# Longtime behavior of completely positively correlated Symbiotic Branching
Model

Patric Karl Glöde, Leonid Mytnik  Technion—Israel Institute of Technology,
Faculty of Industrial Engineering & Management, Haifa 32000, Israel
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: September 2022)

###### Abstract.

We study the longtime behavior of a continuous state Symbiotic Branching Model
(SBM). SBM can be seen as a unified model generalizing the Stepping Stone
Model, Mutually Catalytic Branching Processes, and the Parabolic Anderson
Model. It was introduced by Etheridge and Fleischmann [EF04]. The key
parameter in these models is the local correlation $\rho$ between the driving
Brownian Motions. The longtime behavior of all SBM exhibits a dichotomy
between coexistence and non-coexistence of the two populations depending on
the recurrence and transience of the migration and also in many cases on the
branching rate. The most significant gap in the understanding of the longtime
behavior of SBM is for positive correlations in the transient regime. In this
article we give a precise description of the longtime behavior of the SBM with
$\rho=1$ with not necessarily identical initial conditions.

###### Key words and phrases:

Mutually Catalytic Branching, Symbiotic Branching, Parabolic Anderson Model,
Coexistence

###### 2000 Mathematics Subject Classification:

Primary: 60J80; Secondary: 60J85

P.K. Glöde was partially supported by the Technion Fund; both authors were
partially supported by German Israeli Foundation grant 1170-186.6/2011 and L.
Mytnik was partially supported by Israel Science Foundation Grant 1704/18

##  1\. Introduction and Results

###  1.1. Symbiotic Branching Dynamics

The Symbiotic Branching Model (SBM) is a continuous state model of a two type
branching population living and migrating in a geographic space. The branching
rate of each of the two populations locally depends on the size of the other
population. While the SBM is a continuous state model the dynamics are best
understood by having an informal look at the discrete particle approximation.
Both the continuous and the discrete model depend essentially on the branching
rate $b>0$ and the correlation parameter $\rho\in[-1,1]$. Geographic space is
represented by $\mathbb{Z}^{d}$. So the model describes the evolution in time
of the number of particles of each population at each site
$k\in\mathbb{Z}^{d}$ and let us denote the process by
$(X,Y)=(X_{t},Y_{t})_{t\geq 0}$. We refer to the particles of the two
populations as particles of type $1$ and $2$, respectively. The evolution of
the particle model is as follows:

  * •

At each site, each pair of particles of opposite type share an exponential
clock of rate $b|\rho|$. When the clock rings, both particles die and the
following happens depending on the parameter $\rho$:

If $\rho\in[-1,0)$, with equal probability either the type 1 particle has two
offspring and the type 2 particle has no offspring or vice versa.

If $\rho\in(0,1]$, with equal probability either both particles have zero or
two offspring.

  * •

At time $t>0$, each particle of type 1 living at site $k$ has an exponential
clock of rate $b(1-|\rho|)Y_{t}(k)$, each particle of type 2 at site $k$ has
an exponential clock of rate $b(1-|\rho|)X_{t}(k)$. When the clock rings, the
particle dies and with equal probability has either zero or two offspring.

  * •

All particles perform simple continuous, rate $1$, random walks independently
from each other.

  * •

The migration and the branching dynamics are independent from each other.

The particle system described above has a scaling limit. It can be shown that
if each particle is assigned mass $1/n$ and the initial mass is of order $n$,
then the rescaled particle system converges to a system of interacting
diffusions $(u,v)=(u_{t},v_{t})_{t\geq 0}$ which satisfy the following
stochastic differential equations:

$\begin{array}[]{l}\mathrm{d}u_{t}(i)=\Delta
u_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)v_{t}(i)}\,\mathrm{d}W^{u}_{t}(i)\,,\quad
t\geq 0,\\\\[4.30554pt] \mathrm{d}v_{t}(i)=\Delta
v_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)v_{t}(i)}\,\mathrm{d}W^{v}_{t}(i)\,,\quad
t\geq 0,\end{array}$ (1.1)

where $i\in\mathbb{Z}^{d}$ and $\\{(W^{u}(i),W^{v}(i)):i\in\mathbb{Z}^{d}\\}$
is an independent field of locally correlated planar Brownian motions and the
spatial correlation is given by $\rho$. This model was introduced Etheridge
and Fleischmann [EF04] and is known as the Symbiotic Branching Model (SBM).

The SBM generalizes a couple of famous particle models. If $\rho=-1$, the
Brownian motions $W^{u}(i)$ and $W^{v}(i)$ are totally anti-correlated for
each $i\in\mathbb{Z}^{d}$. Under the additional condition that
$u_{0}+v_{0}=\boldsymbol{1}$ for all $k\in\mathbb{Z}^{d}$, the process
$u_{t}+v_{t}$ solves the heat equation and hence since the initial conditions
are constant, $u_{t}+v_{t}\equiv 1$ for all $t\geq 0$. Therefore
$v_{t}=1-u_{t}$ for all $t\geq 0$. One can therefore rewrite (1.8) to obtain
the well known system of differential equations for the stepping stone model
(also known as interacting Fisher-Wright diffusions), see Shiga [Shi80]. If
$\rho=0$, the Brownians motions $W^{u}(i)$ and $W^{v}(i)$ are independent for
each $i\in\mathbb{Z}^{d}$. This leads to the well known mutually catalytic
branching model, studied by Dawson and Perkins [DP98], Cox, Dawson and Greven
[CDG04], Cox, Klenke and Perkins [CKP00] among others. In continuous space, in
dimension $d=2$, this models was studied in a series of papers by Dawson et.
al. [DEF+02a], [DEF+02b], [DFM+03]. If $\rho=1$, the Brownian motions
$W^{u}(i)$ and $W^{v}(i)$ are totally positively correllated for each
$i\in\mathbb{Z}^{d}$. Under the additional assumption that the identical
initial conditions are the same for both populations, that is, $u_{0}=v_{0}$,
the SBM coincides with the Parabolic Anderson Model (PAM). For more
information about the PAM see Carmona and Molchanov [CM94], Greven and den
Hollander [GdH07], Glöde [Glö06].

Let us note, that after certain limiting procedures, the above models give a
rise to the so called infinite rate mutually catalytic and symbiotic models,
that were studied extensively in the recent years, see Klenke and Mytnik
[KM10], [KM12a], [KM12b], [KM20], Blath, Hammer and Ortgiese [BHO16], Hammer,
Ortgiese and Florian [HOV18].

In our paper we are interested in the longtime behavior of the SBM with
$\rho=1$ in the case of not necessarily equal initial conditions — the case
that has not been studied in the literature. The main question is whether both
populations can survive forever or whether just one population will survive
while the other population will die out. If there is a positive probability
that both populations will survive we will say coexistence is possible.
Otherwise we will say coexistence is impossible.

The longtime behavior of the SBM has been thoroughly studied for different
correlation parameters $\rho$. For correlations $\rho\in(-1,0]$ it has been
proved that there is a clear dichotomy: coexistence is possible if and only if
the migration is transient. See Blath, Döring and Etheridge [BDE11], Dawson
and Perkins [DP98], and Döring and Mytnik [DM13]. For $\rho=-1$, Shiga [Shi80]
has proved that in the recurrent regime coexistence is impossible for the
particular case of $u_{0}+v_{0}=1$. For correlation $\rho\in(0,1)$, Blath,
Döring, and Etheridge [BDE11] have proved that if the migration is recurrent,
coexistence is impossible.

For $\rho=1$ in the case of identical initial conditions (the PAM) the phase
transition between survival and non-survival (note that in the PAM there is
only one population so it does not make sense to speak of coexistence of two
populations) occurs in the transient regime, that is, survival is impossible
if the migration is recurrent. If the migration is transient then there is a
critical branching parameter $b_{\ast}$ such that for $b>b_{\ast}$ survival is
impossible while for $b<b_{\ast}$ survival is possible. See Theorem 1.5 below
(which is essentially the result of Greven and den Hollander [GdH07]).

For correlation $\rho\in(0,1)$, Blath, Döring, and Etheridge [BDE11] have
proved that if the migration is recurrent, coexistence is impossible. It is
believed that for positive correlation there is a critical branching parameter
such for all $b$ larger than this parameter coexistence is impossible also in
the transient regime (like in the case of the PAM). This conjecture however
has not yet been proved.

Thus, note that for $\rho\in(-1,0]$ and for the PAM the longtime behavior is
fully characterized. Also, in the recurrent regime the longtime behavior of
the SBM is fully understood for all $\rho$. However, in the transient regime
there are gaps in the understanding of what happens for $\rho=-1$, with
$u_{0}+v_{0}\neq 1$, for $\rho\in(0,1)$, and for $\rho=1$ with non-identical
initial conditions. In this paper our aim is to contribute towards a more
complete understanding of the longtime behavior for positive correlations.

For $\rho\in(0,1]$ one of the main open questions related to the longtime
behavior of SBM is:

  * If the migration is transient, is there a critical parameter $b_{\\#}(\rho)$ such that for $b<b_{\\#}(\rho)$ coexistence is possible while for $b>b_{\\#}(\rho)$ it is impossible.

Our main result, Theorem 1.6 gives a partial answer to this question: we
characterize coexistence/non-coexistence dichotomy for SBM for the case of
$\rho=1$ and with initial conditions that are not necessarily equal.

The paper is organised as follows. In Section 1.2 we formally introduce the
SBM and related concepts that will be investigated. Section 1.3 states
existing relevant results on the longtime behavior of the PAM which is of
profound importance for studying the long time behavior of the SBM with
correlation $\rho=1$. In Section 1.4 we present our main result for the
longtime behavior of the SBM with correlation $\rho=1$. The proof of our
result is split into two parts. In Section 2 we treat the regime when the
coexistence is possible while in Section 3 we deal with the regime when the
coexistence is impossible.

###  1.2. Definitions

In this section we formally introduce the SBM. For a rigorous definition of
the processes we need to specify an appropriate state space. Let
$\varphi_{\lambda}:\mathbb{Z}^{d}\mapsto\mathbb{R},\varphi_{\lambda}(k)=e^{\lambda|k|}$,
$\lambda\in\mathbb{R}$ and define

$\displaystyle E_{\textup{tem}}$ $\displaystyle:=$
$\displaystyle\\{\phi:\mathbb{Z}^{d}\mapsto\mathbb{R}_{+}:\langle\phi,\varphi_{\lambda}\rangle<\infty\text{
for all }\lambda<0\\}\,,$ (1.2) $\displaystyle E_{\textup{fin}}$
$\displaystyle:=$
$\displaystyle\\{\phi:\mathbb{Z}^{d}\mapsto\mathbb{R}_{+}:\sum_{k\in\mathbb{Z}^{d}}\phi(k)<\infty\\}\,,$
(1.3) $\displaystyle E_{\textup{cpt}}$ $\displaystyle:=$
$\displaystyle\\{\phi:\mathbb{Z}^{d}\mapsto\mathbb{R}_{+}:\phi(k)=0\text{ for
all but finitely many }k\in\mathbb{Z}^{d}\\}\,.$ (1.4)

Following definitions on page 1091 of [DP98], for any $\lambda\in\mathbb{R}$
and $u,v:\mathbb{Z}^{d}\mapsto\mathbb{R}$, set
$|u-v|_{\lambda}:=\langle|u-v|,\varphi_{\lambda}\rangle$, and thus we define
metric on $E_{\textup{tem}}$ as follows:

$d_{\text{tem}}(u,v):=\sum_{n=1}^{\infty}2^{-n}(|u-v|_{-\lambda_{n}}\wedge
1),$ (1.5)

where $\lambda_{n}\downarrow 0$. Also $E_{\textup{fin}}$ is topologized by the
$l^{1}$-norm, $\|u-v|\|_{1}:=\sum_{k\in\mathbb{Z}^{d}}|u(k)-v(k)|$, and
$E_{\textup{cpt}}$ is topologized by the $l^{\infty}$-norm.

Now $\Omega_{\text{tem}}$ (resp. $\Omega_{\text{fin}}$) is the space of
$E_{\text{tem}}^{2}$-valued (resp. $E_{\text{fin}}^{2}$-valued) continuous
paths on $\mathbb{R}_{+}$ with the compact-open topology.

Whenever $x:\mathbb{Z}^{d}\mapsto\mathbb{R}_{+}$ and $x(i)=\theta\geq 0$ for
all $i\in\mathbb{Z}^{d}$ we use bold letters and write
$x=\boldsymbol{\theta}$. We refer to this situation as a flat configuration.

In the following we denote the discrete Laplace operator by $\Delta$, that is,
for $\phi:\mathbb{Z}^{d}\mapsto\mathbb{R}_{+}$ we set

$\Delta\phi(i):=\frac{1}{2d}\sum_{j\sim i}(\phi(j)-\phi(i))\,,\quad
i\in\mathbb{Z}^{d}\,,$ (1.6)

where $j\sim i$ means $j$ is a neighbor of $i$.

We denote the semigroup corresponding to the continuous-time rate $1$ simple
symmetric random walk on $\mathbb{Z}^{d}$ by $(P_{t})_{t\geq 0}$ and the
transition probabilities by $p_{t}(i,j)$, $t\geq 0$, $i,j\in\mathbb{Z}^{d}$.
That is, if $(Z_{t})_{t\geq 0}$ is a simple symmetric random walk on
$\mathbb{Z}^{d}$ and $\phi:\mathbb{Z}^{d}\mapsto\mathbb{R}$ is a bounded
function, then $P_{t}\phi(i)=E[\phi(Z_{t})|Z_{0}=i]$ and
$p_{t}(i,j)=P\\{Z_{t}=j|Z_{0}=i\\}$, where $t\geq 0$ and
$i,j\in\mathbb{Z}^{d}$. Also, we denote the Green’s function of a simple
symmetric random walk on $\mathbb{Z}^{d}$ ($d\geq 3$) by $g(\cdot,\cdot)$,
that is,

$g(x,y)=\int_{0}^{\infty}P_{s}\mathbbm{1}_{\\{y\\}}(x)\,\mathrm{d}s=\int_{0}^{t}p_{s}(x,y)\,\mathrm{d}s,\;x,y\in\mathbb{Z}^{d}.$
(1.7)

Moreover if $X,Y$ are stochastic processes defined on the same probability
space, we denote their quadratic co-variation by $[X,Y]_{t}$ and quadratic
variation by $[X]_{t}$. Equality in distribution is denoted by
$\overset{\textup{d}}{=}$.

######  Definition 1.1.

Let $b>0$, $\rho=1$, and $(x,y)\in E_{\text{tem}}^{2}$. We say that a
stochastic process $(u,v)=(u_{t},v_{t})_{t\geq 0}$ is a Symbiotic Branching
Model SBM$(1,b,x,y)$ on a filtered probability space
$(\Omega,\mathcal{A},\mathcal{F}=(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$ if
the sample paths of $(u,v)$ lie in $\Omega_{\text{tem}}$, and there exists a
family of planar Brownian motions $\\{W(i)=(W_{t}(i))_{t\geq
0}:i\in\mathbb{Z}^{d}\\}$ adapted to the filtration $\mathcal{F}$ such that
the following is satisfied: $(u,v)$ solves the following system of interacting
stochastic differential equations

$\left\\{\begin{array}[]{l}(u_{0},v_{0})=(x,y)\,,\\\\[4.30554pt]
\mathrm{d}u_{t}(i)=\Delta
u_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)v_{t}(i)}\,\mathrm{d}W_{t}(i)\,,\quad
t\geq 0,i\in\mathbb{Z}^{d}\,,\\\\[4.30554pt] \mathrm{d}v_{t}(i)=\Delta
v_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)v_{t}(i)}\,\mathrm{d}W_{t}(i)\,,\quad
t\geq 0,i\in\mathbb{Z}^{d}\,.\end{array}\right.$ (1.8)

In what follows, we assume throughout the paper that

$\rho=1.$

Also, let us note, that for simplicity, whenever confusion is impossible we
omit some or all of the parameters in the abbreviation SBM$(1,b,u_{0},v_{0})$.

Next proposition provides the existence and uniqueness result for (1.8).

######  Proposition 1.2.

Let $b>0$, and $(x,y)\in E_{\text{tem}}^{2}$.The system of stochastic
differential equations (1.8) has a unique strong solution with sample paths in
$\Omega_{\text{tem}}$.

###### Proof.

Existence of a weak solution to (1.8) is proved in Proposition 3.1 of [BDE11].

Now let us show the pathwise uniqueness. First note that
$\eta_{t}:=v_{t}-u_{t}$ is deterministic and solves the discrete heat
equation, that is $\mathrm{d}\eta_{t}=\Delta\eta_{t}\,\mathrm{d}t$. Hence
$(u,v)$ is uniquely determined if $u$ and $\eta$ are. For $\eta$ existence and
uniqueness are known of course. For $u$ we have the following system of
locally $1$-dimensional stochastic differential equations:

$\mathrm{d}u_{t}(i)=\Delta
u_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)^{2}+bu_{t}(i)\eta_{t}(i)}\,\mathrm{d}W^{u}_{t}(i)\,,\quad
i\in\mathbb{Z}^{d}\,.$ (1.9)

For (1.9) pathwise uniqueness follows by the argument similar to the one used
in the proof of Theorem 3.2 in Shiga and Shimizu [SS80]. Note that in [SS80]
the diffusion coefficient does not depend on time. However, the Yamada-
Watanabe argument used in [SS80] obviously works just as well for our time-
dependent diffusion coefficient since $\eta_{t}(i)<\infty$ for all $t\geq 0$.
Also [SS80] assumes that at each site the process takes values in $[0,1]$. It
is easy to seen however that for the argument in [SS80]to work it is
sufficient to have
$\sup_{t\in[0,T]}\mathbb{E}[|u_{t}(i)-u^{\prime}_{t}(i)|]<\infty$ for all
$T\geq 0$, for any two solutions $u,u^{\prime}$ to (1.9). This can be
immediately seen as follows. Recall that $(P_{t})_{t\geq 0}$ denotes the
semigroup corresponding to the simple symmetric random walk on
$\mathbb{Z}^{d}$. Then, by Proposition 3.1 of [BDE11],
$\mathbb{E}[u_{t}(i)]=P_{t}u_{0}(i)$ for every $i\in\mathbb{Z}^{d}$ and thus
$\sup_{t\geq 0}\mathbb{E}[u_{t}(i)]<\infty$.

By Yamada-Watanabe theorem weak existence and pathwise uniqueness imply that
there exists unique strong solution to (1.9) (see e.g. Theorem 2.2 in [SS80]
for the analogous result). ∎

In the study of the SBM an important role is played by the total mass process
and many results can be deduced from the behavior of this simpler process. The
total mass of an element $x:\mathbb{Z}^{d}\mapsto\mathbb{R}_{+}$, is denoted
by

$\bar{x}=\langle x,\boldsymbol{1}\rangle=\sum_{i}x(i)\,.$ (1.10)

######  Proposition 1.3 (Proposition 3.2 of [BDE11]).

If $u_{0},v_{0}\in E_{\textup{fin}}$, then the total mass processes
$\bar{u}=(\bar{u}_{t})_{t\geq 0}$ and $\bar{v}=(\bar{v}_{t})_{t\geq 0}$ are
non-negative, continuous, square integrable martingales and

$\displaystyle\mathrm{d}\bar{u}_{t}$ $\displaystyle=\sqrt{b\langle
u_{t},v_{t}\rangle}\,\mathrm{d}\tilde{W}_{t}\,,\quad t\geq 0,$ (1.11)
$\displaystyle\mathrm{d}\bar{v}_{t}$ $\displaystyle=\sqrt{b\langle
u_{t},v_{t}\rangle}\,\mathrm{d}\tilde{W}_{t}\,,\quad t\geq 0,$ (1.12)

where $\tilde{W}$ is a Brownian motion with the variance
$[\tilde{W}_{\cdot},\tilde{W}_{\cdot}]_{t}=t$.

The lemma also implies that

$[\bar{u},\bar{v}]_{t}=b\int_{0}^{t}\langle
u_{s},v_{s}\rangle\,\mathrm{d}s\,,t\geq 0.$ (1.13)

Another consequence of the lemma is that by the martingale convergence
theorem, the limits

$\lim_{t\to\infty}\bar{u}_{t}=\bar{u}_{\infty}\quad\text{and}\quad\lim_{t\to\infty}\bar{v}_{t}=\bar{v}_{\infty}\,,$
(1.14)

exist, almost surely.

What happens in the case of flat initial conditions, that is, for
$(u_{0},v_{0})=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$ with
$\theta_{1},\theta_{2}\geq 0$? In this case, the existence of limit of
$(u_{t},v_{t})$ in $E_{\text{tem}}^{2}$ is known for the case of
$\rho\in(-1,1)$ (see Proposition 4.1 in [BDE11]). As for the case of $\rho=1$,
since expectations of $\langle u_{t},\varphi_{\lambda}\rangle,\langle
u_{t},\varphi_{\lambda}\rangle$ are constant in $t$ and thus bounded for any
$\lambda>0$, we can immediately get that the family $\\{(u_{t},v_{t})\,,t\geq
0\\}$ is tight in $E_{\textup{tem}}^{2}$ (see Lemma 2.3(c) of [DP98] and its
proof for analogous argument). Thus there exist weak limit points
$(u_{\infty},v_{\infty})\in E_{\textup{tem}}^{2}$.

######  Definition 1.4 (coexistence).

  1. (1)

Assume that $(u_{0},v_{0})\in E_{\textup{fin}}^{2}$. We say that coexistence
is possible if

$\mathbb{P}\\{\bar{u}_{\infty}\bar{v}_{\infty}>0\\}>0\,.$ (1.15)

We say that coexistence is impossible if

$\mathbb{P}\\{\bar{u}_{\infty}\bar{v}_{\infty}>0\\}=0\,.$ (1.16)

  2. (2)

Assume that $(u_{0},v_{0})=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$
with $\theta_{1},\theta_{2}\geq 0$. We say that coexistence is possible if for
any weak limit point $(u_{\infty},v_{\infty})$ of $\\{(u_{t},v_{t})\,,t\geq
0\\}$ there exists $(\phi,\psi)\in E_{\textup{cpt}}^{2}$ such that

$\mathbb{P}\\{\langle u_{\infty},\phi\rangle\langle
v_{\infty},\psi\rangle>0\\}>0\,.$ (1.17)

We say that coexistence is impossible if for any weak limit point
$(u_{\infty},v_{\infty})$ of $\\{(u_{t},v_{t})\,,t\geq 0\\}$ and for all
$\phi,\psi\in E_{\textup{cpt}}$

$\mathbb{P}\\{\langle u_{\infty},\phi\rangle\langle
v_{\infty},\psi\rangle>0\\}=0\,.$ (1.18)

In the case of the PAM where the two populations are identical we use the
notion of survival instead of coexistence.

###  1.3. Longtime behavior of PAM

Before stating our main results for SBM with $\rho=1$, we present results on
the survival/extinction dichotomy for the PAM, that is, for the
SBM$(1,b,u_{0},v_{0}=u_{0})$. In what follows, we will use for this process
the short notation PAM($b,u_{0}$) and drop the parameter $b$ or the initial
condition whenever no confusion may occur.

Recall that $g(\cdot,\cdot)$ denotes the Green’s function of a simple random
walk on $\mathbb{Z}^{d}$ and define

$b_{2}:=\frac{2}{g(0,0)}\,.$ (1.19)

Then the result on the longtime behavior for the PAM reads as follows.

######  Theorem 1.5 (longtime behavior of PAM [GdH07, BS10, BS11]).

Let $b>0$, and $(u_{t})_{t\geq 0}$ be PAM($b,u_{0}$).

  1. (i)

Assume that $u_{0}=\boldsymbol{\theta}>0$. Then

    1. (a)

if $d\in\\{1,2\\}$, then survival is impossible.

    2. (b)

if $d\geq 3$, then there exists $b_{\ast}>b_{2}$ such that for $b<b_{\ast}$,
survival is possible and for $b>b_{\ast}$ survival is impossible.

  2. (ii)

Assume that $u_{0}\in E_{\textup{cpt}}$. Then survival is possible if $d\geq
3$ and $b<b_{\ast}$, and survival is impossible if $d=1,2$ or $d\geq 3$ and
$b>b_{\ast}$.

######  Remark 1.1.

In fact Greven and den Hollander [GdH07] prove only $b_{2}\leq b_{\ast}$. They
conjectured that in fact $b_{2}<b_{\ast}$. The fact that $b_{2}<b_{\ast}$
follows from results of Birkner and Sun [BS10] for dimensions $d\geq 4$ and
[BS11] for dimension $d=3$. [GdH07] also show that the second moments of the
PAM at each site $i\in\mathbb{Z}^{d}$ are bounded in time if and only if
$b<b_{2}$.

###### Proof of Theorem 1.5.

For flat initial conditions, the result is contained in Theorems 1.2, 1.3,
1.4, and 1.5 of [GdH07]. For summable initial conditions the longtime behavior
can be carried over from the flat setting using the self-duality from Lemma
2.1 as follows: for all $\theta>0$ and $u_{0}\in E_{\rm cpt}$

$\displaystyle\mathbb{E}_{u_{0}}[e^{-\theta\bar{u}_{\infty}}]$
$\displaystyle=\lim_{t\to\infty}\mathbb{E}_{u_{0}}[e^{-\theta\langle\mathbf{1},u_{t}\rangle}]=\lim_{t\to\infty}\mathbb{E}_{\boldsymbol{\theta}}[e^{-\tilde{u}_{t},u_{0}\rangle}]\,,$
(1.20)

where $(\tilde{u}_{t})_{t\geq 0}$ is PAM($b,{\boldsymbol{\theta}}$). By
Theorem 1.3 and 1.4 of [GdH07], the limiting law of $\tilde{u}_{t}$ exists, it
is translation invariant and this implies that the right hand side of (1.20)
equals $1$ if and only if
$\operatorname*{w-lim}_{t\to\infty}\tilde{u}_{t}=\mathbf{0}$. Here and
elsewhere $\operatorname*{w-lim}$ denotes weak (in probability sense) limit. ∎

###  1.4. Our Main Result

Recall the parameter $b_{\ast}$ from Theorem 1.5. We establish the longtime
behavior of the SBM for the case of completely positive correlations
($\rho=1$). Recall that, as we mentioned above, if it is not stated otherwise,
we assume that $\rho=1$ in what follows.

######  Theorem 1.6 (Longtime behavior for summable initial conditions).

Let $(u_{0},v_{0})\in E_{\textup{fin}}^{2}$ such that
$\bar{u}_{0}\bar{v}_{0}>0$.

  1. (i)

Let $d\geq 3$. Then for all $b\in(0,b_{\ast})$ coexistence for
SBM($1,b,u_{0},v_{0}$) is possible.

  2. (ii)

Let

     * (a)

$d\in\\{1,2\\}$ and $b>0$,

or

     * (b)

$d\geq 3$ and $b>b_{\ast}$.

Then coexistence for SBM($1,b,u_{0},v_{0}$) is impossible. Moreover, in both
cases (a) and (b), if $\bar{u}_{0}\leq\bar{v}_{0}$, then

$\bar{u}_{t}\to 0\,.$ (1.21)

and

$\bar{v}_{t}\to\bar{v}_{0}-\bar{u}_{0}\,.$ (1.22)

almost surely, as $t\to\infty$.

######  Theorem 1.7 (Longtime behavior for flat initial conditions).

Assume that $(u_{0},v_{0})=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$
with $\theta_{1},\theta_{2}>0$.

  1. (i)

Let $d\geq 3$. Then for all $b\in(0,b_{\ast})$ coexistence for
SBM($1,b,u_{0},v_{0}$) is possible.

  2. (ii)

Let

     * (a)

$d\in\\{1,2\\}$ and $b>0$,

or

     * (b)

$d\geq 3$ and $b>b_{\ast}$.

Then coexistence for SBM($1,b,u_{0},v_{0}$) is impossible. Moreover, in both
cases (a) and (b), if $\theta_{1}\leq\theta_{2}$, then for any $\phi\in
E_{\textup{cpt}}$

$\langle u_{t},\phi\rangle\to 0\,,$ (1.23)

and

$\langle v_{t},\phi\rangle\to(\theta_{2}-\theta_{1})\langle 1,\phi\rangle\,,$
(1.24)

in probability as $t\to\infty$.

Let us comment on the above results. First of all, as we see, the transition
threshold between possible coexistence and non-coexistence for the
SBM($1,b,u_{0},v_{0}$) is exactly the same as for the PAM model which is, in
fact, SBM($1,b,u_{0},v_{0}=u_{0}$) model. This may sound like as a pretty much
well expected result, however we found that the proof of it is surprisingly
not very straightforward. It does use close connection between SBM($1$) and
PAM models, however in the regime of non-coexistence with summable non-
monotone initial conditions (that is, $u_{0}\not\leq v_{0}$ and $v_{0}\not\leq
u_{0}$) on top of comparison with the PAM, one uses non-trivial decomposition
of the SBM and some interesting PDE results (see Section 3.2 for this
argument). We hope that our proofs with give an additional motivation for
studying the open question of existence of the phase transition in the
transient regime for $\rho\in(0,1)$.

##  2\. Proof of Theorems 1.6(i) and 1.7(i): Coexistence Possible

###  2.1. Preparations

In Section 2 we prove the coexistence parts of Theorems 1.6, 1.7. The actual
proof will be carried out in Subsection 2.2. In this subsection we will review
and prove a couple of results for the PAM which we will need for the proof of
Theorems 1.6(i), 1.7(i).

Let $w$ be PAM($b,w_{0}$), that is, $w$ satisfies the following equation:

$\mathrm{d}w_{t}(i)=\Delta
w_{t}(i)\,\mathrm{d}t+\sqrt{bw^{2}_{t}(i)}\,\mathrm{d}W_{t}(i),\quad t\geq
0,i\in\mathbb{Z}^{d}.$ (2.1)

The PAM exhibits the following simple but very usefull self-duality property,
see Section 2 of Cox, Klenke and Perkins [CKP00]: let $w$ and $\tilde{w}$ be
PAM$(b)$ processes such that $w_{0}=\boldsymbol{\theta}$ and
$\tilde{w}_{0}=\phi\in E_{\textup{fin}}$. Then

$\displaystyle\mathbb{E}_{\tilde{w}_{0}}[e^{-\lambda\langle\tilde{w}_{t},\boldsymbol{\theta}\rangle}]=\mathbb{E}_{\tilde{w}_{0}}[e^{-\langle\tilde{w}_{t},\lambda\boldsymbol{\theta}\rangle}]=\mathbb{E}_{\lambda\boldsymbol{\theta}}[e^{-\langle\tilde{w}_{0},u_{t}\rangle}]=\mathbb{E}_{\boldsymbol{\theta}}[e^{-\langle\tilde{w}_{0},\lambda
w_{t}\rangle}]=\mathbb{E}_{\boldsymbol{\theta}}[e^{-\lambda\langle\phi,w_{t}\rangle}]\,,\quad\forall
t\geq 0.$ (2.2)

From the self-duality we immediately obtain the following statement.

######  Lemma 2.1 (self-duality).

Let $w$ and $\tilde{w}$ be PAM(b) such that $w_{0}=\boldsymbol{\theta}$ and
$\tilde{w}_{0}=\phi\in E_{\textup{fin}}$. Then,

$\langle\tilde{w}_{t},\boldsymbol{\theta}\rangle\overset{\textup{d}}{=}\langle\phi,w_{t}\rangle\,,\;\forall
t\geq 0.$ (2.3)

We now prove a result which allows us to bound the moments of appropriate
functions of SBM by those of PAM. Since we know a lot of information about PAM
this will be very useful.

######  Proposition 2.2 (comparison).

Let $(u,v)$ be the solution of SBM$(1,b,u_{0},v_{0})$. Let $w$ be a
PAM$(b,w_{0})$ such that $w_{0}=u_{0}+v_{0}$. Let $\Phi(t)$ be arbitrary non-
negative non-decreasing convex function on $\mathbb{R}_{+}$.

  * (i)

If $(u_{0},v_{0})\in E_{\textup{fin}}$, then

$\mathbb{E}_{(u_{0},v_{0})}[\Phi(\langle u_{t},\boldsymbol{1}\rangle+\langle
v_{t},\boldsymbol{1}\rangle)]\leq\mathbb{E}_{w_{0}}[\Phi(\langle
w_{t},\boldsymbol{1}\rangle)],\quad\forall t\geq 0.$ (2.4)

  * (ii)

If $(u_{0},v_{0})=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$ for
$\theta_{1},\theta_{2}\geq 0$, then for all $\phi\in E_{\textup{fin}}$ such
that $\phi\geq 0$ one has

$\mathbb{E}_{(u_{0},v_{0})}[\Phi(\langle u_{t},\phi\rangle+\langle
v_{t},\phi\rangle)]\leq\mathbb{E}_{w_{0}}[\Phi(\langle
w_{t},\phi\rangle)],\quad\forall t\geq 0.$ (2.5)

###### Proof.

We start with proving (i). The idea is to use comparison result of Greven,
Klenke and Wakolbinger [GKW02] (the earlier version of this result for the
homogeneous case is due to Cox, Fleischmann, and Greven [CFG96]); since the
conditions of the result in [GKW02] on function $\Phi$ are not satisfied in
the proposition we will use the usual technique of approximation. Let
$\Lambda^{N}=[-N,N]^{d}$ be a torus, and $\Delta$ acting on functions on the
torus will be Laplacian with periodic boundary conditions. Let $(u^{N},v^{N})$
be a solution of

$\left\\{\begin{array}[]{l}\mathrm{d}u^{N}_{t}(i)=\Delta
u^{N}_{t}(i)\,\mathrm{d}t+\sqrt{bu^{N}_{t}(i)v^{N}_{t}(i)\left(1-\frac{u^{N}_{t}(i)}{N}\right)\left(1-\frac{v^{N}_{t}(i)}{N}\right)}\,\mathrm{d}W_{t}(i)\,,\quad
t\geq 0,i\in\Lambda_{N}\,,\\\\[4.30554pt] \mathrm{d}v^{N}_{t}(i)=\Delta
v^{N}_{t}(i)\,\mathrm{d}t+\sqrt{bu^{N}_{t}(i)v^{N}_{t}(i)\left(1-\frac{u^{N}_{t}(i)}{N}\right)\left(1-\frac{v^{N}_{t}(i)}{N}\right)}\,\mathrm{d}W_{t}(i)\,,\quad
t\geq 0,i\in\Lambda_{N}\,.\end{array}\right.$ (2.6)

with $u^{N}_{0}(i)=u_{0}(i),v^{N}_{0}(i)=v_{0}(i),i\in\Lambda_{N}\,.$ The
solutions of this equation take values in
$[0,N]^{\Lambda_{N}}\times[0,N]^{\Lambda_{N}}$.

In the following it is understood that $i\in\Lambda_{N}$ and $t\geq 0$. Let

$\eta^{N}_{t}(i):=v^{N}_{t}(i)-u^{N}_{t}(i)$ (2.7)

and

$\xi^{N}_{t}(i):=u^{N}_{t}(i)+v^{N}_{t}(i)=2u^{N}_{t}(i)+\eta^{N}_{t}(i)\;\;\text{so
that}\;\;u^{N}_{t}(i)=\frac{1}{2}(\xi^{N}_{t}(i)-\eta^{N}_{t}(i))\,.$ (2.8)

Then

$\displaystyle
u^{N}_{t}(i)v^{N}_{t}(i)=\frac{1}{4}\left(\xi^{N}_{t}(i)^{2}-\eta^{N}_{t}(i)^{2}\right)\,.$
(2.9)

Therefore, we get that $\xi^{N}$ satisfies the following system of equations:

$\displaystyle\mathrm{d}\xi^{N}_{t}(i)$
$\displaystyle=\Delta\xi^{N}_{t}(i)\,\mathrm{d}t+\sqrt{b(\xi^{N}_{t}(i)^{2}-\eta^{N}_{t}(i)^{2})\left(1-\frac{\xi^{N}_{t}(i)-\eta^{N}_{t}(i)}{2N}\right)\left(1-\frac{\xi^{N}_{t}(i)+\eta^{N}_{t}(i)}{2N}\right)}\,\mathrm{d}W_{t}(i)\,,\;\;t\geq
0,i\in\Lambda_{N}\,.$ (2.10)

Clearly since the noises for $u^{N}$ and $v^{N}$ are the same we get that
$\eta^{N}$ is deterministic that solves the following system of equations:

$\displaystyle\mathrm{d}\eta^{N}_{t}(i)$
$\displaystyle=\Delta\eta^{N}_{t}(i)\,\mathrm{d}t\,,\;\;t\geq
0,i\in\Lambda_{N}\,.$ (2.11)

Let now $w^{N}=(w^{N}_{t})_{t\geq 0}$ be an approximate PAM$(b,w_{0})$. That
is, for an independent family of Brownian motions
$\\{W^{w}(i):i\in\mathbb{Z}^{d}\\}$, let $w^{N}$ on $\Lambda_{N}$ satisfy

$\displaystyle\mathrm{d}w^{N}_{t}(i)=\Delta
w^{N}_{t}(i)\,\mathrm{d}t+\sqrt{bw^{N}_{t}(i)^{2}\left(1-\frac{w^{N}_{t}(i)}{2N}\right)}\,\mathrm{d}W^{w}_{t}(i)\,,\quad
t\geq 0,i\in\Lambda_{N}\,.$ (2.12)

with $w^{N}_{0}(i)=w_{0}(i),i\in\Lambda_{N}\,.$ Let

$\displaystyle g_{N}^{\xi}(i,x,t)$
$\displaystyle:=b(x^{2}-\eta^{N}_{t}(i)^{2})\left(1-\frac{x-\eta^{N}_{t}(i)}{2N}\right)\left(1-\frac{x+\eta^{N}_{t}(i)}{2N}\right)\,,\quad
x\,,t\geq 0\,,i\in\Lambda_{N}\,,$ (2.13) $\displaystyle g^{w}_{N}(i,x)$
$\displaystyle:=bx^{2}\left(1-\frac{x}{2N}\right)\,,\quad x\,,t\geq
0\,,i\in\Lambda_{N}\,.$ (2.14)

Then, for all $x,t\geq 0,i\in\Lambda_{N}$

$g_{N}^{\xi}(i,x,t)\leq g_{N}^{w}(i,x)\,.$ (2.15)

The respective generators of $\xi^{N}$ and $w^{N}$ are given by the closure of
the following operators

$\displaystyle G_{N,t}^{\xi}f(\phi)$
$\displaystyle=\sum_{i\in\Lambda_{N}}\Delta\phi(i)\,\partial_{i}f(\phi)+\frac{1}{2}\sum_{i\in\Lambda_{N}}g_{N}^{\xi}(\phi(i),t)\,\partial_{i}^{2}f(\phi)\,,\quad\phi\in(\mathbb{R}_{+})^{\Lambda_{N}}\,,$
(2.16) $\displaystyle G^{w}_{N}f(\phi)$
$\displaystyle=\sum_{i\in\Lambda_{N}}\Delta\phi(i)\,\partial_{i}f(\phi)+\frac{1}{2}\sum_{i}g_{N}^{w}(\phi(i))\,\partial_{i}^{2}f(\phi)\,,\quad\phi\in(\mathbb{R}_{+})^{\Lambda_{N}}\,,$
(2.17)

acting on functions $f:(\mathbb{R}_{+})^{\Lambda_{N}}\mapsto\mathbb{R}$ such
that all their partial derivatives up to order $2$ exist and are continuous.
Now let $\mathcal{D}_{N}$ consist of functions $f$ as specified in the
previous sentence with the additional requirement that
$\partial_{i}\partial_{j}f\geq 0$ for all $i,j\in\Lambda_{N}$. Let
$C^{2}_{bc}$ be the set of convex functions
$F:\mathbb{R}_{+}\mapsto\mathbb{R}$ with bounded on the compacts continuous
partial derivatives of orders $m=0,1,2$. Fix arbitrary $F\in C^{2}_{bc}$. For
$B_{r}$ the open ball of radius $r$ in $\mathbb{Z}^{d}$, let
$F_{r}:(\mathbb{R}_{+})^{\mathbb{Z}^{d}}\mapsto\mathbb{R}_{+}$, be defined by

$F_{r}(x)=F\biggl{(}\,\sum_{i\in B_{r}}x(i)\biggr{)}\,.$ (2.18)

Then, for all $r\in(0,N)$, we get that $F_{r}\in\mathcal{D}_{N}$, and taking
into account (2.15) we can apply Theorem 2 of [GKW02] to obtain that for each
$r\in(0,N)$,

$\mathbb{E}_{u_{0}^{N}+v_{0}^{N}}[F_{r}(\xi^{N}_{t})]\leq\mathbb{E}_{u_{0}^{N}+v_{0}^{N}}[F_{r}(w^{N}_{t})]\,,\;\forall
t\geq 0.$ (2.19)

It is easy to check that as $N\rightarrow\infty$, $(\xi^{N},\eta^{N})$
converges weakly (in the space of continuous $E^{2}_{\text{tem}}$-valued
paths) to $(\xi,\eta)$, where $\xi_{\cdot}=u_{\cdot}+v_{\cdot},$
$\eta_{\cdot}=v_{\cdot}-u_{\cdot}$ and $(u,v)$ is a solution to the
SBM($(1,b,u_{0},v_{0}$). Moreover $w^{N}$ converges weakly (in the space of
continuous $E_{\text{tem}}$-valued paths) to $w$ which solves
PAM$(b,w_{0}=u_{0}+v_{0})$. Thus, letting $N\to\infty$, passing to the limit
in (2.19) and using continuity of $F_{r}$ on $E_{\text{tem}}$, we get

$\mathbb{E}_{u_{0}+v_{0}}[F_{r}(\xi_{t})]\leq\mathbb{E}_{u_{0}+v_{0}}[F_{r}(w_{t})]\,,\forall
t\geq 0.$ (2.20)

Taking the limit $r\to\infty$ on both sides and using the monotone convergence
theorem yields

$\mathbb{E}_{(u_{0},v_{0})}[F(\langle u_{t},\boldsymbol{1}\rangle+\langle
v_{t},\boldsymbol{1}\rangle)]\leq\mathbb{E}_{u_{0}+v_{0}}[F(\langle
w_{t},\boldsymbol{1}\rangle)]\,,\forall t\geq 0.$ (2.21)

Then by approximating a non-negative non-decreasing convex function $\Phi$ by
functions from $C^{2}_{bc}$ we can easily finish the proof of (2.4) by passing
to the limit.

The proof of (ii) goes along the same lines with

$F_{r}(x)=F\biggl{(}\,\sum_{i\in B_{r}}x(i)\phi(i)\biggr{)}$ (2.22)

and will thus be omitted. ∎

######  Corollary 2.3 (Uniform integrability).

Let $d\geq 3$. Let $b_{\ast}$ be as in Theorem 1.5 and let $b\in(0,b_{\ast})$.
Let $(u,v)$ be the solution of SBM$(1,b,u_{0},v_{0})$.

  * (a)

If $(u_{0},v_{0})\in E_{\textup{fin}}^{2}$, then $\\{\bar{u}_{t}\,,t\geq 0\\}$
and $\\{\bar{v}_{t}\,,t\geq 0\\}$ are uniformly integrable.

  * (b)

If $(u_{0},v_{0})=(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})$ for
$\theta_{1},\theta_{2}\geq 0$, then for all $\phi\in E_{\textup{fin}}$

$\\{\langle u_{t}\,,\phi\rangle\,,t\geq 0\\}\;\text{and}\;\\{\langle
v_{t}\,,\phi\rangle\,,t\geq 0\\}\text{ are uniformly integrable.}$

###### Proof.

Let $w$ be a PAM$(b,w_{0})$ such that $w_{0}=u_{0}+v_{0}$.

Let us first prove (b), that is, the case of flat initial conditions. In this
case the uniform integrability of $\\{\langle w_{t},\phi\rangle,t\geq 0\\}$,
for $\phi\in E_{\textup{fin}}$, follows easily from Theorems 1.3, 1.4, of
[GdH07]. Then by necessary and sufficient criterion for uniform integrability,
for any $\phi\in E_{\textup{fin}}$, we get the existence of non-negative non-
decreasing convex function $\Phi(t)$ on $\mathbb{R}_{+}$ such that
$\lim_{t\to\infty}\frac{\Phi(t)}{t}<\infty$ and

$\sup_{t\geq 0}\mathbb{E}\left[\Phi(\langle
w_{t},\phi\rangle)\rangle\right]=\infty.$

Then Proposition 2.2(ii) and again the criterion for uniform integrability
imply that both $\\{\langle u_{t},\phi\rangle,t\geq 0\\}$ and $\\{\langle
v_{t},\phi\rangle,t\geq 0\\}$ are unifromly integarble for any $\phi\in
E_{\textup{fin}}$.

For the proof of (a) we use the self-duality of PAM proved in Lemma 2.1 and
again Theorems 1.3, 1.4, of [GdH07] to get uniform integrability of
$\\{\bar{w}_{t}\,,t\geq 0\\}$ and then we use Proposition 2.2(i) and follow
the lines of the proof for the case (b). ∎

###  2.2. Proof of Theorem 1.6(i)

With the preparations from the previous section we are now in a position to
prove Theorem 1.6(i), the coexistence result for summable initial conditions.

We need to show that

$\mathbb{P}\\{\bar{u}_{\infty}\bar{v}_{\infty}>0\\}>0\,.$ (2.23)

Without loss of generality assume that $\bar{u}_{0}\leq\bar{v}_{0}$. Note that
$\bar{\eta}_{t}\equiv\bar{v}_{t}-\bar{u}_{t}=\bar{v}_{0}-\bar{u}_{0}\geq 0$
for all $t$ since $\eta$ solves the heat equation and its total mass remains
constant. Therefore, by sending $t$ to infinity we get

$\bar{v}_{\infty}\geq\bar{u}_{\infty}\,,\;\mathbb{P}-{\rm a.s.}$ (2.24)

By Corollary 2.3(a) $\\{\bar{u}_{t}\,,t\geq 0\\}$ is uniformly integrable, and
thus we get

$\displaystyle\mathbb{E}_{u_{0}}[\bar{u}_{\infty}]=\mathbb{E}_{u_{0}}[\lim_{t\to\infty}\bar{u}_{t}]=\lim_{t\to\infty}\mathbb{E}_{u_{0}}[\bar{u}_{t}]=\bar{u}_{0}>0\,.$
(2.25)

Thus $\mathbb{P}\\{\bar{u}_{\infty}>0\\}>0$ and using (2.24) we get (2.23).

∎

###  2.3. Proof of Theorem 1.7(i)

The proof of the coexistence result for flat initial conditions goes along the
similar lines as the proof for summable initial conditions. Let
$(u_{\infty},v_{\infty})$ be an arbitrary weak limit point of
$\\{(u_{t},v_{t})\,,t\geq 0\\}$. We will show a little bit more than required
in the definition of coexistence. Let $(\phi,\phi)$ be an arbitrary element in
$E_{\textup{cpt}}^{2}$. We will show that

$\mathbb{P}\\{\langle u_{\infty},\phi\rangle\langle
v_{\infty},\phi\rangle>0\\}>0\,.$ (2.26)

Without loss of generality assume that $\theta_{1}\leq\theta_{2}$. Note that
$\eta_{t}(x)\equiv v_{t}(x)-u_{t}(x)\equiv\theta_{2}-\theta_{1}\geq 0$ for all
$t\geq 0$ and $x\in\mathbb{Z}^{d}$ since $\eta$ solves the heat equation
starting at constant initial condition
$\boldsymbol{\theta}_{2}-\boldsymbol{\theta}_{1}$. Therefore, by sending $t$
to infinity we get that for an arbitrary weak limit point
$(u_{\infty},v_{\infty})$ the following holds:

$v_{\infty}(x)\geq u_{\infty}(x)\,,\forall
x\in\mathbb{Z}^{d}\,,\;\mathbb{P}-{\rm a.s.}$ (2.27)

By Corollary 2.3(b) $\\{\langle u_{t},\phi\rangle,t\geq 0\\}$ is uniformly
integrable, and thus we get

$\displaystyle\mathbb{E}[\langle
u_{\infty},\phi\rangle]=\lim_{t\to\infty}\mathbb{E}_{(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})}[\langle
u_{t},\phi\rangle]=\theta_{1}\langle 1,\phi\rangle>0\,.$ (2.28)

Thus $\mathbb{P}\\{\langle u_{\infty},\phi\rangle>0\\}>0$ and using (2.27) we
get (2.26).

∎

##  3\. Proof of Theorems1.6(ii) and 1.7(ii): Coexistence Impossible.

###  3.1. Proof of Theorem 1.7(ii)

We are now turning our attention to the proof of Theorem1.7(ii) which states
that coexistence is impossible for flat initial conditions in the recurrent
case ($d\leq 2$) for arbitrary $b>0$ and in the transient case ($d\geq 3$) for
arbitrary $b>b_{\ast}$.

If in addition, we assume $u_{0}=v_{0}=\boldsymbol{\theta}>0$, then the SBM
coincides with the PAM, whose longtime behavior has been studied extensively
in [GdH07], see Theorem 1.5. In this case we immediately get that the
coexistence is impossible, and both populations do not survive.

Now, without loss of generality assume that $\theta_{1}\leq\theta_{2}$. In
what follows, we consider both cases of $d\leq 2$, $b>0$ and $d\geq
3,b>b_{\ast}$ simultaneously. Note that $\eta_{t}(i)\equiv
v_{t}(i)-u_{t}(i)\equiv\theta_{2}-\theta_{1}\geq 0$ for all $t\geq 0$ and
$i\in\mathbb{Z}^{d}$ since $\eta$ solves the heat equation starting at
constant initial condition $\boldsymbol{\theta}_{2}-\boldsymbol{\theta}_{1}$
and thus it remains constant. Note that we can re-write the equation for $u$
as follows:

$\mathrm{d}u_{t}(i)=\Delta
u_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)(u_{t}(i)+(\theta_{2}-\theta_{1}))}\,\mathrm{d}W_{t}(i)\,,\;i\in\mathbb{Z}^{d}.$
(3.1)

Thus, the diffusion coefficient in the generator for $u$ is strictly larger
than the diffusion coefficient for the PAM$(b)$. We can therefore again use a
comparison result for interacting diffusions by Greven, Klenke and Wakolbinger
[GKW02]. If $w$ is PAM$(b)$ with $w_{0}=\boldsymbol{\theta}_{1}$, Theorem 2 in
[GKW02] immediately implies

$\displaystyle\mathbb{E}_{(\boldsymbol{\theta}_{1},\boldsymbol{\theta}_{2})}[e^{-\langle
u_{t},\phi\rangle}]\geq\mathbb{E}_{\boldsymbol{\theta}_{1}}[e^{-\langle
w_{t},\phi\rangle}]\,,\quad\forall t\geq 0,$ (3.2)

for any $\phi\in E_{\textup{cpt}}$. By Theorem 1.5(i), the right hand side of
the above equation converges to $1$ as $t\to\infty$. Therefore the left hand
side also converges to $1$, and thus we get that

$\langle u_{t},\phi\rangle\to 0\,,$ (3.3)

in probability as $t\to\infty$. Since
$v_{t}=u_{t}+(\boldsymbol{\theta}_{2}-\boldsymbol{\theta}_{1})$ we immediately
get

$\langle
v_{t},\phi\rangle\to(\theta_{2}-\theta_{1})\langle{\boldsymbol{1}},\phi\rangle\,,$
(3.4)

in probability as $t\to\infty$, and this finishes the proof. ∎

######  Remark 3.1.

Morally, the comparison result implies that “more noise kills”. In other
words, if a system of (locally 1-dimensional) interacting diffusions weakly
dies out as time tends to infinity, then so does a system which has a larger
diffusion coefficient. Intuitively this makes sense since locally $0$ is a
trap for the system. Very roughly speaking, the process gets close to 0 at
some time almost surely, and the higher the fluctuations the higher the chance
that the process will actually hit zero once it gets close. Therefore, when
initial conditions are monotonic, the comparison result implies that the non-
coexistence regime of the PAM immediately carries over to the SBM. Actually it
implies not only non-coexistence but also that it is the population which has
been smaller at the beginning is the one which will suffer extinction.

###  3.2. Proof of Theorem 1.6(ii)

In the case of monotone initial conditions $v_{0}(\cdot)\geq u_{0}(\cdot)$, to
get the result, we can argue as in the proof of Theorem 1.7(ii) while again
using the comparison theorem from Greven, Klenke and Wakolbinger [GKW02].

Thus, let us consider the general case when initially one population does not
necessary dominate another one. Again, in what follows, we consider both cases
of $d\leq 2$, $b>0$ and $d\geq 3,b>b_{\ast}$ simultaneously. Recall that we
assume without loss of generality that $\bar{u}_{0}=\langle
u_{0},\boldsymbol{1}\rangle\leq\langle
v_{0},\boldsymbol{1}\rangle=\bar{v}_{0}$. Define

$\eta_{t}:=v_{t}-u_{t}\,,\quad t\geq 0\,.$ (3.5)

Then $\mathrm{d}\eta_{t}=\Delta\eta_{t}\mathrm{d}t,\;t\geq 0$. Note that we
can re-write the equation for $u$ as follows:

$\mathrm{d}u_{t}(i)=\Delta
u_{t}(i)\,\mathrm{d}t+\sqrt{bu_{t}(i)(u_{t}(i)+\eta_{t}(i))}\,\mathrm{d}W_{t}(i)\,,\quad
t\geq 0,i\in\mathbb{Z}^{d}.$ (3.6)

For our general initial conditions in $E_{\textup{fin}}$ the perturbation
$\eta_{t}$ in the diffusion coefficient in (3.6) can be positive or negative.
Hence, the comparison argument used above fails since it is no longer true
that $u_{t}(i)(u_{t}(i)+\eta_{t}(i))\geq u_{t}(i)^{2}$ for all $t\geq
0,i\in\mathbb{Z}^{d}$. Nonetheless, we will again use comparison techniques
with the PAM. We will quickly explain the main idea.

We will need the following notation: for any
$\phi:\mathbb{Z}^{d}\mapsto\mathbb{R}$ we denote its positive and negative
part by

$\phi^{+}:=\max\\{\phi,0\\}\,,\quad\phi^{-}:=\max\\{-\phi,0\\}\,.$ (3.7)

The idea is to decompose $u_{t}$ into the mass where it exceeds $v$ and the
minimum of $u$ and $v$. Define the minimum process $w=(w_{t})_{t\geq 0}$ by
setting

$w_{t}(i):=\min\\{u_{t}(i),v_{t}(i)\\}\,,\quad t\geq 0\,,i\in\mathbb{Z}^{d}.$
(3.8)

Then, clearly

$u_{t}(i)=w_{t}(i)+\eta_{t}^{-}(i)\,,\quad t\geq 0\,,i\in\mathbb{Z}^{d}.$
(3.9)

We will show that the total mass of the negative part of the heat equation,
$\eta_{t}^{-}$, vanishes as $t\to\infty$. Moreover it turns out that the
minimum process $w$ exhibits an “approximate” duality with the PAM which
allows us to deduce its extinction from the extinction of the PAM. Then,
clearly if the excess $u$ has over $v$ vanishes and the minimum of $u$ and $v$
also goes to 0, then $u$ will eventually suffer extinction!

Before we proceed to the proof of the non-coexistence result we need a brief
digression on the heat equation.

In order not to confuse with the difference process $v_{t}-u_{t}$, for the
following general arguments we use the generic notation $\zeta$ for the
solution of the heat equation. So, for $f:\mathbb{Z}^{d}\mapsto\mathbb{R}$,
let

$\mathrm{d}\zeta_{f}(t,i)=\Delta\zeta_{f}(t,i)\,\mathrm{d}t\,,\quad\zeta(0,i)=f(i)\,,\quad
t\geq 0,i\in\mathbb{Z}^{d}\,.$ (3.10)

If $M>0$ and $f=M\mathbbm{1}_{\\{0\\}}$, we write $\zeta^{M}$ for the
corresponding solution. In the following, for a differentiable function
$g:\mathbb{R}_{+}\mapsto\mathbb{R},$ we denote the derivative by
$\partial_{t}g$.

For the proof Theorem 1.6(ii) we will need to understand the behavior of the
negative part of $\zeta$. For example, we will prove and use the fact that if
at time $t=0$ there is an overall excess of “positive heat”, that is,
$\langle\zeta_{f}^{+},\boldsymbol{1}\rangle\geq\langle\zeta_{f}^{-},\boldsymbol{1}\rangle$,
the negative heat will eventually disappear:
$\langle\zeta_{f}^{-}(t),\boldsymbol{1}\rangle\to 0$, as $t\to\infty$. Note
that $\langle\zeta_{f}(t),\boldsymbol{1}\rangle=\langle
f,\boldsymbol{1}\rangle$ is constant.

Some of the facts about the discrete heat equation which we will state and
prove in the following are probably well-known in the literature. For
completeness, we present anyway these results including our proofs.

For $T>0$ and $i\in\mathbb{Z}^{d}$, let $\mathcal{Z}_{f}(T,i)$ be the set of
zeros of $\zeta_{f}(t,i)$ in $[0,T]$, that is,

$\mathcal{Z}_{f}(T,i):=\\{t\in[0,T]:\zeta_{f}(t,i)=0\\}\,.$ (3.11)

In what follows for any $A\subset\mathbb{R}$, $|A|$ will denote the number of
points in $A$.

######  Proposition 3.1 ($\zeta$ analytic).

Let $f:\mathbb{Z}^{d}\mapsto\mathbb{R}$ be bounded. Then $\zeta_{f}(\cdot,i)$
is (real) analytic for each $i\in\mathbb{Z}^{d}$. If $f\not\equiv 0$, then
$|\mathcal{Z}_{f}(T,i)|<\infty$ for each $T>0,i\in\mathbb{Z}^{d}$.

###### Proof.

In the following we consider $i\in\mathbb{Z}^{d}$ to be fixed. Recall that
$p_{t}(\cdot,\cdot)$ denotes the transition probability of a continuous-time,
rate $1$, simple symmetric random walk on $\mathbb{Z}^{d}$ and let
$p^{n}(i,j)$ be the probability that a discrete-time simple symmetric random
walk on $\mathbb{Z}^{d}$ jumps from $i$ to $j$ in $n$ steps. Then, for $t\geq
0$,

$\displaystyle\zeta_{f}(t,i)$
$\displaystyle=\sum_{j}p_{t}(i,j)f(j)=e^{-t}\sum_{j}\Bigl{(}\,\sum_{n=0}^{\infty}p^{n}(i,j)\frac{t^{n}}{n!}\,\Bigr{)}f(j)=e^{-t}\sum_{n}\frac{1}{n!}\Bigl{(}\,\sum_{j}p^{n}(i,j)f(j)\,\Bigr{)}t^{n}$
(3.12)

and thus $\zeta_{f}(\cdot,i)$ has a representation as a power series whose
radius of convergence is infinite. Note that if $f$ is bounded, then

$\sum_{n}\sum_{j}\frac{t^{n}}{n!}p^{n}(i,j)|f(j)|<\infty$ (3.13)

and hence we can re-order the series in (3.12).

To prove the second statement, note that it is immediate from (3.12) that
$\zeta_{f}(\cdot,i)$ can be extended to an analytic function on the complex
plane. Assume that for some $T>0$, $|\mathcal{Z}_{f}(T,i)|=\infty$. Then
$\mathcal{Z}_{f}(T,i)$ has an accumulation point. It is well-known that if the
zero set of an analytic function has an accumulation point then the function
is identically zero on its entire domain (see for example Theorem 4.3.7 in
[Con73]). But this is a contradiction to the fact that $f\not\equiv 0$. ∎

######  Corollary 3.2 (Differentiability of $\zeta^{+}$).

Let $f:\mathbb{Z}^{d}\mapsto\mathbb{R}$ be bounded, $i\in\mathbb{Z}^{d}$, and
$T>0$. There are only finitely many points in $[0,T]$ where
$\zeta_{f}^{+}(\cdot,i)$ is not differentiable.

###### Proof.

Clearly the set of points in $[0,T]$ where $\zeta^{+}(\cdot,i)$ is not
differentiable is a subset of the set of points in $[0,T]$ where
$\zeta(\cdot,i)$ changes its sign. But the latter set is a subset of
$\mathcal{Z}_{f}(T,i)$. Therefore the claim is a consequence of Proposition
3.1. ∎

######  Proposition 3.3 (Properties of $\zeta^{-}$).

Let $f:\mathbb{Z}^{d}\mapsto\mathbb{R}$. There is a function
$q_{f}:\mathbb{R}_{+}\times\mathbb{Z}^{d}\mapsto\mathbb{R}$ such that

$\zeta_{f}^{-}(t,i)=f^{-}(i)+\int_{0}^{t}\Delta\zeta_{f}^{-}(s,i)\,\mathrm{d}s-q_{f}(t,i)\,,\quad
t\geq 0\,,i\in\mathbb{Z}^{d}\,.$ (3.14)

For each $i\in\mathbb{Z}^{d}$, the function
$q_{f}(\cdot,i):\mathbb{R}_{+}\mapsto\mathbb{R}$ has the following properties:

  1. (i)

$q_{f}(\cdot,i)$ is non-negative and non-decreasing.

  2. (ii)

For $T>0$, define $\mathcal{Q}_{f}(T,i):=\\{t\in[0,T]:\text{the derivative of
}q(t,i)\text{ exists in }t\\}$. Then
$|T\setminus\mathcal{Q}_{f}(T,i)|<\infty$.

  3. (iii)

For any $t>0$ define

$\partial_{t}q_{f}(t,i):=\left\\{\begin{array}[]{ll}\text{{\rm the derivative
of} $q(\cdot,i)$ {\rm at} $t$},&\text{{\rm if}
$t\in\bigcup_{T>0}\mathcal{Q}_{f}(T,i)$, }\\\ \text{$0,$}&\text{\rm if
$t\not\in\bigcup_{T>0}\mathcal{Q}_{f}(T,i).$}\end{array}\right.$

Then,

$q_{f}(t,i)=\int_{0}^{t}\partial_{s}q_{f}(s,i)\,\mathrm{d}s\,,\quad t\geq
0\,,$ (3.15)

that is, $q_{f}(\cdot,i)$ is absolutely continuous.

For simplicity we will omit the subindex $f$ whenever confusion is impossible.

###### Proof of Poposition 3.3.

Since $f$ is fixed we drop the subindex $f$ in what follows in the proof of
the proposition. Also in the following we consider $i\in\mathbb{Z}^{d}$ to be
fixed and $\zeta$ as a function of $t$ only. Define

$q(t,i):=-\zeta^{+}(t,i)+\zeta^{+}(0,i)+\int_{0}^{t}\Delta\zeta^{+}(s,i)\mathrm{d}s\,,\quad
t\geq 0.$ (3.16)

Then, by writing $\zeta^{-}(t,i)=\zeta^{+}(t,i)-\zeta(t,i)$ and re-arranging
terms in the discrete heat equation we obtain

$\displaystyle\zeta^{-}(t,i)$
$\displaystyle=\zeta^{-}(0,i)+\int_{0}^{t}\Delta\zeta^{-}(s,i)\,\mathrm{d}s-q(t,i),\quad
t\geq 0.$ (3.17)

Corollary 3.2 and the definition of $q$ in (3.16) clearly imply that
$|T\setminus\mathcal{Q}_{f}(T,i)|<\infty$ for each $T>0$. Since in addition we
know that $\zeta(\cdot,i)$ is analytic, it is easy to see that (3.15) holds
true.

We still have to prove that $q(t,i)$ is non-decreasing in $t$. As
$(T\setminus\mathcal{Q}(T,i))\subset\mathcal{Z}_{f}(T,i)$ and
$|\mathcal{Z}_{f}(T,i)|<\infty$ for every $T>0$ it is clearly enough to prove
that $\partial_{t}q(t,i)\geq 0$ for
$t\in\bigcup_{T>0}(T\setminus\mathcal{Z}_{f}(T,i))$. So, let
$t\in\bigcup_{T>0}(T\setminus\mathcal{Z}_{f}(T,i))$. Then (3.16) implies

$\partial_{t}q(t,i)=-\partial_{t}\zeta^{+}(t,i)+\Delta\zeta^{+}(t,i)\,.$
(3.18)

Now, assume that $\zeta(t,i)>0$. Then $\zeta^{+}(t,i)=\zeta(t,i)$ and
$\partial_{t}\zeta^{+}(t,i)=\partial_{t}\zeta(t,i)$. Therefore, since
$\partial_{t}\zeta(t,i)=\Delta\zeta(t,i)$ we get from (3.18) that

$\displaystyle\partial_{t}q(t,i)$
$\displaystyle=\Delta\zeta^{+}(t,i)-\Delta\zeta(t,i)$ (3.19)
$\displaystyle=\frac{1}{2d}\sum_{j\sim
i}\Bigl{(}\zeta^{+}(t,j)-\zeta(t,j)\Bigr{)}$ (3.20) $\displaystyle\geq 0.$
(3.21)

Alternatively if $\zeta(t,i)<0$, we have that $\zeta^{+}(t,i)=0$ and
$\partial_{t}\zeta^{+}(t,i)=0$. Therefore (3.18) implies

$\displaystyle\partial_{t}q(t,i)$ $\displaystyle=\Delta\zeta^{+}(t,i)$ (3.22)
$\displaystyle=\frac{1}{2d}\sum_{j\sim i}\zeta^{+}(t,j)$ (3.23)
$\displaystyle\geq 0.$ (3.24)

The above gives us that $q(t,i)$ is non-decreasing in $t$. Since $q(0,i)=0$
this implies also $q(t,i)\geq 0$. Thus we have proved all claims. ∎

######  Lemma 3.4.

Let $f\in E_{\textup{fin}}$ and denote $M:=\langle f,\boldsymbol{1}\rangle$.
Then,

$\lim_{t\to\infty}\langle|\zeta_{f}(t)-\zeta^{M}(t)|,\boldsymbol{1}\rangle=0\,.$
(3.25)

For a related result for the heat equation on $\mathbb{R}^{d}$ see (1.10) in
[EZ91]. We are grateful to Yehuda Pinchover who pointed us to this reference.

###### Proof.

Recall that $p_{t}(\cdot,\cdot)$ denotes the transition probability of a
simple symmetric random walk on $\mathbb{Z}^{d}$ with the generator $\Delta$.
Note that then
$\zeta_{f}(t,i)=\sum_{j}p_{t}(i,j)f(j)=\sum_{i}p_{t}(0,i-j)f(j)$ and thus
$\zeta^{M}(t,i)=Mp_{t}(0,i)=\sum_{j}f(j)p_{t}(0,i)$. Therefore,

$\displaystyle\langle|\zeta_{f}(t)-\zeta^{M}(t)|,\boldsymbol{1}\rangle$
$\displaystyle=\sum_{i}\Big{|}\sum_{j}(p_{t}(0,i-j)-p_{t}(0,i))f(j)\Big{|}$
(3.26) $\displaystyle\leq\sum_{j}|f(j)|\sum_{i}|p_{t}(0,i-j)-p_{t}(0,i)|,\quad
t\geq 0.$ (3.27)

Now since $\sum_{i}|p_{t}(0,i-j)-p_{t}(0,i)|\leq 2$ and $f\in
E_{\textup{fin}}$ the dominated convergence theorem implies that it is enough
to show that, for each $j$,

$\lim_{t\to\infty}\sum_{i}|p_{t}(0,i-j)-p_{t}(0,i)|=0\,.$ (3.28)

Let $p^{n}(0,i)$ be the probability that a discrete time simple random walk on
$\mathbb{Z}^{d}$ jumps from $0$ to $i$ in $n$ steps. Then

$\displaystyle\sum_{i}|p_{t}(0,i-j)-p_{t}(0,i)|$ $\displaystyle\leq
e^{-t}\sum_{n\geq 0}\frac{t^{n}}{n!}\sum_{i}|p^{n}(0,i-j)-p^{n}(0,i)|,\quad
t\geq 0.$ (3.29)

By Proposition 2.4.1 in Lawler and Limic [LL10], one has for all
$j\in\mathbb{Z}^{d}$

$\sum_{i}|p^{n}(0,i-j)-p^{n}(0,i)|\leq c|j|n^{-1/2}\,,$ (3.30)

for some constant $c$. Hence,

$\sum_{i}|p_{t}(0,i-j)-p_{t}(0,i)|\leq c|j|e^{-t}\sum_{n\geq
0}\frac{t^{n}}{n!}\frac{1}{n^{1/2}}\,,$ (3.31)

which, as can be easily checked, for each $j$ tends to $0$ as $t\to\infty$ and
we are done.

∎

The above lemma implies the following proposition.

######  Proposition 3.5 (longtime behavior of $\zeta^{-}$ and $\zeta^{+}$).

Let $f\in E_{\textup{fin}}$ and denote $M:=\langle f,\boldsymbol{1}\rangle$.
Assume $M\geq 0$. Then,

$\displaystyle\lim_{t\to\infty}\langle|\zeta_{f}(t)|,\boldsymbol{1}\rangle=\lim_{t\to\infty}\langle\zeta^{+}_{f}(t),\boldsymbol{1}\rangle=M\,,\quad\lim_{t\to\infty}\langle\zeta^{-}_{f}(t),\boldsymbol{1}\rangle=0\,.$
(3.32)

Moreover,

$\lim_{t\to\infty}\langle q_{f}(t),\boldsymbol{1}\rangle=\langle
f^{-},\boldsymbol{1}\rangle\,.$ (3.33)

###### Proof.

Recall that $\langle\zeta^{M}(t),\boldsymbol{1}\rangle=M$ for all $t\geq 0$.
Thus by the reverse triangle inequality (applied to the $L^{1}$ norm on
$E_{\textup{fin}}$),

$\displaystyle\bigl{|}\langle|\zeta_{f}(t)|,\boldsymbol{1}\rangle-M\bigr{|}$
$\displaystyle=\bigl{|}\langle|\zeta_{f}(t)|,\boldsymbol{1}\rangle-\langle\zeta^{M}(t),\boldsymbol{1}\rangle\bigr{|}=\bigl{|}\|\zeta_{f}(t)\|_{1}-\|\zeta^{M}(t)\|_{1}\bigr{|}$
(3.34)
$\displaystyle\leq\|\zeta_{f}(t)-\zeta^{M}(t)\|_{1}=\langle|\zeta_{f}(t)-\zeta^{M}(t)|,\boldsymbol{1}\rangle,\quad
t\geq 0.$ (3.35)

By Lemma 3.4,
$\lim_{t\to\infty}\langle|\zeta_{f}(t)-\zeta^{M}(t)|,\boldsymbol{1}\rangle=0$.
Now we use

$\displaystyle\langle\zeta^{\pm}_{f}(t),\boldsymbol{1}\rangle=\frac{1}{2}(\langle|\zeta_{f}(t)|,\boldsymbol{1}\rangle\pm\langle\zeta_{f}(t),\boldsymbol{1}\rangle)=\frac{1}{2}(\langle|\zeta_{f}(t)|,\boldsymbol{1}\rangle\pm
M),\quad t\geq 0.$ (3.36)

to get (3.32). Finally, (3.14) and the fact that
$\lim_{t\to\infty}\langle\zeta^{-}_{f}(t),\boldsymbol{1}\rangle=0$ imply
$\lim_{t\to\infty}\langle q_{f}(t),\boldsymbol{1}\rangle=\langle
f^{-},\boldsymbol{1}\rangle$. ∎

We are now ready to prove the non-coexistence result for SBM(1).

###### Proof of Theorem 1.6(ii).

Recall that $\eta_{t}=v_{t}-u_{t}$ and $u_{t}=w_{t}+\eta_{t}^{-}$, where
$w_{t}=\min\\{u_{t},v_{t}\\}$. Since we assume that $\langle
u_{0},\boldsymbol{1}\rangle\leq\langle v_{0},\boldsymbol{1}\rangle$, we have
$\langle\eta_{0},\boldsymbol{1}\rangle\geq 0$. Therefore, Proposition 3.5
implies that

$\lim_{t\to\infty}\langle\eta_{t}^{-},1\rangle=0\,.$ (3.37)

Hence, if we can also prove that

$\operatorname*{w-lim}_{t\to\infty}\langle w_{t},\boldsymbol{1}\rangle=0\,.$
(3.38)

then $\operatorname*{w-lim}_{t\to\infty}\langle
u_{t},\boldsymbol{1}\rangle=\operatorname*{w-lim}_{t\to\infty}(\langle
w_{t},\boldsymbol{1}\rangle+\langle\eta_{t}^{-},\boldsymbol{1}\rangle)=0$ and
we are done. So we are left with the task to show (3.38). The idea is to use
the fact that $w$ is “approximately” dual to a PAM as $t\to\infty$.

Note again that $\eta_{t}=\zeta_{v_{0}-u_{0}}(t)$ is the solution of the
discrete heat equation started in $v_{0}-u_{0}$. Hence, since
$w_{t}=u_{t}-\eta^{-}_{t}$, by Proposition 3.3, we get that $w$ satisfies the
following equation

$w_{t}(i)=w_{0}(i)+\int_{0}^{t}\Delta
w_{s}(i)\,\mathrm{d}s+\int_{0}^{t}\sqrt{bu_{s}(i)v_{s}(i)}\,\mathrm{d}W_{s}(i)+q(t,i)\,,\quad
t\geq 0\,,i\in\mathbb{Z}^{d}\,.$ (3.39)

Here, actually $q(t,i)=q_{v_{0}-u_{0}}(t,i)$ but for simplicity we omit the
subindex. By (3.39) and the properties of $q$ proved in Proposition 3.3,
$w_{\cdot}(i)$ is a semimartingale, for every $i\in\mathbb{Z}^{d}$. Note that

$w_{t}^{2}(i)\leq u_{t}(i)v_{t}(i)\,,\quad t\geq 0\,,i\in\mathbb{Z}^{d}\,,$
(3.40)

an observation which will be essential for the comparison between $w$ and the
PAM.

Now, fix $\varepsilon>0$ arbitrary small. Recall that we denote
$\bar{q}(t)=\sum_{i}q(t,i)$. By Proposition 3.3,
$\lim_{t\to\infty}\bar{q}(t)=\bar{q}^{\infty}$ exists and is finite. Define

$T^{\ast}:=\inf\\{t\geq 0:0\leq\bar{q}^{\infty}-\bar{q}(t)\leq\varepsilon\\}.$
(3.41)

Now let $\tilde{w}=(\tilde{w}_{t})_{t\geq 0}$ be a PAM starting at
$\tilde{w}_{0}$ and independent of $w$. That is, it satisfies

$\displaystyle\tilde{w}_{t}(i)=\tilde{w}_{0}(i)+\int_{0}^{t}\Delta\tilde{w}_{s}(i)\,\mathrm{d}s+\sqrt{b\tilde{w}_{s}(i)^{2}}\,\mathrm{d}\tilde{W}_{s}(i)\,,\quad
t\geq 0\,,i\in\mathbb{Z}^{d}\,,$ (3.42)

where $\tilde{W}(i)$, $i\in\mathbb{Z}^{d}$, are independent Brownian motions
which are assumed to be also independent of $\\{W(i),i\in\mathbb{Z}^{d}\\}$.

Now recall the following result which is stated in Lemma 4.4.10 of Ethier and
Kurtz [EK86]. Let $f:[0,\infty)\times[0,\infty)\mapsto\mathbb{R}$ be a
function such that $f(\cdot,t)$ is absolutely continuous for each $t$ and
$f(s,\cdot)$ is absolutely continuous for each $s$ and
$\int_{0}^{T}\int_{0}^{T}|f_{1}(s,t)|\,\mathrm{d}s\,\mathrm{d}t,\int_{0}^{T}\int_{0}^{T}|f_{2}(s,t)|\,\mathrm{d}s\,\mathrm{d}t<\infty$,
$T\geq 0$. Here $(f_{1},f_{2})=\nabla f$. Then, for every $R\geq 0$,

$f(T,0)-f(R,T-R)=\int_{R}^{T}(f_{1}(s,T-s)-f_{2}(s,T-s))\,\mathrm{d}s\,,\quad\text{for
almost every $T\geq R$.}$ (3.43)

Lemma 4.4.10 [EK86] actually states this formula only for $R=0$ but it can
easily verified that it holds for all $R\geq 0$. Now we would like to apply
(3.43) to the function

$f(a,b)=\mathbb{E}[\exp(-\langle w_{a},\tilde{w}_{b}\rangle)]\,,\quad a,b\geq
0\,,$ (3.44)

with $R=T^{\ast}$. Note that Itô’s formula implies that $f$ is absolutely
continuous in $a$ for each fixed $b$ and it is absolutely continuous in $b$
for each fixed $a$. Hence the conditions of Lemma 4.4.10 in [EK86] are
satisfied. Now we use Itô’s formula to calculate $f_{1}(s,t)$ and
$f_{2}(s,t)$. Then, for all non-negative $\phi\in E_{\textup{fin}}$, we have

$\displaystyle\exp(-\langle w_{s},\phi\rangle)$ (3.45)
$\displaystyle=\exp(-\langle
w_{0},\phi\rangle)-\sum_{i}\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\phi(i)\,\mathrm{d}w_{r}(i)$ (3.46)
$\displaystyle\quad\,+\frac{1}{2}\sum_{i}\int_{0}^{s}e^{-\langle
w_{u},\phi\rangle}\phi(i)^{2}\,\mathrm{d}[w(i)]_{r}$ (3.47)
$\displaystyle=\exp(-\langle w_{0},\phi\rangle)-\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\sum_{i}\phi(i)\Delta w_{r}(i)\,\mathrm{d}r$ (3.48)
$\displaystyle\quad\,-\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\sum_{i}\phi(i)\sqrt{bu_{r}(i)v_{r}(i)}\,\mathrm{d}W_{r}(i)-\sum_{i}\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\phi(i)\,\mathrm{d}q(r,i)$ (3.49)
$\displaystyle\quad\,+\frac{1}{2}\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\sum_{i}\phi(i)^{2}bu_{r}(i)v_{r}(i)\,\mathrm{d}r,\quad
s\geq 0.$ (3.50)

It is easy to check that $s\mapsto\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\sum_{i}\phi(i)\sqrt{bu_{r}(i)v_{r}(i)}\,\mathrm{d}W_{r}(i)$
is a martingale and not just the local martingale (see Proposition 3.2 of
Blath, Döring and Etheridge [BDE11] for a relevant result). Thus, taking
expectation on both sides yields

$\displaystyle\mathbb{E}[\exp(-\langle w_{s},\phi\rangle)]$ (3.51)
$\displaystyle=\exp(-\langle
w_{0},\phi\rangle)-\mathbb{E}\left[\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\langle\Delta
w_{r},\phi\rangle\,\mathrm{d}r\right]-\mathbb{E}\left[\langle\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\,\mathrm{d}q(r),\phi\rangle\right]$ (3.52)
$\displaystyle\quad\,+\mathbb{E}\left[\frac{1}{2}\int_{0}^{s}e^{-\langle
w_{r},\phi\rangle}\langle
bu_{r}v_{r},\phi^{2}\rangle\,\mathrm{d}r\right],\quad s\geq 0.$ (3.53)

Taking derivative with respect to $s$ (it exits for almost every $s$ for which
$\partial_{s}q(s)$ exists, see Proposition 3.3) yields

$\displaystyle\partial_{s}$ $\displaystyle\mathbb{E}\left[\exp(-\langle
w_{s},\phi\rangle)\right]$ (3.54) $\displaystyle=-\mathbb{E}\left[e^{-\langle
w_{s},\phi\rangle}\langle\Delta
w_{s},\phi\rangle\right]-\mathbb{E}\left[e^{-\langle
w_{s},\phi\rangle}\langle\partial_{s}q(s),\phi\rangle\right]+\mathbb{E}\left[\frac{1}{2}e^{-\langle
w_{s},\phi\rangle}\langle bu_{s}v_{s},\phi^{2}\rangle\right],\;s\geq 0.$
(3.55)

Now assume that $\tilde{w}_{0}\in E_{\textup{fin}}$. In exactly the same way
as above we calculate for every $\psi\in E_{\textup{fin}}$ (use that the
Laplacian is self-adjoint):

$\displaystyle\partial_{t}$
$\displaystyle\mathbb{E}\left[\exp(-\langle\psi,\tilde{w}_{t}\rangle)\right]$
(3.56)
$\displaystyle=-\mathbb{E}\left[e^{-\langle\psi,\tilde{w}_{t}\rangle}\langle\psi,\Delta\tilde{w}_{t}\rangle\right]+\mathbb{E}\left[\frac{1}{2}e^{-\langle\psi,\tilde{w}_{t}\rangle}\langle\psi^{2},b\tilde{w}_{t}^{2}\rangle\right]$
(3.57)
$\displaystyle=-\mathbb{E}\left[e^{-\langle\psi,\tilde{w}_{t}\rangle}\langle\Delta\psi,\tilde{w}_{t}\rangle\right]+\mathbb{E}\left[\frac{1}{2}e^{-\langle\psi,\tilde{w}_{t}\rangle}\langle
b\psi^{2},\tilde{w}_{t}^{2}\rangle\right],\quad t\geq 0.$ (3.58)

For $x=(x_{1},\ldots,x_{d})\in\mathbb{Z}^{d}$ define
$|x|_{\infty}=\sup_{i=1,\ldots,d}|x_{i}|$ and
$B_{n}=\\{x\in\mathbb{Z}^{d},|x|_{\infty}\leq n\\}$. Fix arbitrary $\theta>0$.
Let $\tilde{w}^{n}_{0}=\theta\mathbf{1}_{B_{n}}$ and $\tilde{w}^{n}$ be the
solution to the PAM starting at $\tilde{w}^{n}_{0}$. Now by (3.43), (3.44),
(3.55) and (3.58), for any $T>T^{*}$, we have

$\displaystyle\mathbb{E}\left[\exp(-\langle
w_{T^{*}},\tilde{w}^{n}_{T-T^{*}})\right]-\mathbb{E}\left[\exp(-\langle
w_{T},\tilde{w}^{n}_{0})\right]$ (3.59)
$\displaystyle=\int_{T^{*}}^{T}-\mathbb{E}\left[e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle\Delta
w_{s},\tilde{w}_{T-s}\rangle\right]+\mathbb{E}\left[\frac{1}{2}e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle
bw_{s}^{2},(\tilde{w}^{n}_{T-s})^{2}\rangle\right]\,\mathrm{d}s$ (3.60)
$\displaystyle\quad\,+\mathbb{E}\left[e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle\Delta
w_{s},\tilde{w}^{n}_{T-s}\rangle\right]+\mathbb{E}\left[e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle\partial_{s}q(s),\tilde{w}^{n}_{T-s}\rangle\right]$
(3.61) $\displaystyle\quad\,-\mathbb{E}\left[\frac{1}{2}e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle
bu_{s}v_{s},(\tilde{w}^{n}_{T-s})^{2}\rangle\right]$ (3.62)
$\displaystyle=\int_{T^{*}}^{T}\mathbb{E}\left[\frac{1}{2}e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle
b(w_{s}^{2}-u_{s}v_{s}),(\tilde{w}^{n}_{T-s})^{2}\rangle\right]\,\mathrm{d}s+\mathbb{E}\left[e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle\partial_{s}q(s),\tilde{w}^{n}_{T-s}\rangle\right]$
(3.63) $\displaystyle\leq\int_{0}^{T}\mathbb{E}\left[e^{-\langle
w_{s},\tilde{w}^{n}_{T-s}\rangle}\langle\partial_{s}q(s),\tilde{w}^{n}_{T-s}\rangle\right]\,\mathrm{d}s$
(3.64)
$\displaystyle\leq\int_{T^{*}}^{T}\langle\partial_{s}q(s),\mathbb{E}\left[\tilde{w}^{n}_{T-s}\right]\rangle\,\mathrm{d}s$
(3.65)
$\displaystyle=\int_{T^{*}}^{T}\langle\partial_{s}q(s),P_{T-s}\tilde{w}^{n}_{0}\rangle\,\mathrm{d}s\,.$
(3.66)

where we have used (3.40) in (3.64) and also the fact that
$\partial_{s}q(s,i)\geq 0,$ for all $i\in\mathbb{Z}^{d},t>0$. It is easy to
see that as $n\rightarrow\infty$, $\tilde{w}^{n}$ converges weakly to
$\tilde{w}$, where $\tilde{w}$ is the solution to PAM with initial condition
$\tilde{w}_{0}=\boldsymbol{\theta}$. Also, by monotone convergence theorem, we
get

$P_{T-s}\tilde{w}^{n}_{0}(i)\rightarrow\theta,\;\forall i\in\mathbb{Z}^{d}.$
(3.67)

as $n\to\infty$. Moreover the above convergence of
$P_{T-s}\tilde{w}^{n}_{0}(\cdot)$ to $\theta$ is bounded pointwise convergence
and thus we get,

$\int_{T^{*}}^{T}\langle\partial_{s}q(s),P_{T-s}\tilde{w}^{n}_{0}\rangle\,\mathrm{d}s\to\theta\int_{T^{*}}^{T}\langle\partial_{s}q(s),\mathbf{1}\rangle\,\mathrm{d}s,\;\text{
as $n\to\infty$. }$ (3.68)

Thus, by (3.66), (3.68) and weak convergence of $\tilde{w}^{n}$ to
$\tilde{w}$, we get

$\displaystyle\mathbb{E}\left[\exp(-\langle
w_{T^{*}},\tilde{w}_{T-T^{*}})\right]-\mathbb{E}\left[\exp(-\theta\langle
w_{T},\mathbf{1})\right]$ (3.69)
$\displaystyle\leq\theta\int_{T^{*}}^{T}\langle\partial_{s}q(s),\mathbf{1}\rangle\,\mathrm{d}s$
(3.70) $\displaystyle=\theta(\bar{q}(T)-\bar{q}(T^{*}))$ (3.71)
$\displaystyle\leq\varepsilon\theta,\;\forall T\geq T^{\ast},$ (3.72)

where the last inequality follows by (3.41). We know from Theorem 1.5 that for
any compactly supported function $\phi$,
$\operatorname*{w-lim}_{t\to\infty}\langle\phi,\tilde{w}_{t}\rangle=0$ if
$b>b_{\ast}$. Note that $w_{T^{*}}$ is not necessarily compactly supported, so
we need an additional simple argument. Fix $\tilde{\epsilon}>0$ arbitrary
small. Since $P_{T^{\ast}}w_{0}\in E_{\textup{fin}}$ we can take $n$
sufficiently large such that

$\displaystyle\mathbb{E}\left[\langle
w_{T^{*}}\mathbf{1}_{B^{c}_{n}},\tilde{w}_{T-T^{*}}\rangle\right]$
$\displaystyle=\theta\langle(P_{T^{*}}w_{0})\mathbf{1}_{B^{c}_{n}},\mathbf{1}\rangle$
(3.73) $\displaystyle=\theta\langle
P_{T^{*}}w_{0},\mathbf{1}_{B^{c}_{n}}\rangle$ (3.74)
$\displaystyle\leq\tilde{\epsilon}.$ (3.75)

On the other hand, by Theorem 1.5,

$\langle w_{T^{*}}\mathbf{1}_{B_{n}},\tilde{w}_{T-T^{*}}\rangle\Rightarrow 0,$
(3.76)

at $T\rightarrow\infty$. Since $\tilde{\epsilon}$ was arbitrary small we get
from (3.75), (3.76)

$\langle w_{T^{*}},\tilde{w}_{T-T^{*}}\rangle\Rightarrow 0,$ (3.77)

at $T\rightarrow\infty$. Hence $\lim_{T\to\infty}\mathbb{E}[\exp(-\langle
w_{T*},\tilde{w}_{T-T^{*}}\rangle)]=1$. So we can find $T^{\ast\ast}>T^{\ast}$
such that

$|\mathbb{E}[\exp(-\langle
w_{T*},\tilde{w}_{T-T^{*}}\rangle)]-1|\leq\varepsilon,\;\text{for}\;T>T^{\ast\ast}.$
(3.78)

Hence for $t>T^{\ast\ast}$, we get by (3.72), (3.78)

$\displaystyle\mathbb{E}[\exp(-\theta\langle w_{t},\boldsymbol{1}\rangle)]\geq
1-\varepsilon(1+\theta).$ (3.79)

This implies (3.38) since $\varepsilon$ was arbitrary small. ∎

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