb{Z}$. We only prove that $0\leq
G^{1}_{k}-G^{1}_{k-1}\leq p^{*}$ since an identical argument works for $H$. To
show this, note that for any $\lambda\in\mathbb{R}$,

$\mathcal{S}(a+\lambda,b+\lambda,c+\lambda)=b+\lambda+\frac{1}{2}[(c-b)^{m+1}-(b-a)^{m+1}]=\lambda+\mathcal{S}(a,b,c)\,.$

Since $G^{0}_{k-1}\leq G^{0}_{k}\leq G^{0}_{k-1}+p^{*}$ for all
$k\in\mathbb{Z}$, it follows that

$(G^{0}_{k-2},G^{0}_{k-1},G^{0}_{k})\leq(G^{0}_{k-1},G^{0}_{k},G^{0}_{k+1})\leq(G^{0}_{k-2}+p^{*},G^{0}_{k-1}+p^{*},G^{0}_{k}+p^{*})$

Since $\mathcal{S}$ is nondecreasing in all its arguments on $R$ and all the
above triples lie in $R$, it follows that

$G^{1}_{k-1}=\mathcal{S}(G^{0}_{k-2},G^{0}_{k-1},G^{0}_{k})\leq\mathcal{S}(G^{0}_{k-1},G^{0}_{k},G^{0}_{k+1})=G^{1}_{k}\,,$

and

$\displaystyle G^{1}_{k}$
$\displaystyle=\mathcal{S}(G^{0}_{k-1},G^{0}_{k},G^{0}_{k+1})$
$\displaystyle\leq\mathcal{S}(G^{0}_{k-2}+p^{*},G^{0}_{k-1}+p^{*},G^{0}_{k}+p^{*})$
$\displaystyle=p^{*}+\mathcal{S}(G^{0}_{k-2},G^{0}_{k-1},G^{0}_{k})$
$\displaystyle=p^{*}+G^{1}_{k-1}\,,$

as required. ∎

As a straightforward consequence of the lemma, we obtain that the monotonicity
condition (2.3) holds whenever the functions are “generated” by Lipschitz
continuous extended CDFs. For $M>0$ write

$\mathcal{C}_{M}:=\left\\{w\in L^{\infty}(\mathbb{R}):\text{$w$ is an
$M$-Lipschitz continuous extended CDF}\right\\}.$

The following corollary now verifies that the approximation scheme (4.3) is
monotone on $\mathcal{C}_{M}$.

######  Corollary 4.4.

For all $M>0$ there exists $N_{0}$ such that for $N\geq N_{0}$ the following
holds. Fix $u_{0},w_{0}\in\mathcal{C}_{M}$ and let $u^{N}$ and $w^{N}$ be
defined by (4.3) with initial conditions given by $u_{0}$ and $w_{0}$,
respectively. If $u_{0}\leq w_{0}$, then $u^{N}\leq w^{N}$.

###### Proof.

Define probability distributions $\mu^{N}$ and $\nu^{N}$ on
$\mathbb{Z}\cup\\{-\infty,\infty\\}$ by setting
$\mu^{N}[-\infty,k]=u_{0}(kN^{-1/(m+2)})$ and
$\nu^{N}[-\infty,k]=w_{0}(kN^{-1/(m+2)})$. By the observation of Remark 4.1,
we then have $u^{N}(x,t):=F^{\lfloor Nt\rfloor}_{\lfloor
N^{1/(m+2)}x\rfloor}(\mu^{N})$ and $w^{N}(x,t):=F^{\lfloor Nt\rfloor}_{\lfloor
N^{1/(m+2)}x\rfloor}(\nu^{N})$. Moreover, if $N\geq
N_{0}:=M^{m+2}(m+1)^{(m+2)/m}$ then for all $k\in\mathbb{Z}$ we have

$\mu^{N}(\\{k\\})=u_{0}(kN^{-1/(m+2)})-u_{0}((k-1)N^{-1/(m+2)})\leq
M\frac{1}{N^{1/(m+2)}}\leq p^{*}\,,$

so by Lemma 4.3, it follows that for $N\geq N_{0}$, $u^{N}(x,t)\leq
w^{N}(x,t)$ for all $x\in\mathbb{R}$ and $t\in[0,\infty)$, as required. ∎

Having verified monotonicity, we are now ready to use Theorem 2.1 to prove
Proposition 4.2.

###### Proof of Proposition 4.2.

Define $G:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ by

$G(v_{x},v_{xx}):=-\frac{m+1}{2}|v_{x}|^{m}v_{xx}\,.$

For $a,b\in\mathbb{R}$ with $a\leq b$ we then have $G(p,a)\geq G(p,b)$ for any
$p\in\mathbb{R}$, i.e., $G$ is degenerate elliptic. Since $G$ is also
continuous, it follows that if $v_{0}$ is any Lipschitz continuous extended
CDF, then the initial value problem

$\begin{cases}v_{t}+G(v_{x},v_{xx})=0&\text{in
$\mathbb{R}\times(0,\infty)$,}\\\ v(x,0)=v_{0}(x)&\text{in
$\mathbb{R}$}\end{cases}$ (4.8)

is good.

Writing $\Delta_{x}=\Delta_{x}^{N}$ and $\Delta_{t}=\Delta_{t}^{N}$, define

$\displaystyle\mathcal{G}^{N}(a,b,c)$
$\displaystyle=-\frac{1}{\Delta_{x}}\frac{1}{2}\left[\left|\frac{c-b}{\Delta_{x}}\right|^{m}\left(\frac{c-b}{\Delta_{x}}\right)-\left|\frac{b-a}{\Delta_{x}}\right|^{m}\left(\frac{b-a}{\Delta_{x}}\right)\right]$
$\displaystyle=-\frac{1}{2\Delta_{t}}\left[|c-b|^{m}(c-b)-|b-a|^{m}(b-a)\right]\,,$

where we have used that $\Delta_{x}^{m+2}=\Delta_{t}$ for the second equality.
Recalling the notation $\langle
v^{N}(x,t)\rangle_{N}=(v^{N}(x,t-\Delta_{t}),v^{N}(x,t),v^{N}(x,t+\Delta_{t}))$,
if $v^{N}$ is defined by (4.3), then since $v^{N}$ is nondecreasing in $x$ for
all $t$, we have

$v^{N}(x,t+\Delta_{t})=v^{N}(x,t)-\Delta_{t}\mathcal{G}^{N}\langle
v^{N}(x,t)\rangle_{N}\,,$

so we may rewrite (4.3) in the form given by (2.2):

$\begin{cases}\frac{v^{N}(x,t+\Delta_{t})-v^{N}(x,t)}{\Delta_{t}}+\mathcal{G}^{N}\langle
v^{N}(x,t)\rangle_{N}=0,&\text{in $\mathbb{R}\times[\Delta_{t},\infty)$}\,,\\\
v^{N}(x,0)=v_{0}\big{(}\big{\lfloor}\frac{x}{\Delta_{x}}\big{\rfloor}\Delta_{x}\big{)},&\text{in
$\mathbb{R}\times[0,\Delta_{t})$}\,.\end{cases}$ (4.9)

Since $v_{0}$ is a Lipschitz continuous extended CDF, Corollary 4.4 then
ensures that the monotonicity condition (2.3) is satisfied for $N$
sufficiently large. Moreover, for all $t$, $v^{N}(\cdot,t)$ is a piecewise
constant approximation of an extended CDF, so $\sup_{x\in\mathbb{R},t\geq
0}|v^{N}(x,t)|\leq 1$, which verifies stability. To check consistency, note
that for $\varphi:\mathbb{R}\times(0,\infty)\to\mathbb{R}$ and $\varepsilon>0$
we have

$\displaystyle\mathcal{G}^{N}\langle\varphi(y,s)+\varepsilon\rangle_{N}$
$\displaystyle=-\frac{1}{\Delta_{x}}\frac{1}{2}\left[\left|\frac{\varphi(y+\Delta_{x},s)-\varphi(y,s)}{\Delta_{x}}\right|^{m}\frac{\varphi(y+\Delta_{x},s)-\varphi(y,s)}{\Delta_{x}}\right.$
$\displaystyle\quad\quad\quad\quad-\left.\left|\frac{\varphi(y,s)-\varphi(y-\Delta_{x},s)}{\Delta_{x}}\right|^{m}\frac{\varphi(y)-\varphi(y-\Delta_{x},s)}{\Delta_{x}}\right],$
$\displaystyle=\mathcal{G}^{N}\langle\varphi(y,s)\rangle_{N}\,,$

so if $\varphi$ is smooth and bounded then

$\displaystyle\lim_{(N,y,s,\varepsilon)\to(\infty,x_{0},t_{0},0)}\mathcal{G}^{N}\langle\varphi(y,s)+\varepsilon\rangle_{N}$
$\displaystyle=\lim_{(N,y,s)\to(\infty,x_{0},t_{0})}\mathcal{G}^{N}\langle\varphi(y,s)\rangle_{N}$
$\displaystyle=-\frac{1}{2}((\varphi_{x}(x_{0},t_{0}))^{m+1})_{x}$
$\displaystyle=-\frac{m+1}{2}\varphi_{x}(x_{0},t_{0})\varphi_{xx}(x_{0},t_{0})$
$\displaystyle=G(\varphi_{x}(x_{0},t_{0}),\varphi_{xx}(x_{0},t_{0})),$
$\displaystyle=-\frac{1}{2}(|\varphi_{x}(x_{0},t_{0})|^{m}\varphi_{x}(x_{0},t_{0}))_{x}$
$\displaystyle=-\frac{m+1}{2}|\varphi_{x}(x_{0},t_{0})|^{m}\varphi_{xx}(x_{0},t_{0})$
$\displaystyle=G(\varphi_{x}(x_{0},t_{0}),\varphi_{xx}(x_{0},t_{0}))$

and this establishes consistency.

We have just verified that (4.3), or equivalently (4.9), is a good
approximation scheme for (4.8). It follows by Theorem 2.1 that for every
$T>0$, for every $K\subseteq\mathbb{R}$ compact, and for $N\geq N_{0}$
sufficiently large, we have

$\sup_{0\leq t\leq T}\sup_{x\in K}|v^{N}(x,t)-v(x,t)|\leq\varepsilon,$

where $v$ is the viscosity solution of

$\begin{cases}v_{t}-\frac{m+1}{2}|v_{x}|^{m}v_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\ v(x,0)=v_{0}(x)&\text{in
$\mathbb{R}$.}\end{cases}$ (4.10)

For the second assertion of Proposition 4.2, we note that since

$v^{N}(x,t)=F^{\lfloor Nt\rfloor}_{\lfloor
N^{1/(m+2)}x\rfloor}={\mathbf{P}}\left(X_{\lfloor Nt\rfloor}\leq\lfloor
N^{1/(m+2)}x\rfloor\right)={\mathbf{P}}\left(X_{\lfloor Nt\rfloor}\leq
N^{1/(m+2)}x\right),$

we have that for $t=1$,

$\displaystyle\sup_{x\in
K}\left|{\mathbf{P}}\left(\frac{X_{N}}{N^{1/(m+2)}}\leq
x\right)-v(x,1)\right|\leq\varepsilon.$

Finally, since $v(\cdot,1)$ is nondecreasing and continuous and is the
pointwise limit of extended CDFs, it follows that $v(\cdot,1)$ itself is an
extended CDF. ∎

##  5\. Convergence under $p^{*}$-bounded initial conditions

Recall from Section 4 that $p^{*}=\tfrac{1}{(m+1)^{1/m}}$ and that we say a
probability distribution $\mu$ on $\mathbb{Z}$ is $p^{*}$-bounded if
$\mu(\\{z\\})\leq p^{*}$ for all $z\in\mathbb{Z}$. The main result of this
section is that $\mathrm{SCM}(m,\mu)$-distributed cooperative motions converge
in distribution to a scaled and shifted Beta random variable when $\mu$ is
$p^{*}$-bounded.

######  Proposition 5.1.

Let ($X_{n}$, $n\geq 0$) be $\mathrm{SCM}(m,\mu)$-distributed with $\mu$ a
probability distribution on $\mathbb{Z}$. If $\mu$ is $p^{*}$-bounded, then

$\lim_{n\to\infty}\mathbf{P}\left(\frac{X_{n}}{n^{1/(m+2)}}\leq
x\right)=v(x,1)$

locally uniformly in $x$, where $v(x,1)$ is the CDF of

$\frac{2^{\frac{m+1}{m+2}}(m+1)^{1/(m+2)}}{D^{\frac{m}{m+2}}\rho^{1/2}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right].$
(5.1)

###### Proof of Proposition 5.1.

Fix an $\varepsilon>0$ and let $(X_{n},n\geq 0)$ be
$\mathrm{SCM}(m,\mu)$-distributed. Let $F^{0}_{k}:=\mathbf{P}(X_{0}\leq
k)=\mu(-\infty,k]$. Using (4.3) with $v_{0}(x)=F^{0}_{\lfloor
N^{1/(m+2)}x\rfloor}$, we have that $v^{N}(x,t)=F^{\lfloor Nt\rfloor}_{\lfloor
N^{1/(m+2)}x\rfloor}$. We will choose schemes $v^{-,N}$ and $v^{+,N}$ with
smoother initial conditions which are $p^{*}$-bounded and which sandwich
$v_{0}$ (i.e. bound it from below and above). Proposition 4.2 will allow us to
conclude that $v^{-,N}$ and $v^{+,N}$ converge as $N\to\infty$, while Lemma
4.3 will guarantee that $v^{-,N}$ and $v^{+,N}$ sandwich $v^{N}$ for all
times.

By (3.1), we have

$\hat{U}(x,t;1-\varepsilon)=\frac{1}{t^{\frac{1}{m+2}}}\left[\left(\frac{\sqrt{2}(1-\varepsilon)}{D\sqrt{m+1}}\right)^{\frac{2m}{m+2}}-\frac{2\rho|x|^{2}}{(m+1)t^{\frac{2}{m+2}}}\right]^{1/m}_{+}\,,$
(5.2)

We then define

$v_{\varepsilon}(x,t):=\int_{-\infty}^{x}\hat{U}(y,t+\varepsilon;1-\varepsilon)\,dy.$
(5.3)

Let

$\Pi^{1-\varepsilon}(\varepsilon):=\frac{(1-\varepsilon)^{\frac{m}{m+2}}(m+1)^{\frac{1}{m+2}}}{2^{\frac{1}{m+2}}\rho^{\frac{1}{2}}D^{\frac{m}{m+2}}}\varepsilon^{\frac{1}{m+2}},$
(5.4)

so that the support of $\hat{U}(\cdot,\varepsilon;1-\varepsilon)$ is given by
$[-\Pi^{1-\varepsilon}(\varepsilon),\Pi^{1-\varepsilon}(\varepsilon)]$. (A
more general function $\Pi^{\theta}(t)$ is used in the Appendix, and this
notation is chosen to agree with that notation.) In particular, this implies
that $v_{\varepsilon}(x,0)$ is constant on
$(-\infty,-\Pi^{1-\varepsilon}(\varepsilon))$ and on
$(\Pi^{1-\varepsilon}(\varepsilon),\infty)$. Since
$\hat{U}(\cdot,\varepsilon;1-\varepsilon)$ belongs to
$L^{\infty}(\mathbb{R})$, $v_{\varepsilon}(x,0)$ is Lipschitz continuous. Note
that $v_{\varepsilon}(\Pi^{1-\varepsilon}(\varepsilon),0)=1-\varepsilon$, by
the choice of constant in the definition of
$\hat{U}(\cdot\,,\cdot\,;1-\varepsilon)$.

Define

$\displaystyle L$ $\displaystyle:=L(\varepsilon)=\max\\{k\leq
0:F^{0}_{k}\leq\varepsilon\\}$ $\displaystyle R$
$\displaystyle:=R(\varepsilon)=\min\\{k\geq 0:F^{0}_{k}\geq 1-\varepsilon\\}.$

These values are both finite because $\mu$ is a probability distribution on
$\mathbb{R}$. Then for $\tilde{n}\in\mathbb{N}$, let $v^{+,N}(x,t;\tilde{n})$
be defined by (4.3) with $(\Delta_{x})^{m+2}=\Delta_{t}=1/N$ and initial
condition

$v^{+,N}(x,t;\tilde{n})=v_{\varepsilon}(\lfloor(x-L\tilde{n}^{-1/(m+2)}+2\Pi^{1-\varepsilon}(\varepsilon))/\Delta_{x}\rfloor\Delta_{x},0)+\varepsilon$
(5.5)

for $t\in[0,\Delta_{t})$.

Similarly, define $v^{-,N}(x,t;\tilde{n})$ by (4.3) with
$(\Delta_{x})^{m+2}=\Delta_{t}=1/N$ and initial condition

$v^{-,N}(x,t;\tilde{n})=v_{\varepsilon}(\lfloor(x-R\tilde{n}^{-1/(m+2)}-\Pi^{1-\varepsilon}(\varepsilon))/\Delta_{x}\rfloor\Delta_{x},0)$
(5.6)

for $t\in[0,\Delta_{t})$.

Since the initial condition is bounded and uniformly continuous, by Theorem
2.1 we know that $v^{+,N}(x,t;\tilde{n})$ converges to the unique viscosity
solution $v^{+}(x,t;\tilde{n})$ of

$\begin{cases}v^{+}_{t}-\frac{m+1}{2}\left|v^{+}_{x}\right|^{m}v^{+}_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\
v^{+}(x,0)=v_{\varepsilon}(x-L\tilde{n}^{-1/(m+2)}+2\Pi^{1-\varepsilon}(\varepsilon),0)+\varepsilon&\text{in
$\mathbb{R}$.}\end{cases}$ (5.7)

Similarly, $v^{-,N}(x,t;\tilde{n})$ converges to the unique viscosity solution
$v^{-}(x,t;\tilde{n})$ of

$\begin{cases}v^{-}_{t}-\frac{m+1}{2}\left|v^{-}_{x}\right|^{m}v^{-}_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\
v^{-}(x,0)=v_{\varepsilon}(x-R\tilde{n}^{-1/(m+2)}-\Pi^{1-\varepsilon}(\varepsilon),0)&\text{in
$\mathbb{R}$.}\end{cases}$ (5.8)

Notice that the initial conditions in (5.7) and (5.8) are simply shifts of
$v_{\varepsilon}(x,0)$. Therefore, by Corollary 3.4 and Lemma A.4, we know the
profiles of $v^{+}(x,t;\tilde{n})$ and $v^{-}(x,t;\tilde{n})$ are the
corresponding shifts of
$\int_{-\infty}^{x}\hat{U}(y,t+\varepsilon;1-\varepsilon)\,dy$ for all times
$t$ (shifted left by
$|L|\tilde{n}^{-1/(m+2)}+2\Pi^{1-\varepsilon}(\varepsilon)$ and up by
$\varepsilon$, or right by
$R\tilde{n}^{-1/(m+2)}+\Pi^{1-\varepsilon}(\varepsilon)$, respectively).

Next, because $v^{-,N}(x,0;\tilde{n}),v^{+,N}(x,0;\tilde{n})$ are
discretizations of Lipschitz functions, there exists $\eta>0$ such that for if
$\Delta_{x}\leq\eta$, then for all $x$,

$\displaystyle|v^{-,N}(x+\Delta_{x},0)-v^{-,N}(x,0)|<p^{*},$
$\displaystyle|v^{+,N}(x+\Delta_{x},0)-v^{+,N}(x,0)|<p^{*}.$

We claim that if
$\Delta_{x}\leq\min(\Pi^{1-\varepsilon}(\varepsilon),\tilde{n}^{-1/(m+2)})$,
then $v^{+,N}(x,0)\geq v^{N}(x,0)$ for all $x$. To see this, first note that
by choice of $L$, when $x<L\Delta_{x}$, we have

$v^{N}(x,0)\leq\varepsilon.$

Because $v^{+,N}(x,0;\tilde{n})\geq\varepsilon$ by definition, it follows that
for $x<L\Delta_{x}$,

$v^{N}(x,0)\leq\varepsilon\leq v^{+,N}(x,0;\tilde{n}).$ (5.9)

For $x\geq L\Delta_{x}$, notice that since
$\Delta_{x}\leq\tilde{n}^{-1/(m+2)}$ and $L\leq 0$, we have
$L(\Delta_{x}-\tilde{n}^{-1/(m+2)})\geq 0$. Since $v_{\varepsilon}$ is
nondecreasing, it follows that

$\displaystyle v^{+,N}(x,0;\tilde{n})$
$\displaystyle=v_{\varepsilon}(\lfloor(x-L\tilde{n}^{1/(m+2)}+2\Pi^{1-\varepsilon}(\varepsilon))/\Delta_{x}\rfloor\Delta_{x},0)+\varepsilon$
$\displaystyle\geq
v_{\varepsilon}(\lfloor(L(\Delta_{x}-\tilde{n}^{-1/(m+2)})+2\Pi^{1-\varepsilon}(\varepsilon))/\Delta_{x}\rfloor\Delta_{x},0)$
$\displaystyle\geq v_{\varepsilon}(\lfloor
2\Pi^{1-\varepsilon}(\varepsilon)/\Delta_{x}\rfloor\Delta_{x},0)+\varepsilon$
$\displaystyle\geq
v_{\varepsilon}(\Pi^{1-\varepsilon}(\varepsilon),0)+\varepsilon=1.$

The last inequality holds since $\lfloor
2\Pi^{1-\varepsilon}(\varepsilon)/\Delta_{x}\rfloor\Delta_{x}\geq\Pi^{1-\varepsilon}(\varepsilon)$
when $\Delta_{x}\leq\Pi^{1-\varepsilon}(\varepsilon)$.

Therefore, we have that for $x\geq L\Delta_{x}$ and
$\Delta_{x}\leq\min(\Pi^{1-\varepsilon}(\varepsilon),\tilde{n}^{-1/(m+2)})$,

$v^{N}(x,0)\leq 1\leq v^{+,N}(x,0;\tilde{n}).$ (5.10)

Combining (5.9) and (5.10), we see that when
$\Delta_{x}\leq\min(\Pi^{1-\varepsilon}(\varepsilon),\tilde{n}^{-1/(m+2)})$,

$v^{N}(x,0)\leq v^{+,N}(x,0;\tilde{n})$

for all $x$.

Similarly, if $\Delta_{x}\leq\tilde{n}^{-1/(m+2)}$, then for all
$x<R\Delta_{x}$, we have

$\displaystyle v^{-,N}(x,0;\tilde{n})$
$\displaystyle=v_{\varepsilon}(\lfloor(x-R\tilde{n}^{-1/(m+2)}-\Pi^{1-\varepsilon}(\varepsilon))/\Delta_{x}\rfloor\Delta_{x},0)$
$\displaystyle<v_{\varepsilon}(R\Delta_{x}-R\tilde{n}^{-1/(m+2)}-\Pi^{1-\varepsilon}(\varepsilon),0)$
$\displaystyle<v_{\varepsilon}(-\Pi^{1-\varepsilon}(\varepsilon),0)$
$\displaystyle=0$ $\displaystyle\leq v^{N}(x,0).$

Moreover, if $x\geq R\Delta_{x}$, then by the choice of $R$,

$v^{-,N}(x,0;\tilde{n})\leq 1-\varepsilon\leq v^{N}(x,0),$

where the first inequality comes from the fact that
$v^{-,N}(x,0;\tilde{n})\leq 1-\varepsilon$ for all $x$. Therefore, if
$\Delta_{x}\leq\tilde{n}^{-1/(m+2)}$, then for all $x$,

$v^{-,N}(x,0;\tilde{n})\leq v^{N}(x,0).$

Together, the above bounds give us an ordering of the initial conditions of
the three schemes whenever
$\Delta_{x}\leq\min(\Pi^{1-\varepsilon}(\varepsilon),\tilde{n}^{-1/(m+2)})$:

$v^{-,N}(x,0;\tilde{n})\leq v^{N}(x,0)\leq v^{+,N}(x,0;\tilde{n}).$

If $N$ is large enough that
$\Delta_{x}\leq\min(\Pi^{1-\varepsilon}(\varepsilon),\tilde{n}^{-1/(m+2)},\eta)$,
then we also have that $v^{-,N}(x,0;\tilde{n}),$ $v^{N}(x,0),$
$v^{+,N}(x,0;\tilde{n})$ are each bounded above by $p^{*}$ and ordered. For
such $N$, it then follows by Lemma 4.3 that for all $x\in\mathbb{R}$, $t>0$,

$v^{-,N}(x,t;\tilde{n})\leq v^{N}(x,t)\leq v^{+,N}(x,t;\tilde{n}).$

Now fix $T>1$, $K\subseteq\mathbb{R}$ compact, and $\delta>0$. We can then
apply Proposition 4.2 at time $t=1<T$ to see that there exists a constant
$N_{0}=N_{0}(m,\delta,M,K,T)$ such that for all $N\geq N_{0}$, for all
$x\in\mathbb{R}$,

$\displaystyle v^{+,N}(x,1;\tilde{n})$ $\displaystyle\leq
v^{+}(x,1;\tilde{n})+\delta\hskip 14.45377pt\text{and}$ $\displaystyle
v^{-,N}(x,1;\tilde{n})$ $\displaystyle\geq v^{-}(x,1;\tilde{n})-\delta.$

Combining the above inequalities, and recalling that
$\Delta_{x}=\Delta_{x}^{N}=N^{-1/(m+2)}$, we see that if $N\geq N_{0}$ is
large enough that
$\Delta_{x}\leq\min(\Pi^{1-\varepsilon}(\varepsilon),\tilde{n}^{-1/(m+2)},\eta)$,
then

$\mathbf{P}(X_{N}\leq
xN^{1/(m+2)})=v^{N}(x,1)\in[v^{-}(x,1;\tilde{n})-\delta,v^{+}(x,1;\tilde{n})+\delta]\,.$

For
$\tilde{n}\geq\max(1/\Pi^{1-\varepsilon}(\varepsilon),N_{0},\eta^{-(m+2)})$,
requiring that $N\geq\tilde{n}$ will automatically satisfy the other
constraints on $N$.

Furthermore, by Theorem 3.1, we know that $v_{\varepsilon}$ is the unique
viscosity solution of

$\begin{cases}(v_{\varepsilon})_{t}-\frac{m+1}{2}\left|(v_{\varepsilon})_{x}\right|^{m}(v_{\varepsilon})_{xx}=0&\text{in
$\mathbb{R}\times(0,T]$,}\\\
v_{\varepsilon}(x,0)=v_{\varepsilon}(x,0)&\text{in $\mathbb{R}$,}\end{cases}$

and since the PDE is invariant to shifts in space and addition by constants,
this implies that

$v^{+}(x,t;\tilde{n})=\varepsilon+\int_{-\infty}^{x-L\tilde{n}^{-1/(m+2)}+2\Pi^{1-\varepsilon}(\varepsilon)}\hat{U}(y,t+\varepsilon;1-\varepsilon)\,dy,$

and

$v^{-}(x,t;\tilde{n})=\int_{-\infty}^{x-R\tilde{n}^{-1/(m+2)}-\Pi^{1-\varepsilon}(\varepsilon)}\hat{U}(y,t+\varepsilon;1-\varepsilon)\,dy.$

The above bounds then imply that

$\displaystyle\limsup_{N\to\infty}\mathbf{P}(X_{N}\leq xN^{1/(m+2)})$
$\displaystyle\leq\inf_{\tilde{n}\geq\max(1/\Pi^{1-\varepsilon}(\varepsilon),N_{0},\eta^{-(m+2)})}\left(\varepsilon+\int_{-\infty}^{x-L\tilde{n}^{-1/(m+2)}+2\Pi^{1-\varepsilon}(\varepsilon)}\hat{U}(y,1+\varepsilon;1-\varepsilon)\,dy\right)+\delta$
$\displaystyle=\varepsilon+v_{\varepsilon}(x+2\Pi^{1-\varepsilon}(\varepsilon),1)+\delta$

and

$\displaystyle\liminf_{N\to\infty}\mathbf{P}(X_{N}\leq xN^{1/(m+2)})$
$\displaystyle\geq\sup_{\tilde{n}\geq\max(1/\Pi^{1-\varepsilon}(\varepsilon),N_{0},\eta^{-(m+2)})}\left(\int_{-\infty}^{x-R\tilde{n}^{-1/(m+2)}-\Pi^{1-\varepsilon}(\varepsilon)}\hat{U}(y,1+\varepsilon;1-\varepsilon)\,dy\right)-\delta$
$\displaystyle=v_{\varepsilon}(x-\Pi^{1-\varepsilon}(\varepsilon),1)-\delta.$

Now recalling by the explicit formula for $\Pi^{1-\varepsilon}(\varepsilon)$
in (5.4) and the formula for $v_{\varepsilon}$ in (5.3), we have

$\lim_{\varepsilon\to 0}\Pi^{1-\varepsilon}(\varepsilon)=0$

and

$\lim_{\varepsilon\to
0}v_{\varepsilon}(x-\Pi^{1-\varepsilon}(\varepsilon),1)=\int_{-\infty}^{x}\hat{U}(y,1;1)\,dy=\lim_{\varepsilon\to
0}v_{\varepsilon}(x+2\Pi^{1-\varepsilon}(\varepsilon),1)\,.$

Therefore, taking the limit as $\delta,\varepsilon\to 0$, we see that

$\displaystyle\int_{-\infty}^{x}\hat{U}(y,1;1)\,dy$
$\displaystyle\leq\liminf_{N\to\infty}\mathbf{P}(X_{N}\leq xN^{1/(m+2)})$
$\displaystyle\leq\limsup_{N\to\infty}\mathbf{P}(X_{N}\leq xN^{1/(m+2)})$
$\displaystyle\leq\int_{-\infty}^{x}\hat{U}(y,1;1)\,dy,$

and so

$\lim_{N\to\infty}\mathbf{P}\left(\frac{X_{N}}{N^{1/(m+2)}}\leq
x\right)=\int_{-\infty}^{x}\hat{U}(y,1;1)\,dy.$

The fact that $\hat{U}(y,1;1)$ is the density of the scaled and recentered
Beta distribution (5.1) was explained in our discussion in Section 3.1; this
completes the proof. ∎

##  6\. Relaxation of $p^{*}$-bounded and the Proof of Theorem 1.1

###  6.1. Eventually $p^{*}$-bounded distributions

In this section, we show that every symmetric cooperative motion eventually
has a $p^{*}$-bounded distribution.

######  Proposition 6.1.

Let $(X_{n},n\geq 0)$ be $\mathrm{SCM}(m,\mu)$-distributed. There exists a
constant $N:=N(m)$ such that for all $n\geq N$,
$\max_{k\in\mathbb{Z}}\mathbf{P}(X_{n}=k)\leq p^{*}$.

###### Proof.

By Lemma 4.3 we know that if $\max_{k\in\mathbb{Z}}p^{n}_{k}\leq p^{*}$ then
also $\max_{k\in\mathbb{Z}}p^{n+1}_{k}\leq p^{*}$. It therefore suffices to
show that there exists $C>0$ such that for all $n\geq 0$ and $k\in\mathbb{Z}$,
if $\max_{k\in\mathbb{Z}}p^{n}_{k}>p^{*}$ then

$p^{n+1}_{k}\leq\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-C.$ (6.1)

Indeed, if this holds, then for all $n\geq 0$ and all $k\in\mathbb{Z}$,

$\displaystyle\mathbf{P}(X_{n+1}=k)$
$\displaystyle\leq\max\big{(}p^{*},\max(\mathbf{P}(X_{n}=k-1),\mathbf{P}(X_{n}=k),\mathbf{P}(X_{n}=k+1))-C\big{)}$
$\displaystyle\leq\max(p^{*},\max_{k\in\mathbb{Z}}\mathbf{P}(X_{n}=k)-C)\,,$

which implies that $\max_{k\in\mathbb{Z}}\mathbf{P}(X_{n}=k)\leq p^{*}$ for
all $n\geq(1-p^{*})/C$.

We first treat the case that
$\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})>p^{*}\geq 1/2$. This implies that
$m\geq 1$, and we will use this fact in the argument. By definition we have

$p^{n+1}_{k}-p^{n}_{k}=\frac{1}{2}\left((p^{n}_{k+1})^{m+1}-2(p^{n}_{k})^{m+1}+(p^{n}_{k-1})^{m+1}\right).$

We can write the right-hand side as
$-(z^{m+1}-\tfrac{1}{2}x^{m+1}-\tfrac{1}{2}y^{m+1})$ with
$z=p^{n}_{k},x=p^{n}_{k+1},y=p^{n}_{k-1}$. Since $z^{m+1}$ is increasing in
$z$ and also $a^{m+1}+b^{m+1}\leq(a+b)^{m+1}$ for $a,b\geq 0$ by convexity, if
$x+y\leq\tfrac{1}{2}\leq z$, then $z^{m+1}\geq 2^{-(m+1)}\geq
x^{m+1}+y^{m+1}$. This implies that

$z^{m+1}-\frac{1}{2}x^{m+1}-\frac{1}{2}y^{m+1}\geq
z^{m+1}-\frac{1}{2}z^{m+1}=\frac{1}{2}z^{m+1}\geq\frac{1}{2^{m+2}}\,,$

which shows that if $p^{n}_{k}\geq 1/2$, then $p^{n+1}_{k}\leq
p^{n}_{k}-1/2^{m+2}$.

Next, by definition we have

$\displaystyle p^{n+1}_{k}-p^{n}_{k+1}$
$\displaystyle=p^{n}_{k}-p^{n}_{k+1}+\frac{1}{2}\Big{(}(p^{n}_{k+1})^{m+1}-2(p^{n}_{k})^{m+1}+(p^{n}_{k-1})^{m+1}\Big{)}$
$\displaystyle=-\Big{(}z-\frac{1}{2}z^{m+1}-y+y^{m+1}-\frac{1}{2}x^{m+1}\Big{)},$

with $z=p^{n}_{k+1},y=p^{n}_{k},x=p^{n}_{k-1}$. Moreover, since $x+y\leq 1/2$,
$\min(x,y)<1/3$. (Indeed, $\min(x,y)\leq 1/4$, but there are enough factors of
$2$ floating around that we instead use the bound $1/3$ to reduce potential
confusion over what factors come from where.) Combined with the fact that when
$m\geq 1$, $x-x^{m+1}$ and $x^{m+1}$ are increasing on $[0,1/2]$, we thus have

$-y+y^{m+1}-\frac{1}{2}x^{m+1}\geq-\frac{1}{2}+\frac{1}{2^{m+1}}-\frac{1}{2^{m+2}}+C_{1}\,,$

where
$C_{1}:=\min\left(\frac{1}{2}-\left(\frac{1}{2}\right)^{m+1}-\frac{1}{3}+\left(\frac{1}{3}\right)^{m+1},\frac{1}{2}\left[\left(\frac{1}{2}\right)^{m+1}-\left(\frac{1}{3}\right)^{m+1}\right]\right)$.
Since $z\geq\tfrac{1}{2}$ and $z-\frac{z^{m+1}}{2}$ is concave down on
$[1/2,1]$, we also have
$z-\frac{z^{m+1}}{2}\geq\tfrac{1}{2}-\tfrac{1}{2^{m+2}}$, so

$z-\frac{1}{2}z^{m+1}-y+y^{m+1}-\frac{1}{2}x^{m+1}\geq C_{1}>0,$

and therefore if $p^{n}_{k+1}\geq 1/2$ then $p^{n+1}_{k}\leq
p^{n}_{k+1}-C_{1}$.

By a symmetric argument, if $p^{n}_{k-1}\geq 1/2$ then $p^{n+1}_{k}\leq
p^{n}_{k-1}-C_{1}$. Combining the three cases, it follows that if
$\max(p^{n}_{k-1},p^{n}_{k},p^{n+1}_{k})\geq p^{*}\geq 1/2$, then
$p^{n+1}_{k}\leq\max(p^{n}_{k-1},p^{n}_{k},p^{n+1}_{k})-\min(C_{1},1/2^{m+2})$.

We now turn to the case that $p^{*}<1/2$ and
$\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})>p^{*}$, and again show that in this
case

$p^{n+1}_{k}\leq\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-C$

for some absolute constant $C>0$. Whenever $p^{*}<\frac{1}{2}$, we
automatically have $m<1$, and we will use this fact throughout the rest of the
proof. The arguments are similar to those above, with the addition of new
cases.

Before beginning our analysis, we remark that

$x\mapsto x-x^{m+1}$ is increasing on $[0,p^{*})$ and decreasing on
$(p^{*},1)$, (6.2)

and for $m<1$,

$x\mapsto x-\frac{1}{2}x^{m+1}$ is increasing on $[0,1)$, (6.3)

We will use this fact several times in the argument.

Suppose that $p^{n}_{k}>p^{*}$. First, note that since $p^{*}>1/3$, at least
one of $p^{n}_{k-1}$ and $p^{n}_{k+1}$ is less than $1/3$. By the symmetry of
the recurrence, we may assume that $p^{n}_{k-1}<1/3$. If also $p^{n}_{k}\geq
p^{n}_{k+1}$, it follows that

$\displaystyle\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}$
$\displaystyle=p^{n}_{k}-p^{n+1}_{k}$
$\displaystyle=(p^{n}_{k})^{m+1}-\frac{1}{2}(p^{n}_{k-1})^{m+1}-\frac{1}{2}(p^{n}_{k+1})^{m+1}$
$\displaystyle\geq(p^{n}_{k})^{m+1}-\frac{1}{2}(p^{n}_{k-1})^{m+1}-\frac{1}{2}(p^{n}_{k})^{m+1}$
$\displaystyle\geq\frac{1}{2}(p^{*})^{m+1}-\frac{1}{2}\left(\frac{1}{3}\right)^{m+1}$

so
$p^{n+1}_{k}<\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-\tfrac{1}{2}((p^{*})^{m+1}-(1/3)^{m+1})$.
On the other hand, if $p^{n}_{k+1}>p^{n}_{k}$ then it follows that

$\displaystyle\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}$
$\displaystyle=p^{n}_{k+1}-p^{n+1}_{k}$
$\displaystyle=p^{n}_{k+1}-\frac{1}{2}(p^{n}_{k+1})^{m+1}-p^{n}_{k}+(p^{n}_{k})^{m+1}-\frac{1}{2}(p^{n}_{k-1})^{m+1}\,.$

Since $p^{n}_{k+1}\geq p^{*}$, by (6.3) we have
$p^{n}_{k+1}-\tfrac{1}{2}(p^{n}_{k+1})^{m+1}\geq
p^{*}-\tfrac{1}{2}(p^{*})^{m+1}$. Using (6.2), we have
$-p^{n}_{k}+(p^{n}_{k})^{m+1}\geq-p^{*}+(p^{*})^{m+1}$. Finally, since
$p^{n}_{k+1}>p^{n}_{k}>1/3$, we have
$\tfrac{1}{2}(p^{n}_{k-1})^{m+1}<\tfrac{1}{2}(1/3)^{m+1}$. Using all these
bounds together yields that

$\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}\geq\frac{1}{2}\left((p^{*})^{m+1}-\left(\frac{1}{3}\right)^{m+1}\right),$

so again
$p^{n+1}_{k}\leq\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-\tfrac{1}{2}((p^{*})^{m+1}-(1/3)^{m+1})$.

Finally, suppose that
$p^{n}_{k}<p^{*}<\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})$. As noted above,
this implies that $m<1$. We suppose without loss of generality that
$\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})=p^{n}_{k+1}$. We argue in two cases,
depending on whether or not $p^{n}_{k-1}>p^{*}$. In both cases, we have

$\displaystyle\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}$
$\displaystyle=p^{n}_{k+1}-p^{n+1}_{k}$
$\displaystyle=p^{n}_{k+1}-\frac{1}{2}(p^{n}_{k+1})^{m+1}-p^{n}_{k}+(p^{n}_{k})^{m+1}-\frac{1}{2}(p^{n}_{k-1})^{m+1}\,,$
(6.4)

and we aim to bound the right-hand side away from zero from below.

If $p^{n}_{k-1}\leq p^{*}$, then since both $x-x^{m+1}$ and $x^{m+1}/2$ are
increasing on $[0,p^{*})$ and $\min(p^{n}_{k-1},p^{n}_{k})<1/3<p^{*}$, it
follows that

$\displaystyle\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}$
$\displaystyle\geq
p^{n}_{k+1}-\frac{1}{2}(p^{n}_{k+1})^{m+1}-p^{*}+(p^{*})^{m+1}-\frac{1}{2}(p^{*})^{m+1}+C_{2}$
$\displaystyle=p^{n}_{k+1}-\frac{1}{2}(p^{n}_{k+1})^{m+1}-p^{*}+\frac{1}{2}(p^{*})^{m+1}+C_{2}\,,$

where we have taken

$C_{2}:=\min\left(p^{*}-(p^{*})^{m+1}-\frac{1}{3}+\left(\frac{1}{3}\right)^{m+1},\frac{1}{2}(p^{*})^{m+1}-\frac{1}{2}\left(\frac{1}{3}\right)^{m+1}\right)>0\,.$

By (6.3), since $p^{n}_{k+1}=\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})$, it
follows that

$\displaystyle\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}>C_{2}>0\,.$

If $p^{n}_{k-1}>p^{*}$, then we bound the right-hand side of (6.4) away from
zero by showing that in this case,
$\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}$ is minimized when
$p^{n}_{k-1}=p^{n}_{k+1}=p^{*}$ and $p^{n}_{k}=1-2p^{*}$. To accomplish this,
we reparametrize the equation, letting $p^{n}_{k-1}+p^{n}_{k+1}=2p^{*}+T$ and
$p^{n}_{k+1}=p^{*}+\frac{T}{2}+\Lambda$, so that
$p^{n}_{k-1}=p^{*}+\frac{T}{2}-\Lambda$. Notice that $T\in[0,1-2p^{*}]$ since
$p^{n}_{k+1}=\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})$, and
$\Lambda\in\left[0,T/2\right]$, since $p^{n}_{k+1}\geq p^{n}_{k-1}$. With
these new variables, we have

$\displaystyle g(T,\Lambda,p^{n}_{k})$
$\displaystyle:=\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-p^{n+1}_{k}$
$\displaystyle=p^{*}+\frac{T}{2}+\Lambda-\frac{1}{2}\left(p^{*}+\frac{T}{2}+\Lambda\right)^{m+1}-\frac{1}{2}\left(p^{*}+\frac{T}{2}-\Lambda\right)^{m+1}-p^{n}_{k}+\left(p^{n}_{k}\right)^{m+1}\,.$

It follows that

$\frac{\partial
g}{\partial\Lambda}=1-\frac{m+1}{2}(p^{*}+\tfrac{T}{2}+\Lambda)^{m}+\frac{m+1}{2}(p^{*}+\tfrac{T}{2}-\Lambda)^{m}>0$

the last inequality holding for $\Lambda\in[0,T/2]$ since $m<1$. This implies
that

$g(T,\Lambda,p^{n}_{k})\geq g(T,0,p^{n}_{k}).$

By (6.2) and since $p^{n}_{k}\in[0,1-2p^{*}-T]\subset[0,p^{*})$, it follows
that

$\displaystyle g(T,\Lambda,p^{n}_{k})$ $\displaystyle\geq g(T,0,1-2p^{*}-T)$
$\displaystyle=p^{*}+\frac{T}{2}-\left(p^{*}+\frac{T}{2}\right)^{m+1}-(1-2p^{*}-T)+\left(1-2p^{*}-T\right)^{m+1}\,.$

Finally, taking a derivative of $g(T,0,1-2p^{*}-T)$ in $T$, we get

$\frac{dg(T,0,1-2p^{*}-T)}{dT}=\frac{3}{2}-\frac{m+1}{2}\left(p^{*}+\frac{T}{2}\right)^{m}-(m+1)\left(1-2p^{*}-T\right)^{m}.$

Since $p^{*}=\frac{1}{(m+1)^{1/m}}$, we have

$\displaystyle\left.\frac{dg(T,0,1-2p^{*}-T)}{dT}\right|_{T=0}$
$\displaystyle=\frac{3}{2}-\frac{m+1}{2}\left(p^{*}\right)^{m}-(m+1)\left(1-2p^{*}\right)^{m}$
$\displaystyle=\frac{3}{2}-\frac{1}{2}-(m+1)\left(1-2p^{*}\right)^{m}$
$\displaystyle=1-(m+1)\left(1-2p^{*}\right)^{m}>0$

the inequality holding since $1-2p^{*}<p^{*}$ and $m<1$. To see that
$\frac{dg(T,0,1-2p^{*}-T)}{dT}>0$ for all $T\in[0,1-2p^{*}]$, we show that
$\frac{d^{2}g(T,0,1-2p^{*}-T)}{dT^{2}}>0$. We compute

$\displaystyle\frac{d^{2}g(T,0,1-2p^{*}-T)}{dT^{2}}$
$\displaystyle=-\frac{m(m+1)}{4}\left(p^{*}+\frac{T}{2}\right)^{m-1}+m(m+1)(1-2p^{*}-T)^{m-1}$
$\displaystyle=m(m+1)\left[(1-2p^{*}-T)^{m-1}-\frac{1}{4}\left(p^{*}+\frac{T}{2}\right)^{m-1}\right]>0,$

where the last inequality is due to the fact that $1-2p^{*}-T<p^{*}+T/2$ and
$x^{m-1}$ is decreasing on $[0,1]$.

Because the second derivative is positive and $\tfrac{dg}{dT}|_{T=0}>0$, we
know that $\tfrac{dg}{dT}>0$ for all $T\in[0,1-2p^{*}]$ and thus that
$g(T,0,1-2p^{*}-T)\geq g(0,0,1-2p^{*})$. Putting this all together, we get

$\displaystyle g(T,D,p^{n}_{k})$ $\displaystyle\geq g(0,0,1-2p^{*})$
$\displaystyle=p^{*}-\left(p^{*}\right)^{m+1}-(1-2p^{*})+\left(1-2p^{*}\right)^{m+1}=:C_{3}>0\,,$

the inequality holding by (6.2), since $p^{*}>1-2p^{*}$. Combining the above
cases, it follows that when $p^{*}<1/2$ and
$\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})>p^{*}$, we have

$p^{n+1}_{k}\leq\max(p^{n}_{k-1},p^{n}_{k},p^{n}_{k+1})-C$

for $C=\min(C_{2},C_{3})>0$, where $C_{2},C_{3}$ are as defined above. ∎

###  6.2. The proof of Theorem 1.1

We conclude the section with the proof of Theorem 1.1.

###### Proof of Theorem 1.1.

Fix $m>0$ and $\mu$ an initial probability distribution. As $(X_{n},n\geq 0)$
is $\mathrm{SCM}(m,\mu)$-distributed, by Proposition 6.1, there exists
$N=N(m)$ so that for all $n\geq N$,
$\max_{k\in\mathbb{Z}}\mathbf{P}(X_{n}=k)\leq p^{*}$. Define $\tilde{\mu}$ to
be the distribution of $X_{N}$, so that $\tilde{\mu}$ is $p^{*}$-bounded. If
$\tilde{X}_{n}:=X_{N+n}$, then $(\tilde{X}_{n},n\geq 0)$ is
SCM$(m,\tilde{\mu})$-distributed. By Proposition 5.1, this implies that

$\lim_{n\to\infty}\mathbf{P}\left(\frac{\tilde{X}_{n}}{n^{1/(m+2)}}\leq
x\right)=v(x,1)$

locally uniformly in $x$, where $v(x,1)$ is the CDF of

$\frac{2^{(m+1)/(m+2)}(m+1)^{1/(m+2)}}{D^{m/(m+2)}\rho^{1/2}}\left[\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)-\frac{1}{2}\right].$

Since $N$ is fixed and $v(x,1)$ is continuous, this implies that for all
$x\in\mathbb{R}$,

$\lim_{n\to\infty}\mathbf{P}\left(\frac{X_{n}}{n^{1/(m+2)}}\leq
x\right)=\lim_{n\to\infty}\mathbf{P}\left(\frac{\tilde{X}_{n}}{n^{1/(m+2)}}\leq
x\left(\frac{n-N}{n}\right)^{1/(m+2)}\right)=v(x,1),$

so

$\frac{X_{n}}{n^{1/(m+2)}}\xrightarrow{d}\frac{2^{\frac{m+1}{m+2}}(m+1)^{1/(m+2)}}{D^{\frac{m}{m+2}}\rho^{1/2}}\left(B-\frac{1}{2}\right),$

where $B$ is
$\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)$-distributed, as
required. ∎

##  7\. Generalizations and Extensions

###  7.1. Robustness of the Method

While we have focused on proving Theorem 1.1, we believe the approach and
method of proof are rather general, and can potentially be applied to other
interesting probabilistic models. We next present a summary of the general
ingredients needed, by our method, to prove a distributional convergence
result for a generic sequence of random variables $(X_{n},n\geq 0)$ on
$\mathbb{Z}$, whose laws are specified by a recursive distributional equation
(RDE). Throughout the discussion, we highlight where each of these ingredients
appears in our proof of the Bernoulli central limit theorem in Section 2.2 and
the proof of Theorem 1.1.

Given an (RDE) for the CDF $F^{n}_{k}=\mathbf{P}[X_{n}\leq k]$, we seek the
following ingredients:

  1. (1)

_The RDE is a discrete approximation of a well-posed PDE_. Presented with an
RDE, one may attempt to “guess” a PDE which the RDE approximates. It is then
crucial to identify whether this PDE is well-posed; in particular whether it
is equipped with a solution theory which has existence and uniqueness for
initial conditions which are (extended) CDFs. In the context of the Bernoulli
central limit theorem, we relied on the theory of classical solutions; in the
context of Theorem 1.1, we relied on viscosity solutions for PDEs with
Lipschitz continuous, extended CDF initial conditions.

  2. (2)

_A self-similar scaling_. Since the RDE describes the dynamics at integer
times and integer locations, one further needs to identify a self-similar
scaling of space and time which keeps the PDE limit of the CDF invariant. In
the case of the Bernoulli central limit theorem, this was the classical
Brownian/parabolic scaling, and in the case of Theorem 1.1, this was given by
$t\mapsto ct$, $x\mapsto c^{1/(m+2)}x$. Upon identifying a self-similar
scaling, one can hope to translate the RDE into a finite difference scheme, as
we did in transitioning from (4.1) to (4.3).

  3. (3)

_An established theory of convergence results for finite difference schemes of
the PDE._ In these works, we relied on the convergence results of Barles and
Souganidis [3] for degenerate parabolic PDEs with continuous initial
conditions; equipped with convergence results, we had to ensure that our
finite difference scheme satisfied the appropriate hypotheses of the Barles-
Souganidis theorem. For other models, different convergence results for finite
difference schemes approximating solutions of PDEs may be used. (For example,
[1] uses results of Crandall and Lions [9]; [2] uses results of Evje and
Karlsen [11].)

  4. (4)

_Coupling different initial conditions._ It is possible that either the
solution theory of the well-posed PDE or the convergence results for the
associated finite difference schemes are not robust enough to handle
discretizations of a PDE with a Heaviside initial condition. This was the case
in both the Bernoulli central limit theorem and Theorem 1.1. We resolved this
by developing a “sandwiching argument,” which ultimately relied on coupling
stochastic processes with different initial conditions.

  5. (5)

_(Optional.) A way of identifying the explicit solution of the PDE with a
Heaviside initial condition_. The prior ingredients yield that the CDF of a
rescaled random variable converges to a function which solves a PDE with a
Heaviside initial condition. This function (evaluated at time $1$) is
precisely the CDF of the limiting random variable. To obtain a more explicit
description of the limiting distribution, one may attempt to identify an
explicit representation for the solution of this PDE with a Heaviside initial
condition. For the Bernoulli central limit theorem, this was automatic due to
the theory of fundamental solutions. In the setting of Theorem 1.1, we needed
to develop the results of Section 3 in order to identify the symmetric Beta
random variable in the limit.

While this discussion is not a precise general theorem, we hope that it
clarifies the main ingredients needed in order to adapt our approach to
different models.

One such adaptation we mention here is cooperative motion with “$q$-lazy
dynamics”, where as before,

$X_{n+1}=\begin{cases}X_{n}+E_{n}&\mbox{ if }X_{n}=X_{n,1}=\ldots=X_{n,m}\\\
X_{n}&\mbox{ otherwise,}\end{cases}$

for $X_{n,1},\ldots,X_{n,m}$ IID copies of $X_{n}$, but now $(E_{n},n\geq 0)$
are IID with
${\mathbf{P}}\left(E_{0}=1\right)=q/2={\mathbf{P}}\left(E_{0}=-1\right)$,
${\mathbf{P}}\left(E_{0}=0\right)=1-q$. In this case, a straightforward
extension of our method yields that the appropriate PDE governing the limiting
probability distribution is given by

$u_{t}-\frac{q}{2}(u^{m+1})_{xx}=0.$

An analysis of the corresponding ZKB solution of the above PDE yields that

$\frac{X_{n}}{n^{1/(m+2)}}\xrightarrow{d}\frac{2^{\frac{m+1}{m+2}}(m+1)^{1/(m+2)}q^{1/(m+2)}}{D^{\frac{m}{m+2}}\rho^{1/2}}\cdot\left(B-\frac{1}{2}\right),$

where $B$ is again
$\operatorname{Beta}\left(\frac{m+1}{m},\frac{m+1}{m}\right)$-distributed.

It would of course be natural to consider step sizes with support not
contained in $\\{-1,0,1\\}$; however in the setting of cooperative motion,
monotonicity becomes an issue for more general step sizes. (This is explained
in more detail in [1, Section 5.2].) There is only one other family of cases
we can handle — when the step sizes have support contained in $\\{-R,0,R\\}$
for some fixed $R\in\mathbb{N}$. We turn to this in the next subsection.

###  7.2. Persistent Lattice Effects

Fix $q\in(0,1]$ and $r\in[1/2,1]$, and consider a variant of the cooperative
motion process defined by

$X_{n+1}=\begin{cases}X_{n}+E_{n}&\mbox{ if }X_{n}=X_{n,1}=\ldots=X_{n,m}\\\
X_{n}&\mbox{ otherwise}\end{cases}$ (7.1)

where $(E_{n},n\geq 0)$ are IID with ${\mathbf{P}}\left(E_{0}=R\right)=rq$,
${\mathbf{P}}\left(E_{0}=-R\right)=(1-r)q$ and
${\mathbf{P}}\left(E_{0}=0\right)=1-q$. (As before, $X_{n,1},\ldots,X_{n,m}$
are IID copies of $X_{n}$.) If the initial distribution $\mu$ is supported by
$R\mathbb{Z}+k$ for some $1\leq k\leq R$ then this process simply looks like a
cooperative motion with $\\{-1,0,1\\}$ steps (which is symmetric if $r=1/2$
and asymmetric otherwise). However, for other initial distributions,
alternative asymptotic behaviour can appear. The reason for this is explained
in detail in the setting of asymmetric cooperative motion in our previous work
[1], in which we prove the following result.

######  Theorem 7.1 ([1], Theorem 5.1).

Consider the cooperative motion process defined in (7.1), with $r>1/2$ and
with $r$ and $q$ not both $1$. Let $\pi_{k}={\mathbf{P}}\left(X_{0}=k\mod
R\right)$ for $k\in\\{1,2,\ldots,r\\}$. Then

$\frac{X_{n}}{n^{1/(m+1)}}\xrightarrow{d}R(m+1)\left(\frac{(2r-1)q}{m^{m}}\right)^{\frac{1}{m+1}}B\cdot\sum_{k=1}^{R}(\pi_{k})^{\frac{m}{m+1}}\mathds{1}_{\left\\{A=k\right\\}},$

where $A$ is a random variable with $\mathbf{P}(A=k)=\pi_{k}$ for
$k\in\\{1,2,\ldots,R\\}$, and $B$ is
$\operatorname{Beta}(\tfrac{m+1}{m},1)$-distributed and independent of $A$.

The analogous result for symmetric cooperative motion is as follows.

######  Theorem 7.2.

Consider the cooperative motion process defined in (7.1), with $r=1/2$. Let
$\pi_{k}={\mathbf{P}}\left(X_{0}=k\mod R\right)$ for $k\in\\{1,2,\ldots,r\\}$.
Then

$\frac{X_{n}}{n^{1/(m+2)}}\xrightarrow{d}\frac{2^{\frac{m+1}{m+2}}(m+1)^{\frac{1}{m+2}}q^{\frac{1}{m+2}}}{D^{\frac{m}{m+2}}\rho^{1/2}}\left(B-\frac{1}{2}\right)\cdot\sum_{k=1}^{R}(\pi_{k})^{\frac{m}{m+1}}\mathds{1}_{\left\\{A=k\right\\}},$

where $A$ is a random variable with $\mathbf{P}(A=k)=\pi_{k}$ for
$k\in\\{1,2,\ldots,R\\}$, and $B$ is
$\operatorname{Beta}(\tfrac{m+1}{m},\tfrac{m+1}{m})$-distributed and
independent of $A$.

We omit the proof of Theorem 7.2 as it is both straightforward and precisely
analogous to the (rather short) proof of Theorem 7.1 from [1].

##  Appendix A Facts about Relevant Solutions

We begin with a review of distributional solutions.

######  Definition A.1.

Let $\Psi:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and nondecreasing and
let $u_{0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$. We say that $u\in
C([0,T];L^{1}(\mathbb{R}))\cap L^{\infty}(\mathbb{R}\times[0,T])$ is a
distributional solution of

$\begin{cases}u_{t}-(\Psi(u))_{xx}=0&\text{in $\mathbb{R}\times(0,T]$}\,,\\\
u(x,0)=u_{0}(x)&\text{in $\mathbb{R}$}\,,\end{cases}$ (A.1)

if and only if for all $\varphi\in C^{\infty}_{c}(\mathbb{R}\times[0,T))$,

$\int_{0}^{T}\int_{-\infty}^{\infty}u\varphi_{t}+\Psi(u)\varphi_{xx}\,dxdt+\int_{-\infty}^{\infty}u_{0}(x)\varphi(x,0)\,dx=0.$
(A.2)

We also define viscosity solutions of a certain class of PDEs.

######  Definition A.2.

Consider the good initial value problem

$\begin{cases}v_{t}+G(v_{x},v_{xx})=0&\text{in $\mathbb{R}\times(0,T]$,}\\\
v(x,0)=v_{0}(x)&\text{in $\mathbb{R}$}.\end{cases}$ (A.3)

We say that a bounded, uniformly continuous function
$v:\mathbb{R}\times[0,T]\rightarrow\mathbb{R}$ is a viscosity subsolution of
(A.3) if for any function $\varphi\in C^{2,1}(\mathbb{R}\times(0,T])$ such
that $v-\varphi$ has a local maximum at
$(x_{0},t_{0})\in\mathbb{R}\times(0,T]$, we have

$\varphi_{t}(x_{0},t_{0})+G(\varphi_{x}(x_{0},t_{0}),\varphi_{xx}(x_{0},t_{0}))\leq
0.$

We say that $v$ is a viscosity supersolution of (A.3) if for any function
$\varphi\in C^{2,1}(\mathbb{R}\times(0,T])$ such that $v-\varphi$ has a local
minimum at $(x_{0},t_{0})\in\mathbb{R}\times(0,T]$, we have

$\varphi_{t}(x_{0},t_{0})+G(\varphi_{x}(x_{0},t_{0}),\varphi_{xx}(x_{0},t_{0}))\geq
0.$

Finally, we say that $v$ is a viscosity solution of (A.3) if and only if $v$
is both a viscosity subsolution and supersolution of (A.3), and additionally,
for all $x\in\mathbb{R}$, $v(y,t)\to v_{0}(x)$ as $(y,t)\to(x,0)$.

######  Remark A.3.

If $u$ and $v$ belong to $C^{2,1}(\mathbb{R}\times(0,T])\cap
C(\mathbb{R}\times[0,T])$ and are classical solutions of (A.1) and (A.3) on
$\mathbb{R}\times(0,T]$, respectively, then they are also distributional and
viscosity solutions, respectively. To see this, first note that in the case of
(A.1), if $u$ is sufficiently regular then (A.2) is automatically satisfied by
integration by parts.

In the case when $v\in C^{2,1}(\mathbb{R}\times(0,T])\cap
C(\mathbb{R}\times[0,T])$, verifying that $v$ is a viscosity solution is also
straightforward since $G$ is degenerate elliptic. Whenever $v-\varphi$ has a
local maximum at $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$, we have

$v_{t}(x_{0},t_{0})=\varphi_{t}(x_{0},t_{0})\quad\text{and}\quad
v_{x}(x_{0},t_{0})=\varphi_{x}(x_{0},t_{0})\quad\text{and}\quad
v_{xx}(x_{0},t_{0})\leq\varphi_{xx}(x_{0},t_{0}).$

If $(x_{0},t_{0})\in\mathbb{R}\times\left\\{t=T\right\\}$, we have

$v_{t}(x_{0},t_{0})\geq\varphi_{t}(x_{0},t_{0})\quad\text{and}\quad
v_{x}(x_{0},t_{0})=\varphi_{x}(x_{0},t_{0})\quad\text{and}\quad
v_{xx}(x_{0},t_{0})\leq\varphi_{xx}(x_{0},t_{0}).$

This implies that

$\varphi_{t}(x_{0},t_{0})+G(\varphi_{x}(x_{0},t_{0}),\varphi_{xx}(x_{0},t_{0}))\leq
v_{t}(x_{0},t_{0})+G(v_{x}(x_{0},t_{0}),v_{xx}(x_{0},t_{0}))=0,$

and this verifies the subsolution property. The supersolution property is
analogous.

A similar computation to the one above can be used to show that the ZKB
solution shifted by any $\varepsilon>0$ amount of time (so given by
$\hat{U}(x,t+\varepsilon;1)$, where $\hat{U}$ is as in (3.1)) is in fact the
unique distributional solution of the porous medium equation with initial
condition $\hat{U}(x,\varepsilon;1)$.

######  Lemma A.4.

Let

$\hat{U}(x,t;\theta)=\frac{1}{t^{1/(m+2)}}\left[\left(\frac{\sqrt{2}\theta}{D\sqrt{m+1}}\right)^{\frac{2m}{m+2}}-\frac{2\rho|x|^{2}}{(m+1)t^{2/(m+2)}}\right]^{1/m}_{+}\,.$

For every $\varepsilon>0$, the function
$(x,t)\mapsto\hat{U}(x,t+\varepsilon;\theta)$ is the unique distributional
solution of

$\begin{cases}u_{t}-\frac{1}{2}(u^{m+1})_{xx}=0&\text{in
$\mathbb{R}\times(0,\infty)$,}\\\
u(x,0)=\hat{U}(x,\varepsilon;\theta)&\text{in $\mathbb{R}$.}\end{cases}$ (A.4)

###### Proof.

Throughout the proof, to be consistent with our prior notation, we let
$\Psi(r)=\frac{1}{2}r^{m+1}$. Define

$[-\Pi^{\theta}(t),\Pi^{\theta}(t)]\\\
:=\left[-\frac{\theta^{\frac{m}{m+2}}(m+1)^{1/(m+2)}}{2^{1/(m+2)}\rho^{1/2}D^{m/(m+2)}}t^{1/(m+2)},\frac{\sqrt{\theta}(m+1)^{1/(m+2)}}{2^{1/(m+2)}\rho^{1/2}D^{m/(m+2)}}t^{1/(m+2)}\right],$

which is the support of $\hat{U}(\cdot,t;\theta)$. A routine calculation
verifies that $\hat{U}(\cdot\,,\cdot\,;\theta)$ solves (A.4) for all $t>0$ and
$x\in(-\Pi^{\theta}(t),\Pi^{\theta}(t))$. For
$x\notin(-\Pi^{\theta}(t),\Pi^{\theta}(t))$ we have $\hat{U}(x,t;\theta)=0$,
and hence $\hat{U}(\cdot\,,\cdot\,;\theta)$ is also a classical solution on
$\\{(x,t):x\not\in[-\Pi^{\theta}(t),\Pi^{\theta}(t)]\\}$.

This implies that $(x,t)\mapsto\hat{U}(x,t+\varepsilon;\theta)$ is a piecewise
smooth function which satisfies the porous medium equation in each of its
subdomains of smoothness. Moreover, letting
$u(x,t):=\hat{U}(x,t+\varepsilon;\theta)$, we have that $u(\cdot,t)$ and
$\left(\frac{1}{2}u^{m+1}(\cdot,t)\right)_{x}$ are both 0 at the endpoints of
the support,
$\left\\{-\Pi^{\theta}(t+\varepsilon),\Pi^{\theta}(t+\varepsilon)\right\\}$.
By a standard integration by parts calculation, this implies that $u$ is a
distributional solution. ∎

We conclude the appendix by stating the stability property of viscosity
solutions.

######  Proposition A.5.

[7, Theorem 8.3] Let $(v_{n},1\leq n<\infty)$ denote a collection of viscosity
subsolutions of

$(v_{n})_{t}+G_{n}\left((v_{n})_{x},(v_{n})_{xx}\right)\leq 0\quad\text{in
$\mathbb{R}\times(0,T]$},$

where $G_{n}$ is degenerate elliptic and continuous for each $n$. If
$v_{n}\rightarrow\bar{v}$ and $G_{n}\rightarrow\bar{G}$ locally uniformly,
then $\bar{v}$ is a viscosity subsolution of

$\bar{v}_{t}+\bar{G}\left(\bar{v}_{x},\bar{v}_{xx}\right)\leq 0\quad\text{in
$\mathbb{R}\times(0,T]$}.$

The analogous statement holds for viscosity supersolutions.

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