# Towards understanding the performance of IEEE 802.11p MAC in heterogeneous
traffic conditions

M S Gayathree
School of Computing and Electrical Engineering
Indian Institute of Technology Mandi
Himachal Pradesh, India
<EMAIL_ADDRESS>
&Sreelakshmi Manjunath
School of Computing and Electrical Engineering
Indian Institute of Technology Mandi
Himachal Pradesh, India
<EMAIL_ADDRESS>

###### Abstract

Motivated by the need to study the performance of vehicular communication
protocols as applicable to heterogeneous traffic conditions, we study the
performance of IEEE 802.11p medium access protocol under such a traffic setup.
We consider a setup comprising connected vehicles and human-driven Motorised
Two Wheelers (MTWs), where the connected vehicles are required to move as
platoon with a desired constant headway despite interruptions from the two
wheelers. We invoke specific mobility models for the movement of the
vehicles—car following models for connected vehicle platoons and gap-
acceptance model to capture the movement of the MTWs—and use them to configure
(i) the traffic setup and (ii) the rate at which data packets related to
safety-critical messages need to be transmitted. A control-theoretic analysis
of the car-following models yields a bound on the admissible communication
delay to ensure non-oscillatory convergence of the platoon headway. We then
use suitable Markov chain models to derive the distribution of the MAC access
delay experienced by packets pertaining to safety-critical events as well as
routine safety messages. The distribution along with the bound on the
admissible delay enables us to derive the reliability of the 802.11p MAC
protocol in terms of traffic and EDCA parameters. Our study highlights the
need for redesign of MAC protocols for vehicular communications for safety-
critical applications in heterogeneous conditions.

_Keywords_ Heterogeneous traffic, Vehicular communication, Full Velocity
Difference (FVD) model, Modified Optimal Velocity Model (MOVM), Gap acceptance
model, Performance analysis.

##  1 Introduction

Road safety remains a significant developmental challenge, a matter of public
health concern, and a primary cause of fatalities and injuries globally.
Annually, road traffic accidents result in the loss of about 1.3 million
lives, with low and middle-income countries contributing to 93% of the global
road fatalities [1]. Connected vehicles and Intelligent Transportation Systems
(ITS) are expected to play a vital role in the future of road transportation.
Equipped with vehicular communication, connected vehicles are capable of
planning and performing cooperative movement for improved safety, better ride
quality and shorter ride times. This ability to perform cooperative movement
is dependent on the efficiency of vehicular communication protocols.
Therefore, understanding the performance of these protocols is central to the
design of connected vehicles and ITS. Further, when deployed, particularly in
developing nations, connected vehicles would have to co-exist and interoperate
with conventional vehicles of various types and dimensions, and with varying
levels of automation. This necessitates the study of vehicular communication
protocols in the context of heterogeneous traffic conditions.

Platooning, arguably the most basic cooperative movement of vehicles, refers
to vehicles moving in a single lane to maintain a constant spacing between
each other. Such cooperative movement requires decentralised decision making
and is hence sensitive to time delays. Connected vehicles deploy the IEEE 1609
protocol stack, and communicate using Dedicated Short Range Communication
(DSRC). At the MAC layer, the IEEE 802.11p standard with Enhanced Distributed
Channel Access (EDCA) mechanism is used. In congested environments, with large
number of vehicles, large values of channel access delays can cause huge
packet delays, thus impacting time-critical applications such as platooning.
Hence, it is desirable to study the performance of this EDCA mechanism in
terms of delay.

Performance analysis of the IEEE 802.11p MAC protocol has indeed attracted
significant research attention. In [2], the performance of IEEE $802.11$p EDCA
is analyzed using a $2$-D Markov chain model that considers varying Contention
Window (CW), internal collision scenarios and freezing mechanism. The
performance metrics, including normalized throughput and time delay, has been
calculated and derived. An analytical model for the IEEE $802.11$p EDCA
mechanism, has been evaluated through extensive simulations and introduced in
[3]. The model was created by merging two Markov chain models — a $2$-D model
representing the backoff procedure of each Access Category (AC) queue and a
$1$-D model describing contention period resulting from varying Arbitrary
Inter-Frame Space (AIFSs) and Contention Windows (CWs).

In [4], a comprehensive analytical model for assessing the performance of the
protocol has been developed. The model incorporates two Markov chains for
different priority ACs, which are later used to analyze the delay distribution
in broadcast mode. The accuracy of the model in its prediction of standard
deviation, mean, and probability distribution of the delay is verified through
simulations conducted on Network Simulator $2$ (NS$2$). Further, the idea was
extended to compute the reliability of IEEE $802.11$p in terms of packet delay
and packet delivery ratio in [5]. In the other hand, in [6], a dual-slope
propagation model is employed to determine the critical ranges in vehicular
communications such as the carrier sensing range and interference range, which
are typically assumed to be constant. Further, the authors of [4, 6],
established that the shifted exponential distribution can serve as an
effective approximation of the MAC access delay. A performance analysis of
IEEE $802.11$p for continuous backoff freezing is conducted in [7]. In [8],
IEEE $802.11$p MAC sublayer is studied in the presence of hidden terminals.
The backoff procedure for each AC is modeled using a $3$-D Markov chain.
Additionally, the model considers data packets and control packets separately.
Most of these studies adopt the approach of mathematical modelling and
analysis, followed by simulations, for performance analysis. A comparison of
the existing work is provided in Table 1. Of these studies, only [9, 10, 11]
consider mobility models for vehicular movement, while the others assume that
the vehicle positions follow certain stochastic processes. Notably, all these
studies focus on homogeneous traffic settings with a common assumption that
all the vehicles are equipped with vehicular communication. Performance of the
IEEE 802.11p in the context of heterogeneous traffic environments, and its
reliability for specific cooperative movement such as platooning in such
scenarios merits investigation.

This paper presents a study aimed at understanding the performance of the IEEE
802.11p medium access protocol, for platooning of connected vehicles in the
presence of motorised two wheelers. We consider the (i) Full Velocity
Difference (FVD) model and the (ii) Modified Optimal Velocity Model (MOVM)
[12] for the connected vehicle platoon and the gap acceptance model [13] to
capture the probability that a two wheeler interrupts the platoon by taking up
the gap between two consecutive connected vehicles. These models are used to
configure the traffic setup and the rate at which safety-critical/routine
messages need to be sent, _i.e._ the rate at which data packets need to be
transmitted. Stability analysis of the FVD and MOVM yields a bound on the time
delay that serves as the necessary and sufficient condition for non-
oscillatory convergence of the platoon headways , _i.e._ for each vehicle the
headway converges to a desired constant without exhibiting any oscillations.
We define the reliability of IEEE 802.11p as the probability that the delay
experienced by a given packet is within the critical delay defined by the
aforementioned bound. We use MATLAB to carry out the mathematical analysis and
protocol simulation to evaluate the mean packet delay and reliability, for
various headway requirements of the platoon. We consider two specific cases
for generation of packets corresponding to safety-critical events: packets are
generated as per a Poisson process with (i) a constant rate, (ii) rate as a
function of the platoon headway, based on the probability of a two wheeler
interrupting the platoon thereby generating a safety-critical event. Figure 1
gives an overview of our work.

Traffic SetupMobility Model Platoon of cars • FVD, MOVM Model • Packets
generated at a constant rate $\lambda_{1}$ pkts/sec Heterogeneous Traffic
2-wheeler interrupting the platoon of cars • Modelled using Gap Acceptance
Model • Packets generated at rate $\lambda_{0}$ pkts/sec Eqm.
headway($y^{\ast}$)Rate of pkt generation($\lambda_{0}$, $\lambda_{1}$)V2V
Communication IEEE 802.11p Protocol Obtain mean MAC access delay by
performing: (i) Mathematical Analysis (ii) Protocol Simulation Local Stability
Analysis Bound on delay for non-oscillatory convergence Reliability
$\mathbb{P}$ (MAC Access Delay < Critical Delay) Critical Delay

Figure 1: Overview of our work

The rest of the paper is organised as follows. The mobility models are
described in Section 2. The performance evaluation of 802.11p using the
mathematical model and simulation results are outlined in Section 3. The
conclusions and future work are discussed in Section 4.

Table 1: Comparison of existing studies References | Steady/Time Varying | Platoon | Mobility Model | Traffic Type
---|---|---|---|---
[5] |  Steady-state |  $\times$ |  Not considered |  Homogeneous
[6] |  Steady-state |  $\times$ |  Not considered |  Homogeneous
[9] |  Steady-state |  $\checkmark$ |  Intelligent Driver Model |  Homogeneous
[14] |  Steady-state |  $\times$ |  Constant Position Model of NS3 |  Homogeneous
[15] |  Steady-state |  $\times$ |  Not considered |  Homogeneous
[10] |  Time-varying |  $\times$ |  Freeway Mobility Model |  Homogeneous
[11] |  Time-varying |  $\checkmark$ |  Intelligent Driver Model |  Homogeneous
[16] |  Time-varying |  $\times$ |  Not considered |  Homogeneous
Our Work |  Steady-state |  $\checkmark$ |  FVD, MOVM & Gap Acceptance Model |  Heterogeneous

##  2 Traffic setup and Mathematical models

We now describe the traffic setup considered in our work, and briefly discuss
the priority-based packet classification used in the IEEE 802.11p MAC
protocol. This is followed by a description of the mobility models for the
connected vehicle platoon and the probability that a two wheeler maneuvers
into the platoon to fill the gap caused by the headway between two consecutive
vehicles in the platoon. We then use this _gap-acceptance_ probability to
model the rate of generation of packets corresponding to event-driven traffic.

###  2.1 Traffic setup and vehicular communication

The objective of our work is to study the performance of the IEEE 802.11p
protocol in the context of platooning of connected vehicles in heterogeneous
traffic conditions. To that end, we consider a traffic setup that comprises
two types of vehicles; namely connected vehicles and human-driven motorised
two wheelers. The connected vehicles are assumed to be traversing on a single-
lane road, as a platoon, with no overtaking. Human-driven motorised two
wheelers are assumed to surround the platoon throughout the road, and may
interrupt the platoon by overtaking, filling the gap or cutting across the
platoon. The connected vehicles must maintain their platoon despite these
interruptions by the two wheelers.

The IEEE 802.11p protocol allows prioritised channel access in order to ensure
that packets pertaining to safety-related applications are prioritised over
others. Generally, there are four access categories; namely AC0 – AC3, with
AC0 being the highest priority category and AC3 being the lowest. Of these,
AC0 and AC1 are reserved for safety-related applications while the other two
may be used for infotainment. Among the two, it is a general practice to
reserve the AC0 category for safety-critical messages and allot AC1 category
for routine safety-related applications.

In order to ensure safe platooning, the connected vehicles must communicate
their position/velocity information to other connected vehicles in their
vicinity. Therefore, each vehicle needs to send packets periodically at a pre-
defined rate. Additionally, in the heterogeneous traffic setting that we
consider, each vehicle will have to communicate its position/velocity as and
when a two wheeler interrupts with the platoon. The rate at which these
packets are generated would have to depend on the probability of such an
interruption. As such interruptions can be considered safety-critical, we
classify packets pertaining to these events as high priority AC0 category,
while the periodic updates are assigned the lower priority AC1. Figure 2
illustrates the traffic setup. Here, we assume that the connected vehicles
encounter no other safety-critical events, apart from the interruptions by the
motorised two wheelers. Therefore, packets pertaining to the AC1 traffic are
generated periodically at pre-defined rate; and those of the AC0 traffic are
event-based and are generated based on an appropriate mathematical model.

$y^{\ast}$

Figure 2: Schematic diagram of the heterogeneous traffic setup under study.

###  2.2 Connected vehicle platoon: Car following models

The Optimal Velocity Model (OVM) [17] is one of the earliest models for
platooning, in which each vehicle updates its velocity according to the
position of the vehicle ahead. The position of the preceding vehicle can
either be obtained through driver perception or through V2V communication. In
case of the former, the model must account for the reaction delay of the
driver, while for the latter communication delays must be accounted for. In
addition to these delays, one must also consider mechanical delays such as
actuator delay, braking delay etc. The OVM models the acceleration of a
vehicle as a function of its headway, _i.e_ the distance between the preceding
vehicle and itself. In this work, we consider two variants of the OVM, namely
the (i) Modified Optimal Velocity Model (MOVM) [18] and the (ii) Full Velocity
Difference (FVD) model. The MOVM, like the OVM, describes the acceleration of
a vehicle as a function of its headway. However, unlike the OVM it accounts
for the reaction delay of the driver, which makes the feedback regarding the
vehicle’s headway delayed. The FVD, on the other hand, considers the influence
of the velocity difference between itself and the preceding vehicle in
addition to the influence of its headway. This model too can be enhanced to
consider the delayed feedback. Which is the approach we take in our work.

For a platoon of $N$ traversing on an infinite highway, with the relative
velocities and headways chosen as the state variables, the FVD can be written
as

$\displaystyle\dot{v}_{1}(t)=$
$\displaystyle\,\ddot{x}_{0}(t)+a(\dot{x}_{0}(t)-V(y_{1}(t-\tau))-v_{1}(t-\tau))-lv_{1}(t-\tau),$
(1) $\displaystyle\dot{v}_{k}(t)=$
$\displaystyle\,a(V(y_{k-1}(t-\tau))-V(y_{k}(t-\tau))-v_{k}(t-\tau))+\,l\left(v_{k-1}(t-\tau-
v_{k}(t-\tau))\right),$ $\displaystyle\dot{y}_{i}(t)=$
$\displaystyle\,{v}_{i}(t)$

for $i=1,2,\dots,N$ and for $k=2,3,\dots,N$. Here, $a$ and $l$ are the
sensitivity parameters, $x_{0}(t),\dot{x}_{0}(t)$ and $\ddot{x}_{0}(t)$
represent the position, velocity and acceleration, respectively, of the lead
vehicle. Further, $v_{i}(t)$ and $y_{i}(t)$ indicates the relative velocity
and the headway between the $i^{th}$ and ${(i-1)}^{th}$ vehicle, respectively.
The feedback delay $\tau$ represents the cumulative delay in the communication
channel and other delays such as actuation and braking delays in the vehicle.
Note that for the sake of this analysis, we make the simplifying assumption
that the delay experienced by all the vehicles is the same, $\tau$. The
function $V(\cdot)$ represents the Optimal Velocity Function (OVF), which
models the acceleration of a follower vehicle as a function of the available
headway. One such OVF, known as the Bando OVF model is given by [17]

$V(y)=V_{0}\left(\tanh\left(\frac{y-y_{m}}{\tilde{y}}\right)+\tanh\left(\frac{y_{m}}{\tilde{y}}\right)\right)$
(2)

where $V_{0}$, $y_{m}$ and $\tilde{y}$ are model parameters. The model (1)
boils down to the MOVM for the case $l=0$. While the FVD is known to be a more
accurate representation for a platoon of autonomous vehicles, the MOVM
captures the scenario of human-driven vehicles better. The analysis that
follows would apply in either case subject to appropriate choice of the
parameter $l$.

Recall that for successful platooning, we require that the spacing between
each consecutive pair of vehicles, _i.e._ each vehicle’s headway, be constant.
This condition also implies that the relative velocities of the vehicles must
converge to zero. This would ensure collision-free platooning and jerk-less
movement of each vehicle. For a platoon represented by system (1), successful
platooning can be guaranteed when the headways $y_{i}(t)$ converge to a
constant value and the relative velocities $v_{i}(t)$ converge to zero. In
systems-theoretic terms, we require the dynamical system (1) to converge to a
stable equilibrium. Generally, in order to find the conditions on the system
parameters that ensure convergence to stable equilibrium one would carry out a
local stability analysis. Such analysis and the resulting analytical
conditions can be found in [yu2013full] for the FVD model and in [18] for the
MOVM.

While such conditions for local stability indeed ensure that the headways
converge to a constant, in practical situations it is often desired that the
headway does not undergo any oscillations while it converges. Such _non-
oscillatory_ convergence ensures smooth and jerk free movement of the vehicles
in the platoon. We now derive a bound on the time delay $\tau$ that ensures
non-oscillatory convergence of the platoon headways. Below we present the
analysis for the FVD model, from which one can also derive required bound for
the MOVM by substituting $l=0.$ The ensuing analysis closely follows [12].

Equating the first derivative of the state variables in system (1) to zero
yields the equilibrium point $y_{i}^{\ast}$ = $V^{-1}(\dot{x})$ and
$v_{i}^{\ast}=0.$ Given the velocity of the lead vehicle at equilibrium, i.e.,
$\dot{x}_{0}$, the equilibrium headway, $y_{i}^{\ast}$ can be obtained for
given OVF parameters. Consider a perturbation from the equilibrium headway
$u_{i}(t)$ = $y_{i}(t)$ \- $y_{i}^{\ast}(t)$. For small values of the
perturbation, we may take a Taylor series expansion about the equilibrium to
obtain a linearised approximation of (1). This yields

$\displaystyle\dot{v}_{1}(t)=$
$\displaystyle\,-aV^{\prime}(y^{\ast})u_{1}(t-\tau)-(a+l)v_{1}(t-\tau),$ (3)
$\displaystyle\dot{v}_{k}(t)=$
$\displaystyle\,aV^{\prime}(y^{\ast})\left(u_{k-1}(t-\tau)-u_{k}(t-\tau)\right)-av_{k}(t-\tau)+\,l(v_{k-1}(t-\tau)-v_{k}(t-\tau)),$
$\displaystyle\dot{u}_{i}(t)=$ $\displaystyle\,v_{i}(t),$

where prime denotes differentiation with respect to the state variable. The
above linearised system has the following characteristic equation

$f(\lambda)=\big{(}\lambda^{2}+(a+l)\lambda
e^{-\lambda\tau}+aV^{\prime}(y^{\ast})e^{-\lambda\tau}\big{)}^{N}=0,$ (4)

which can simply be reduced to

$f(\lambda)=\lambda^{2}+(a+l)\lambda
e^{-\lambda\tau}+aV^{\prime}(y^{\ast})e^{-\lambda\tau}=0,$ (5)

In order to obtain the necessary and sufficient condition for non-oscillatory
convergence, we look for the bound on the delay $\tau$ and the system
parameters that ensures that the roots of the characteristic equation (5) lies
on the negative real axis. To find this, we begin by assuming
$\lambda=\sigma+i\omega,$ and then deriving the condition that ensures
$\omega=0.$

Substituting $\lambda=\sigma+i\omega$ in (5) and separating the real and
imaginary parts, we obtain

$\displaystyle\left(\sigma(a+l)+aV^{\prime}(y^{\ast})\right)e^{-\sigma\tau}\,\cos(\omega\tau)+(a+l)\omega
e^{-\sigma\tau}\sin(\omega\tau)=$ $\displaystyle\,\omega^{2}-\sigma^{2},$
$\displaystyle(a+l)\omega
e^{-\sigma\tau}\cos(\omega\tau)-\big{(}\sigma(a+l)+aV^{\prime}(y^{\ast})\big{)}e^{-\sigma\tau}\,\sin(\omega\tau)=$
$\displaystyle\,-2\sigma\omega.$

Squaring and adding these equations, we obtain

$\displaystyle\left(\sigma(a+l)+aV^{\prime}(y^{\ast})\right)^{2}+\big{(}(a+l)\omega\big{)}^{2}=\,\big{(}(\omega^{2}-\sigma^{2})^{2}+4\sigma^{2}\omega^{2}\big{)}e^{2\sigma\tau}$
(6)

For $\lambda$ to be a real root, we require $\omega=0$. Substituting
$\omega=0$ in (6) yields

$\displaystyle\left(\sigma(a+l)+aV^{\prime}(y^{\ast})\right)^{2}=\sigma^{4}\,e^{2\sigma\tau},$

which serves as the necessary condition for non-oscillatory convergence. For
this to also be the sufficient condition, we must ensure that $\omega=0$
serves as unique solution to (6). Substituting for
$\left(\sigma(a+l)+aV^{\prime}(y^{\ast})\right)^{2}$ in equation (6), we get

$\displaystyle\sigma^{4}e^{2\sigma\tau}+(a+l)^{2}\omega^{2}=(\omega^{2}+\sigma^{2})^{2}e^{2\sigma\tau},$

which upon simplification yields

$\displaystyle\omega^{2}\big{(}\omega^{2}+2\sigma^{2}-(a+l)^{2}e^{-2\sigma\tau}\big{)}=0.$

For $\omega=0$ to be a unique solution, we require
$2\sigma^{2}\,e^{2\sigma\tau}=(a+l)^{2}$. Therefore, the characteristic
equation (5) has purely real roots if and only if

$\displaystyle\left(\sigma(a+l)+aV^{\prime}(y^{\ast})\right)^{2}=\sigma^{4}\,e^{2\sigma\tau}$
and $\displaystyle 2\sigma^{2}\,e^{2\sigma\tau}=(a+l)^{2}.$

Using these two conditions and solving for $\sigma$ yields

$\displaystyle\sigma=\tilde{d}(-2\pm\sqrt{2}),$

where $\tilde{d}=aV^{\prime}(y^{\ast})/(a+l)$. Substituting this value of
$\sigma$ and $\omega=0$ in the characteristic equation (5) yields the critical
value of $\tau$ which marks the necessary and sufficient condition for non-
oscillatory convergence. We denote this critical value by $\tau_{cr}$, and
this value is given by

$\displaystyle\tau_{cr}=\frac{1}{\tilde{d}(-2-\sqrt{2})}\ln\bigg{(}\frac{-\tilde{d}(a+l)(-2-\sqrt{2})-aV^{\prime}(y^{\ast})}{\big{(}\tilde{d}(-2-\sqrt{2})\big{)}^{2}}\bigg{)}.$
(7)

Therefore, any value of feedback delay that satisfies $\tau<=\tau_{cr}$
ensures non-oscillatory convergence. Note that the value of critical delay
depends on the model parameters $a,V_{0},y_{m},\tilde{y}$.

We now present a graphical representation of the critical delay for non-
oscillatory convergence for the two models that we consider. For our study, we
consider $\tilde{y}=10$, $y_{m}=5,$ as in [12] and $a=5$ while $\dot{x}_{0}$
is set to be $25$ m/s. For FVD, we consider $l=2,$ and for the MOVM we use
$l=0$. The desired headway is varied in the range $[2,10]$ m, and the
corresponding value of $V_{0}$ that satisfies the equilibrium condition
$y_{i}^{\ast}=V^{-1}(\dot{x_{0}})$ is calculated. Using these parameter
values, we compute the value of the critical delay using the condition (7).
The plot showing the critical delay as a function of the headway is presented
in Figure 3. The region below the curve represents the values of time delay
for non-oscillatory convergence of the platoon.

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.02$$0.04$$0.06$$0.08$$0.1$Headway
(m)Critical Delay (s)FVD MOVM Figure 3: Critical delay for non-oscillatory
convergence of the platoon headways as a function of the equilibrium headway
($y^{\ast}$) computed using equation (7).

So far we have derived a bound on the feedback delay that ensures non-
oscillatory convergence of the platoon headway. The feedback delay, in
general, is an accumulation of the communication delay, the delay in the on-
board computers and the throttle, brake and steering actuator delays. Studies
suggest that the computational and mechanical delays are much larger than
communication delays, and typically comprise 85-90% of the total feedback
delay [19]. To that end, we consider the bound on packet delay as $10\%$ of
the critical delay for non-oscillatory convergence.

###  2.3 Two Wheeler Maneuver: Gap Acceptance Model

In a heterogeneous traffic setting, one must consider the possibility of other
vehicles interrupting the platoon—by overtaking, cutting across the platoon or
taking up the gaps in the platoon. Such interruptions can be viewed as safety-
critical events that are required to be communicated by vehicle experiencing
it to the other vehicles in the platoon. The frequency of these safety-
critical events would thus govern the number of packets corresponding to the
event-driven (AC0) traffic that need to transmitted by the connected vehicle.
We now outline a model that captures the event of a motorised two wheeler
interrupting with the connected vehicle platoon. Further, we use it to model
the rate at which event-driven traffic is generated at each transmitting
vehicle.

For our work, we consider a gap acceptance model that is shown to be
applicable specifically to the context of heterogeneous traffic in India.
Probability that a two wheeler will accept gap between any two consecutive
cars in the platoon is given by [13]

$P_{i}=\frac{\exp{\left(\alpha+\beta_{0}\,\frac{y_{i}^{\ast}}{\dot{x}_{0}}\right)}}{1+\exp{\left(\alpha+\beta_{0}\,\frac{y_{i}^{\ast}}{\dot{x}_{0}}\right)}}$
(8)

where, $y_{i}^{\ast}$ is the equilibrium headway of the platoon and
$\dot{x}_{0}$ is the velocity of the vehicles in the platoon. The ratio
$y_{i}^{\ast}/\dot{x}_{0}$, often termed as time gap, is the time that a
vehicle in the platoon will take to cover the headway distance. Further,
$\alpha$ and $\beta$ are the model parameters that must be calibrated
according to the type of the vehicle that is considered to be filling the gap
in the platoon. For the case of motorised two wheelers interrupting with a
platoon of cars, it is proposed that $\alpha=-1.933$ and $\beta_{0}=0.652$
[13].

Equipped with the probability of occurrence of a safety-critical event, we now
model the rate of packet generation for the AC0 category. To begin with, as
these packets are generated by events, we assume that the packets are
generated as per a Poisson process. Let the rate of this process be
$\lambda_{0}$ pkts/sec. It is intuitive that $\lambda_{0}$ would depend on the
gap-acceptance probability given by (8). Further, with increase in the
probability of two wheelers interrupting the platoon, the rate $\lambda_{0}$
must increase. Hence, $\lambda_{0}$ can be modelled as a non-decreasing
function of $P_{i}$. The non-decreasing functions that have been used to model
AC0 traffic in our study are listed in Table 2. Here, $k$ is a tunable
parameter. We use $k=500$.

Table 2: Mathematical models for rate of generation of AC0 packets ($\lambda_{0}$) as a function of gap acceptance probability ($P_{i}$) Non–decreasing Function | $\lambda_{0}(P_{i})$
---|---
Linear | $kP_{i}$
Quadratic | $k\left(P_{i}^{2}+P_{i}\right)$
Sigmoidal | $k\left(\frac{e^{P_{i}}-e^{-P_{i}}}{e^{P_{i}}+e^{-P_{i}}}\right)$
Logarithmic | $1-k\,\log\left(1-P_{i}\right)$

##  3 Performance Evaluation

Consider a straight single-lane road with connected vehicles spaced as per the
desired headway, which need to communicate their position/velocity information
in order to maintain the platoon. We consider a single platoon as in Figure 2.
These vehicles try to access the communication channel using the IEEE 802.11p
protocol, and send the packets in broadcast mode wherein a transmitted packet
is expected to be received by all vehicles within a pre-defined range. In a
single vehicle, both AC0 and AC1 compete to access the communication channel.
However, only one of them can access the channel at any given time. Each AC
maintains its own Contention Window (CW), and if both ACs try to access the
channel simultaneously, a virtual collision occurs. In this scenario, AC0
takes priority and is transmitted, while AC1 enters a backoff stage with its
CW doubled. This process continues until AC1 gets a chance to access the
channel, or until it reaches the maximum retransmission limit, at which point
the packets will be discarded. As a result, only one vehicle (with only one
AC) can access the channel within a given time slot to transmit information to
its neighbouring vehicles.

Let $M$ be the maximum number of times that the CW of $AC_{1}$ is doubled.
Then, $M$ can be given as

$M=\log_{2}\frac{CW_{1,max}+1}{CW_{1,min}+1}$ (9)

Then, the CW of $AC_{1}$ in the $j^{th}$ backoff stage is given by:

$\displaystyle W_{1,j}=\begin{cases}2^{j}W_{1,0}~{}~{},j\leq M\\\
2^{M}W_{1,0}~{}~{},<j\leq L\end{cases}$ (10)

where, $W_{1,j}$ is the maximum CW size of $AC_{1}$ in the $j^{th}$ backoff
stage and $L$ is the retry limit of $AC_{1}$. For each AC within a vehicle, if
the channel is idle for an amount of time equal to an Arbitrary Inter-Frame
Space (AIFS), it transmits. In any other case, the AC will continue to monitor
the channel until it is idle for a period of time up to AIFS. The AIFS for the
two AC’s is given as:

$AIFS[i]=SIFS+AIFSN[i]\times\sigma~{}~{},i=0,1$ (11)

where, $SIFS$ is the Short Inter-Frame Space and $\sigma$ represents one slot
time.
Since AC1, being of lower priority will have to wait more compared to AC0 to
start contending for the channel, the Arbitration Inter-Frame Space Number
(AIFSN) for AC1 is greater than that of AC0, i.e., $AIFSN1>AIFSN0$. The
difference between the two is given by $A_{1}$ _i.e._

$A_{1}=AIFSN[1]-AIFSN[0]$ (12)

Let $\omega_{i}$ denote the internal transmission probability and $p_{vi}$
represent the virtual collision probability of $AC_{i}$. Then,

$\begin{cases}p_{v0}=0\\\ p_{v1}=\omega_{0}\end{cases}$ (13)

The external transmission probability i.e $\tau_{i}$ is given as:

$\begin{cases}\tau_{0}=\omega_{0}\\\
\tau_{1}=\omega_{1}(1-p_{v1})=\omega_{1}(1-\omega_{0})\end{cases}$ (14)

Then, the total transmission probability is the sum of the internal
transmission probabilities given by:

$\tau=\tau_{0}+\tau_{1}$ (15)

The average transmission time of a packet is given by:

$T_{tr}=\frac{PHY_{H}}{R_{b}}+\frac{MAC_{H}+E[P]}{R_{d}}+\delta$ (16)

where, $PHY_{H}$ and $MAC_{H}$ are the header length of the physical and MAC
layer respectively. $\delta$ is the propagation delay, $R_{b}$ is the basic
rate, $R_{d}$ is the data rate and $E[P]$ is the mean packet size. The AC0
messages arrive with a Poisson point process (with rate $\lambda_{0}$) whereas
the AC1 messages arrive periodically at a rate of $\lambda_{1}$ (packets per
second). The packet arrival probability of AC0 and AC1 is given as:

$\begin{cases}p_{a0}=\sum\limits_{k=1}^{\infty}\frac{(\lambda_{0}\sigma)^{k}}{k!}e^{-\lambda_{0}\sigma}=1-e^{-\lambda_{0}\sigma}\\\
p_{a1}=\lambda_{1}\sigma\end{cases}$ (17)

Higher priority AC0 can be modelled as a one-dimensional Markov chain and the
lower priority AC1 can be modelled as a two-dimensional Markov chain [5].
$N_{cs}$ is the number of vehicles which are in the carrier sensing range of
the given vehicle. $p_{bi}$ is the backoff blocking probability, _i.e._ , the
probability that the channel is occupied by the other surrounding vehicles or
there is a transmission attempted by the other ACs within the same vehicle.
$p_{bi}$ is given by:

$\displaystyle p_{bi}=1-\left[P(k)\prod_{\begin{subarray}{c}j=0\\\ j\neq
i\end{subarray}}^{1}(1-\omega_{j})\right]^{A_{i}+1}~{}~{},i=0,1$ (18)
$\displaystyle
P(k)=\sum_{k=0}^{\infty}(1-\tau)^{k}\frac{(N_{cs}-1)^{k}}{k!}e^{-(N_{cs}-1)}$

$\rho_{i}$ is the server utilization factor of $AC_{i}$ and is given as:

$\rho_{i}=\frac{\lambda_{i}}{\mu_{i}}~{}~{},i=0,1$ (19)

Using the one-step transition probabilities, it is possible to express all
stationary probabilities as $b_{0,0}$ and $b_{1,0,0}$ [5] _i.e._ ,

$\begin{cases}b_{k}=\frac{(W_{0,0}-k)}{W_{0,0}(1-p_{b0})}b_{0}\\\
b_{IDLE0}=\frac{1-\rho_{0}}{p_{a0}}b_{0}\\\ \end{cases}$ (20)

$\begin{cases}b_{1,j,0}=p_{v1}^{j}b_{1,0,0},~{}~{}~{}j\in(1,L)\\\
b_{1,0,k}=\frac{(W_{1,0}-k)}{W_{1,0}(1-p_{b1})}b_{1,0,0},~{}~{}~{}k\in(1,W_{1,0}-1)\\\
b_{1,j,k}=\frac{(W_{1,j}-k)p_{v1}^{j}}{W_{1,j}(1-p_{b1})}b_{1,0,0},~{}~{}~{}j\in(1,M),~{}~{}~{}k\in(1,W_{1,j}-1)\\\
b_{1,j,k}=\frac{(W_{1,M}-k)p_{v1}^{j}}{W_{1,M}(1-p_{b1})}b_{1,0,0},~{}~{}~{}j\in(M,L),~{}~{}~{}k\in(1,W_{1,M}-1)\\\
b_{IDLE1}=\frac{1-\rho_{1}}{p_{a1}}b_{1,0,0}\\\ \end{cases}$ (21)

The probability that the higher priority AC0 successfully transmits a packet
is equal to the stationary probability of being in state 0. Furthermore, for
AC1, $\omega_{1}$ can be viewed as the sum of the stationary probabilities of
all the possible backoff states. $\omega_{0}$ and $\omega_{1}$ are given by
equation 22.

$\begin{cases}\omega_{0}=b_{0,0}=\left[1+\frac{(W_{0,0}-1)}{2(1-p_{b0})}+\frac{1-\rho_{0}}{p_{a0}}\right]^{-1}\\\
\omega_{1}=\sum_{j=0}^{L}b_{1,j,0}=\frac{1-p_{v1}^{L+1}}{1-p_{v1}}{\left[\frac{1-p_{v1}^{L+1}}{1-p_{v1}}+\frac{W_{1,0}-1}{2(1-p_{b1})}+\frac{1}{2(1-p_{b1})}\left(\frac{W_{1,0}(2p_{v1})[1-(2p_{v1})^{M_{1}}]}{1-2p_{v1}}-\frac{p_{v1}(1-p_{v1}^{M_{1}})}{1-p_{v1}}+\frac{(2^{M_{1}}W_{1,0}-1)(1-p_{v1}^{L-M_{1}})p_{v1}^{M_{1}+1}}{1-p_{v1}}\right)+\frac{1-\rho_{1}}{p_{a1}}\right]^{-1}}\end{cases}$
(22)

The channel access parameters are given in Table 3 and the other traffic and
network parameters are presented in Table 4. The parameters are similar to the
ones used by authors in [5]. The overall delay experienced by a packet would
comprise of two components: (i) the queuing and channel access delay, (ii) the
propagation delay. The former is the sojourn time spent by a packet in the MAC
layer, waiting in the AC queue and attempting to access the channel through
contention and backoff procedure. The latter is the time spent by the channel
in the physical wireless channel. In our traffic setup, we consider a
deterministic propagation delay of $2\mu$s, which forms a very small component
of the overall delay observed. Consequently, one may conclude that the MAC
access delay forms the larger component of the packet delay and its PGF is
given by [6]:

$P_{Tsi}(z)=\sum\limits_{k=0}^{\infty}q_{i,k}z^{t_{{s}_{i,k}}}$ (23)

Thus, the delay experienced by a packet can be termed as the MAC access delay
and used as a performance metric for evaluating the IEEE 802.11p protocol. The
mean of the MAC access delay can be evaluated by solving equations (18) and
(22) and is given by [5]:

$T_{Si}=\frac{dP_{Tsi}(z)}{dz}\mid_{z=1}={P_{Tsi}}^{\prime}(1)~{}~{},i=0,1$
(24)

where, the Probability Generating Function (PGF) $P_{Tsi}(z)$ is given as [5]:

$\begin{cases}P_{Ts0}(z)=\frac{TR(z)}{W_{0,0}}\sum\limits_{k=0}^{W_{0,0}-1}[H_{0}(z)]^{k}\\\
\begin{gathered}P_{Ts1}(z)=(1-p_{v1})TR(z)\sum\limits_{n=0}^{L}\left[p_{v1}^{n}\prod\limits_{j=0}^{n}B_{1,j}(z)\right]\\\
+p_{v1}^{L+1}\prod\limits_{j=0}^{L}B_{1,j}(z)\end{gathered}\end{cases}$ (25)

PGF $TR(z)$, $H_{i}(z)$ and $B_{1,j}(z)$ are given by [5]:

$TR(z)=z^{T_{tr}}$ (26)
$H_{i}(z)=(1-p_{bi})z^{\sigma}+p_{bi}z^{T_{tr}+AIFS[i]}~{}~{},i=0,1$ (27)
$B_{1,j}(z)=\begin{cases}\frac{1}{W_{1,j}}\sum\limits_{k=0}^{W_{1,j}-1}[H_{1}(z)]^{k}~{}~{},j\in(1,M_{1})\\\
\frac{1}{W_{1,M_{1}}}\sum\limits_{k=0}^{W_{1,M_{1}}-1}[H_{1}(z)]^{k}~{}~{},j\in(M_{1},L)\end{cases}$
(28)

The standard deviation of the MAC access delay can be obtained as [5]:

$D_{Tsi}={P_{Tsi}}^{\prime\prime}(1)+{P_{Tsi}}^{\prime}(1)-\left[{P_{Tsi}}^{\prime}(1)\right]^{2}~{}~{},i=0,1$
(29)

Figure 4 shows the mean MAC access delay obtained for various values of
headway. Simulation of the protocol is carried on the MATLAB 2021a. For the
simulation scenario, we consider vehicles distributed evenly on a road and
spaced by a desired headway (as in Figure 2). The parameters of 802.11p are
set up in accordance with the IEEE 802.11p standard [20]. From Figure 4, it is
clear that the MAC access delay obtained using the simulation matches with the
analytical results. As expected, the mean delay for the AC0 traffic is lower
than that of the AC1 traffic. It can be observed that the mean access delay
decreases as a function of headway. At a given instant, only a single vehicle
can access the communication channel to broadcast packets to the vehicles
which are in its transmission range. The access delay is composed of the
backoff time (which is a function of $p_{bi}$) and the transmission time of
which the transmission time is deterministic. When the headway is less, the
vehicles are close to each other. With this $\rho_{i}$ increases. This
technically means that the mac queue has more packets to serve. This increases
the backoff time and the contention for the channel hence, increasing the mac
access delay.
Figure 5 illustrates the standard deviation of the MAC access delay. The
simulation results match with the analytical results. From the figure, it is
evident that the deviation is more for lower values of headway. It is
therefore important to study the distribution of the MAC access delay as the
mean value may not give us the true picture about the actual traffic
condition.

By utilizing the characteristics of PGF, one can obtain the Probability Mass
Function (PMF) of the MAC access delay as:

$p_{k}(t_{si})=P\\{t_{si}=k\\}=\frac{{P_{T_{si}}^{(k)}}{(0)}}{k!}$ (30)

Once we obtain the PMF for each headway value, it is easy to obtain the
Cumulative Distribution Function (CDF). For this, we use the curve fitting
approach wherein we fit the curve according to shifted exponential
distribution [6]. Figure 6 illustrates the CDF for Linear model of AC0
generation rate. The CDF plots for the other three models follow similar trend
as the Linear model. The equations for CDF can be obtained in terms of
headway. Table 5 illustrates the CDF for different models of AC0 generation
rate in terms of headway. For ensuring the stability of the platoon, it is
desirable that every packet be delivered within the critical time delay.
Therefore, one may quantify the performance of the IEEE 802.11p protocol in
terms of its ability to ensure packet delivery times lesser than the critical
delay. From the CDF, we can obtain the reliability of the protocol which is
defined as the probability that a packet is delivered within a duration
defined by the critical delay. The reliability obtained for various values of
headway is shown in Figure 7 and Figure 8 when vehicles in the platoon follow
FVD and MOVM model respectively.

Table 3: EDCA access parameters AC | CWmin | CWmax | AIFSN
---|---|---|---
$1$ | 15 | 31 | $3$
Table 4: Traffic and network parameter settings Parameter | Value | Parameter | Value
---|---|---|---
Basic rate |  $1$ Mbps | Data rate |  $3$ Mbps
PHY header |  $48$ bits | MAC header |  $112$ bits
Transmission range |  $500$ m | Basic rate |  $1$ Mbps
Carrier sensing range |  $700$ m | Slot time |  $13$ $\mu$s
$E[P]$ |  $500$ bytes | SIFS |  $32$ $\mu$s
Retry limit | $2$ | Propagation delay |  $2$ $\mu$s

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$6$$8$$10$$12$$14$Headway (m)Mean MAC
Access Delay (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (a)
Linear

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$6$$8$$10$$12$$14$Headway (m)Mean MAC
Access Delay (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (b)
Quadratic

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$6$$8$$10$$12$$14$Headway (m)Mean MAC
Access Delay (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (c)
Sigmoidal

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$6$$8$$10$$12$$14$Headway (m)Mean MAC
Access Delay (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (d)
Logarithmic

Figure 4: Mean MAC Access Delay for various models of packet generation rate
for AC0 traffic based on the gap-acceptance probability of motorised two
wheelers for each headway value.

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$1$$2$$3$$4$$5$$6$$7$Headway (m)Standard
Deviation (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (a) Linear

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$1$$2$$3$$4$$5$$6$$7$Headway (m)Standard
Deviation (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (b)
Quadratic

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$1$$2$$3$$4$$5$$6$$7$Headway (m)Standard
Deviation (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (c)
Sigmoidal

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$1$$2$$3$$4$$5$$6$$7$Headway (m)Standard
Deviation (ms)AC0 AnalysisAC1 AnalysisAC0 SimulationAC1 Simulation (d)
Logarithmic

Figure 5: Standard Deviation (ms) observed for various models of AC0 traffic.

$0$$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$55$$60$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Access
Delay (ms)Probability$AC0$$AC1$ (a) Headway = 3m

$0$$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$55$$60$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Access
Delay (ms)Probability$AC0$$AC1$ (b) Headway = 5m

$0$$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$55$$60$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$Access
Delay (ms)Probability$AC0$$AC1$ (c) Headway = 10m

Figure 6: CDF observed when AC0 packets are generated as per Linear model. Table 5: CDF obtained for different AC0 packet generating functions in terms of headway | AC0 | AC1
---|---|---
Linear | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0566y^{\ast}+0.1277\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$ | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0156y^{\ast}+0.057\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$
Quadratic | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0499y^{\ast}+0.1221\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$ | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0131y^{\ast}+0.0588\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$
Sigmoidal | $\displaystyle\begin{cases}1-\left(e^{-\left(0.057y^{\ast}+0.1276\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$ | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0158y^{\ast}+0.0567\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$
Logarithmic | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0885y^{\ast}+0.1724\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$ | $\displaystyle\begin{cases}1-\left(e^{-\left(0.0254y^{\ast}+0.0661\right)\left(x-T_{tr}\right)}\right),~{}~{}x\geq T_{tr}\\\ 0,~{}~{}x<T_{tr}\end{cases}$

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (a) Linear

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (b) Quadratic

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (c) Sigmoidal

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (d) Logarithmic

Figure 7: Reliability observed for various models of AC0 traffic when
vehicular mobility follows FVD model.

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (a) Linear

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (b) Quadratic

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (c) Sigmoidal

$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$0.2$$0.4$$0.6$$0.8$$1$Headway
(m)ReliabilityAC0AC1 (d) Logarithmic

Figure 8: Reliability observed for various models of AC0 traffic when
vehicular mobility follows MOVM model.

##  4 Conclusions and future work

We studied the performance of IEEE 802.11p medium access protocol in the
context of a heterogeneous traffic environment. We specifically investigated a
traffic setup comprising a platoon of connected vehicles and human-driven
motorised two-wheelers. FVD and MOVM were employed to model the mobility of
the vehicles and the gap acceptance model was used to account for the
likelihood of a two-wheeler cutting across the platoon. Using the above
models, we configured the traffic setup and the rate of generation of data
packets. Further, we conducted a local stability analysis of the FVD to
determine a critical delay which serves as an upper bound on the delay
experienced by a packet such that platoon stability can be maintained. We then
obtained the mean MAC access delay and standard deviation for various values
of the platoon headway. CDF was obtained for different AC0 packet generating
functions. Using the CDF, reliability of IEEE 802.11p protocol in context of
heterogeneous traffic setup was derived.

The same study could be extended to incorporate vehicles of other types
cutting across multiple platoons. In addition reliability can be computed by
considering different mobility models (like the Intelligent driver model) of
the vehicles in the platoon. While we carried out the performance analysis of
IEEE 802.11p, other options like C-V2X, NR-V2X, or IEEE 802.11bd could be
explored to analyze performance in heterogeneous traffic scenarios.

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# Ergodic theory of multi-layer interacting particle systems

Frank Redig111Delft Institute of Applied Mathematics, TU Delft, Delft, The
Netherlands<EMAIL_ADDRESS>, Hidde van Wiechen222Delft Institute of
Applied Mathematics, TU Delft, Delft, The Netherlands<EMAIL_ADDRESS>
## Abstract

  * We consider a class of multi-layer interacting particle systems and characterize the set of ergodic measures with finite moments. The main technical tool is duality combined with successful coupling.

##  1 Introduction

In this paper we consider a class of multi-layer particle systems with duality
properties.

The study of multi-layer random walks and interacting particle systems is
motivated by the study of active particles, where the motion is determined by
an internal degree of freedom (determining the direction of motion) and a
random component which models the influence of collisions with surrounding
particles. Other terms for such random motions are e.g. persistent random walk
and run-and-tumble motion.

When the internal degree of freedom takes a finite number of values and
evolves autonomously as a finite state space continuous Markov chain, then one
can view the motion as a random walk on a multi-layer system, where the layers
are indexed by the internal states. There are various studies of the
asymptotic properties of such random walks including law of large numbers
(asymptotic speed), invariance principle (central limit behavior), and large
deviations. See e.g. [12, 10, 4, 7, 13] and references therein.

In this paper, we study a system of particles performing multi-layer random
walks which possibly have interaction of inclusion or exclusion type (see
Section 2.1 below for a precise description of the models). We characterize
the set of invariant measures with finite moments. More precisely we prove
that under an appropriate condition of moment growth, the only ergodic
invariant measures are homogeneous product measures, indexed by the first
moment (particle density). In the case of independent particles, these
measures are Poisson product measures, in the case of interacting systems they
are products of binomial (exclusion) resp. negative binomial (inclusion)
distributions.

The characterization of the invariant measures of multi-layer exclusion
processes has been obtained recently in [1]. The study of the hydrodynamic
limit of a system of active particles has been studied in [8], and for a two-
layer system with duality in [5].

The important ingredient in our setting here is duality combined with the
existence of a successful coupling for the dual process. This road is followed
for the symmetric exclusion process in [11] chapter 8. Duality allows to
characterize the invariant measures via the characterization of bounded
harmonic functions of the dual process, which is a countable state space
Markov chain. If this Markov chain admits a successful coupling, then the
bounded harmonic functions are constants, indexed by the number of dual
particles. The proof of the existence of a successful coupling is a
combination of coupling the finite state space internal state process and the
Ornstein coupling of random walks. These two ingredients are sufficient in the
non-interacting case. In the interacting case, we use the approach of [9]
which consists of “spreading out” the particles combined with Ornstein
coupling of symmetric random walks.

The rest of our paper is organized as follows. In Section 2 we provide the
general setup of multi-layer particle systems, after which we define the three
types of processes (exclusion, inclusion and independent walkers) we will
study in this paper. Afterwards, we study the duality properties and invariant
measures of these processes.

In Section 3 we provide a characterization of the ergodic invariant measures
in a slightly more general setting, where the only assumptions are duality
with polynomial duality functions and the existence of a successful coupling.
This unifies and generalizes earlier results from chapter 8 of Ligget [11] and
Kuoch [9].

Section 4 is devoted to the proof of a successful coupling for the models
under consideration. For independent particles this amounts to generalize the
Ornstein coupling to the multi-layer setting. For interacting particles, it
amounts to generalize the approach of Liggett for the exclusion process [11]
and Kuoch for the inclusion process [9].

##  2 Models and main theorem

###  2.1 Models

In this paper we will look at models of configurations where the coordinates
of individual particles are $(X(t),\sigma(t))$, with $X(t)\in\mathbb{Z}^{d}$
the _position_ of the particle, and $\sigma(t)\in S$ the _internal state_ ,
where $S$ is some finite set. We will denote the particle state space as
$V:=\mathbb{Z}^{d}\times S$.

We consider a configuration process $\\{\eta(t):t\geq 0\\}$ on the state space
$\Omega_{s}$ with generator of the following type,

$\mathscr{L}_{s}f(\eta)=\sum_{v,w\in
V}p(v,w)\eta_{v}(\alpha+s\eta_{w})\nabla_{v,w}f(\eta).$ (2.1)

Here $\eta_{v}$ is equal to the number of particles at site $v\in V$ and, if
we denote $\eta^{v\to w}$ as the configuration $\eta$ where a single particle
has moved from $v$ to $w$ (if possible), we have

$\nabla_{v,w}f(\eta)=f(\eta^{v\to w})-f(\eta).$

The value of $s\in\\{-1,0,1\\}$ determines the type of the process
(interacting or independent particles), the state space $\Omega_{s}$ we
consider, and the transition rates $p(v,w)$ and constants
$\alpha\in\mathbb{R}$ we allow.

The main characteristic of our multi-layer particle systems is that the
transtition rates are determined by the layer in which a particle resides.
Therefore, for every $\sigma\in S$ we define the translation invariant
transition rates $\\{\pi_{\sigma}(x):x\in\mathbb{Z}^{d}\\}$ of a simple
symmetric random walk on $\mathbb{Z}^{d}$, and we define the transition rates
$\\{c(\sigma,\sigma^{\prime}):\sigma,\sigma^{\prime}\in S\\}$ which we will
assume to be symmetric and irreducible. Then we define the following
processes:

  1. 1.

Symmetric exclusion process ($s=-1$). Every site contains at most
$\alpha\in\mathbb{N}$ particles and jumps to sites where there are already
many other particles are less likely. The state space of the multi-layer SEP
is given by $\Omega_{-1}=\\{0,1,...,\alpha\\}^{V}$, and the transition rates
$p(v,w)$ in (2.1) we will study for this model are of the following form,

$p\big{(}(x,\sigma),(y,\sigma^{\prime})\big{)}=\pi_{\sigma}(y-x)\delta_{\sigma,\sigma^{\prime}}+c(\sigma,\sigma^{\prime})\delta_{x,y},$
(2.2)

with $\delta_{\cdot}$ the Kronecker delta.

  2. 2.

Symmetric inclusion process ($s=1$). In contrast to the exclusion process,
this process actually encourages jumps to sites where other particles already
reside. The state space of the multi-layer SIP is given by
$\Omega_{1}=\mathbb{N}^{V}$, and the transition rates are again given by
(2.2). Furthermore, we allow for any $\alpha>0$.

  3. 3.

Independent particles ($s=0$). For this paper, our model for independent
particles will be the run-and-tumble particle process (RTP). A run-and-tumble
particle is a particle with the following dynamics.

     * _Random walk jumps._ With rate $\kappa$, a particle at $(x,\sigma)$ performs a simple symmetric random walk jump on $\mathbb{Z}^{d}$ according to the transition rates $\pi_{\sigma}(\cdot)$, i.e., $(x,\sigma)\to(x+e_{i},\sigma)$ with rate $\kappa\pi_{\sigma}(e_{i})$.

     * _Active jumps._ With rate $\lambda$, a particle at $(x,\sigma)$ performs an active jump in the direction determined by the internal state $\sigma$, i.e., there exists a function $v:S\to\mathbb{Z}^{d}$ such that $(x,\sigma)\to(x+v(\sigma),\sigma)$ with rate $\lambda$.

     * _Internal state jumps._ A particle changes its internal state according to the transition rates $\\{c(\sigma,\sigma^{\prime}):\sigma,\sigma^{\prime}\in S\\}$.

The state space of this process is $\Omega_{0}=\mathbb{N}^{V}$, and from the
dynamics we conclude that the transition rates are of the following form,

$p\big{(}(x,\sigma),(y,\sigma^{\prime})\big{)}=\kappa\pi_{\sigma}(y-x)\delta_{\sigma,\sigma^{\prime}}+\lambda\delta_{\sigma,\sigma^{\prime}}\delta_{y,x+v(\sigma)}+c(\sigma,\sigma^{\prime})\delta_{x,y}.$
(2.3)

We furthermore allow for any $\alpha>0$, however without loss of generality we
can assume that $\alpha=1$.

######  REMARK 2.1.

Notice that for the interacting models we only allow for symmetric transitions
on every layer while in the case of independent particles we introduced a
drift. This is due to the fact that the duality results that we will use in
this paper will not hold in the case of asymmetric transition rates for the
interacting models.

###  2.2 Duality

We will state and prove duality results for the processes we just defined.
Recall the following definition of duality of Markov processes.

######  DEFINITION 2.1.

Let $\\{\eta(t):t\geq 0\\}$ and $\\{\xi(t):t\geq 0\\}$ be two Markov processes
on the state spaces $\Omega$ and $\Omega^{\prime}$ respectively, and let
$D:\Omega\times\Omega^{\prime}\to\mathbb{R}$ be a bounded measurable function.
We say $\\{\eta(t):t\geq 0\\}$ and $\\{\xi(t):t\geq 0\\}$ are _dual_ to one
another, with respect to $D$, if

$\mathbb{E}_{\eta}\left[D(\xi,\eta(t))\right]=\widehat{\mathbb{E}}_{\xi}\left[D(\xi(t),\eta)\right].$

where $\mathbb{E}_{\eta}$ denotes expectation in $\\{\eta(t):t\geq 0\\}$
starting from $\eta$, and $\widehat{\mathbb{E}}_{\xi}$ expectation in the dual
process $\\{\xi(t):t\geq 0\\}$ starting from $\xi$. We then call $D$ the
_duality function_.

####  2.2.1 Duality results

Let $\Omega_{s,f}:=\\{\xi\in\Omega:|\xi|<\infty\\}$ be the subspace of
$\Omega_{s}$ consisting of only those configurations with a finite number of
particles. In the following theorem we will give duality results of the
process defined in section 2.1 with duality functions
$D_{s}:\Omega_{s,f}\times\Omega_{s}\to\mathbb{R}$ of the following form:

$D_{s}(\xi,\eta)=\prod_{v\in V}d_{s}(\xi_{v},\eta_{v}).$ (2.4)

######  THEOREM 2.1.

  1. 1.

If $s=-1$ then the process generated by $\mathscr{L}_{-1}$ is self-dual with
duality function

$\displaystyle D_{-1}(\xi,\eta)$ $\displaystyle=\prod_{v\in
V}\frac{\eta_{v}!}{(\eta_{v}-\xi_{v})!}\cdot\frac{(\alpha-\xi_{v})!}{\alpha!}\cdot
I(\xi_{v}\leq\eta_{v}).$

  2. 2.

If $s=1$, then the process generated by $\mathscr{L}_{1}$ is self-dual with
duality function

$\displaystyle D_{1}(\xi,\eta)$ $\displaystyle=\prod_{v\in
V}\frac{\eta_{v}!}{(\eta_{v}-\xi_{v})!}\cdot\frac{\Gamma(\alpha)}{\Gamma(\alpha+\xi_{v})}\cdot
I(\xi_{v}\leq\eta_{v}).$

  3. 3.

If $s=0$ then the process generated by $\mathscr{L}_{0}$, with transition
rates $p(v,w)$ given by (2.3), is dual to its time-reversed process, i.e., the
process with transition rates

$\widehat{p}\big{(}(x,\sigma),(y,\sigma^{\prime})\big{)}=\kappa\pi_{\sigma}(y-x)\delta_{\sigma,\sigma^{\prime}}+\lambda\delta_{\sigma,\sigma^{\prime}}\delta_{y,x-v(\sigma)}+c(\sigma,\sigma^{\prime})\delta_{x=y}.$
(2.5)

We denote this as the $\widehat{\text{RTP}}$ process. The corresponding
duality function is given by

$D_{0}(\xi,\eta)=\prod_{v\in V}\frac{\eta_{v}!}{(\eta_{v}-\xi_{v})!}\cdot
I(\xi_{v}\leq\eta_{v}).$

The proof of the first two statements can be found in for example [3, 6]. For
the third statement, we will make use of another duality result, with the so-
called _associated deterministic system_ , that is introduced in [2].

####  2.2.2 Proof of duality for the RTP process

Let $\\{v(t):t\geq 0\\}$ be a particle in $V$ following the RTP dynamics
starting from $v(0)=v$. The deterministic system we will consider is the
following: for a function $f:V\to\mathbb{R}$, define

$f_{t}(v):=\sum_{w\in
V}p_{t}(v,w)f(w)=\mathbb{E}\left[f\big{(}v(t)\big{)}\right],$

where $p_{t}(v,w)$ is the transition kernel of a single RTP particle. In other
words, the process $\\{f_{t}:t\geq 0\\}$ follows the Kolmogorov backwards
equation of the RTP process. We now have the following duality result:

######  PROPOSITION 2.1.

For the deterministic processes $\\{f_{t}:t\geq 0\\}$ and the process
$\\{\eta(t):t\geq 0\\}$ generated by $\mathscr{L}_{0}$, it holds that

$\mathbb{E}\left[\prod_{v\in V}f(v)^{\eta_{v}(t)}\right]=\prod_{v\in
V}f_{t}(v)^{\eta_{v}},$ (2.6)

i.e., the two processes are dual to one another with duality function

$\mathscr{D}(f,\eta)=\prod_{v\in V}f(v)^{\eta_{v}}.$

In order to make sure that infinite product in (2.6) is well-defined, we will
only consider $f:V\to\mathbb{R}$ such that $f(v)\leq 1$ for all $v\in V$. The
proof of this result is straightforward and only relies on the fact that the
particles move independently.

PROOF. Define $\\{v_{i}(t):i\in I,t\geq 0\\}$ as the particles of the
configuration $\eta(t)$ with $I$ an arbitrary set of labels, i.e.,
$\eta_{v}(t)=\sum_{i\in I}I(v_{i}(t)=v)$ for all $v\in V$ and $t\geq 0$. We
then have that

$\mathbb{E}\left[\prod_{v\in
V}f(v)^{\eta_{v}(t)}\right]=\mathbb{E}\left[\prod_{i}f(v_{i}(t))\right]=\prod_{i}\mathbb{E}\left[f(v_{i}(t))\right]=\prod_{i}f_{t}(v_{i})=\prod_{v\in
V}f_{t}(v)^{\eta_{v}},$

where in the second step we used the independence of particles. $\;\qed$

We will now prove Theorem 2.1 from the duality result in (2.6).

PROOF. Let $\\{\xi(t):t\geq 0\\}$ denote an $\widehat{\text{RTP}}$ process. We
will first prove the result for the process starting from $n$ particles at
position $v_{i}\in V$, i.e., $\xi=n\cdot\delta_{v_{i}}$, with $\delta_{v_{i}}$
the configuration with a single particle at $v_{i}$. By taking the $n$-th
order derivative with respect to $f(v_{i})$ on the left-hand side of (2.6) and
afterwards setting $f\equiv 1$, we find that

$\left.\frac{\partial^{n}}{\partial
f(v_{i})^{n}}\mathbb{E}_{\eta}\left[\prod_{v\in
V}f(v)^{\eta_{v}(t)}\right]\right|_{f\equiv
1}=\left.\mathbb{E}_{\eta}\left[\frac{\partial^{n}}{\partial
f(v_{i})^{n}}\prod_{v\in V}f(v)^{\eta_{v}(t)}\right]\right|_{f\equiv
1}=\mathbb{E}_{\eta}\big{[}d_{0}(\eta_{v_{i}}(t),n)\big{]},$ (2.7)

where we were able to interchange the derivatives and the expectation using
dominated convergence.

Applying the same operations to the right hand side of (2.6) we obtain

$\displaystyle\left.\frac{\partial^{n}}{\partial f(v_{i})^{n}}\prod_{v\in
V}\left(\sum_{w\in V}p_{t}(v,w)f(w)\right)^{\eta_{v}}\right|_{f\equiv 1}$
$\displaystyle\ \ \ =\left.\sum_{m=1}^{n}\sum_{v^{(1)},...,v^{(m)}\in V\atop
v^{(i)}\neq
v^{(j)}}\sum_{k_{1}+...+k_{m}=n}\binom{n}{k_{1},...,k_{m}}\prod_{j=1}^{m}\frac{\partial^{k_{j}}}{\partial
f(v_{i})^{k_{j}}}\left(\sum_{w\in
V}p_{t}(v^{(j)},w)f(w)\right)^{\eta(v^{(j)})}\right|_{f\equiv 1}$
$\displaystyle\ \ \ =\sum_{m=1}^{n}\sum_{v^{(1)},...,v^{(m)}\in V\atop
v^{(i)}\neq
v^{(j)}}\sum_{k_{1}+...+k_{m}=n}\binom{n}{k_{1},...,k_{m}}\prod_{j=1}^{m}d_{0}\left(\eta(v^{(j)}),k_{j}\right)p_{t}\left(v^{(j)},v_{i}\right)^{k_{j}}$
$\displaystyle\ \ \ =\sum_{m=1}^{n}\sum_{v^{(1)},...,v^{(m)}\in V\atop
v^{(i)}\neq
v^{(j)}}\sum_{k_{1}+...+k_{m}=n}\binom{n}{k_{1},...,k_{m}}\prod_{j=1}^{m}d_{0}\left(\eta(v^{(j)}),k_{j}\right)\widehat{p}_{t}\left(v_{i},v^{(j)}\right)^{k_{j}},$

Here $\widehat{p}_{t}(w,v)$ is the transition kernel of a single
$\widehat{\text{RTP}}$ particle, and we have used that
$p_{t}(v,w)=\widehat{p}_{t}(w,v)$ for all $v,w\in V$. Notice that this last
line is the expected value of $D_{0}(\xi(t),\eta)$, i.e.,

$\left.\frac{\partial^{n}}{\partial f(v_{i})^{n}}\prod_{v\in
V}\left(\sum_{w\in V}p_{t}(v,w)f(w)\right)^{\eta_{v}}\right|_{f\equiv
1}=\widehat{\mathbb{E}}_{\xi}\big{[}D_{0}(\xi(t),\eta)\big{]}.$ (2.8)

Combining (2.6), (2.7) and (2.8), we indeed find that

$\mathbb{E}_{\eta}\big{[}D_{0}(\xi,\eta(t))\big{]}=\widehat{\mathbb{E}}_{\xi}\big{[}D_{0}(\xi(t),\eta)\big{]}.$

If we consider any finite configuration of particles $\xi\in\Omega_{0,f}$,
i.e., $\xi=\sum_{i=1}^{n}\delta_{v_{i}}$ for some $n\in\mathbb{N}$ and
$v_{i}\in V$, then the duality result can be found by taking the derivative
with respect to each $f(v_{i})$ on both the left-hand side and right-hand side
of the equation (2.6). $\;\qed$

####  2.2.3 Invariant measures

######  PROPOSITION 2.2.

For the processes defined in section 2.1, the following measures denoted by
$\mu_{\rho,s}$ are invariant.

  1. 1.

If $s=-1$, then $\mu_{\rho,-1}$ with $\rho\in[0,1]$ is distributed according
to a product Binomial distribution, i.e.,

$\mu_{\rho,-1}(\eta)=\prod_{v\in
V}\binom{\alpha}{\eta_{v}}\rho^{\eta_{v}}(1-\rho)^{\alpha-\eta_{v}}.$

  2. 2.

If $s=1$, then $\mu_{\rho,1}$ with $\rho\in[0,1)$ is distributed according to
a product Negative Binomial distribution, i.e.,

$\mu_{\rho,1}(\eta)=\prod_{v\in
V}\frac{\Gamma\big{(}\alpha+\eta_{v}\big{)}}{\Gamma(\alpha)\cdot\eta_{v}!}\rho^{\eta_{v}}(1-\rho)^{\alpha}.$

  3. 3.

If $s=0$, then $\mu_{\rho,0}$ with $\rho\geq 0$ is distributed according to a
product Poisson distribution, i.e.,

$\mu_{\rho,0}(\eta)=\prod_{v\in
V}\cfrac{\rho^{\eta_{v}}}{\eta_{v}!}e^{-\rho}.$

The following proposition provides the relation between the measures of
Proposition 2.2 and the duality functions of Theorem 2.1. From this relation,
the invariance of the measures follows from the conservation of particles.

######  PROPOSITION 2.3.

Let $\mu\in\mathscr{P}(\Omega_{s})$, then $\mu=\mu_{\rho,s}$ if and only if
for every $\xi\in\Omega_{f}$ and an arbitrary $v\in V$,

$\int D_{s}(\xi,\eta)d\mu(\eta)=\left(\int
D_{s}(\delta_{v},\eta)d\mu(\eta)\right)^{|\xi|}.$ (2.9)

PROOF. A straightforward calculation shows that (2.9) holds for
$\mu_{\rho,s}$. The uniqueness property follows from the fact that
$D_{s}(\xi,\eta)$ is a (multivariate) polynomial of order at most $|\xi|$.
This implies that (2.9) is actually a moment problem, which in the case of
$\mu_{\rho,s}$ has a unique solution. $\;\qed$

##  3 Ergodic theory of particle systems with homogeneous factorized duality
polynomials

In this section we provide a characterization of the ergodic invariant
measures satisfying a certain moment growth condition in a general setting
where we assume the existence of homogenous factorized duality polynomials,
and the existence of successful coupling for the dual process. This
generalizes earlier results from [11] chapter 8 for the symmetric exclusion
process, and [9] for the inclusion process. The characterization will be
applied in Section 4 to our models (see Theorem 4.1 below).

###  3.1 Basic assumptions

####  3.1.1 Configurations

We consider a configuration process $\\{\eta(t):t\geq 0\\}$ on (a subset of)
the state space $\Omega=\mathbb{N}^{G}$, where $G$ is assumed to be an
infinite countable set. We denote by $S(t)$ the semigroup of this process,
i.e., $S(t)f(\eta)=\mathbb{E}_{\eta}f(\eta(t))$. We further denote by
$\Omega_{f}$ the set of finite configurations, i.e., elements of
$\xi\in\Omega$ such that $|\xi|:=\sum_{x}\xi_{x}<\infty$.

####  3.1.2 Factorized duality functions

We assume that there exists a duality function

$D:\Omega_{f}\times\Omega\to\mathbb{R}_{+}$ (3.1)

such that we have the duality relation

$\mathbb{E}_{\eta}D(\xi,\eta(t))=\widehat{\mathbb{E}}_{\xi}D(\xi(t),\eta).$
(3.2)

We assume that $D(\emptyset,\eta)=1$ where $\emptyset$ denotes the empty
configuration. Moreover we assume that the duality functions are in
homogeneous factorized form, i.e., of the form

$D(\xi,\eta)=\prod_{x\in G}d(\xi_{x},\eta_{x}),$ (3.3)

where $d(0,n)=1$, i.e., in the product only a finite number of factors is
different from one, and where $d(k,\cdot)$ is a non-negative polynomial of
degree $k$. Moreover, we assume that every polynomial $p(n)$ of degree $k$ can
be expressed as a linear combination of the polynomials $d(r,n)$ with $0\leq
r\leq k$.

We assume that both in the process $\\{\eta(t):t\geq 0\\}$ and in the dual
process $\\{\xi(t):t\geq 0\\}$ the number of particles is conserved.

In the examples of this paper, the duality functions are multivariate
polynomials with degree at most $|\xi|$, and the dual process $\\{\xi(t):t\geq
0\\}$ is or the same process (for the interacting examples) or the process
obtained by reverting the velocities. In this section we take an abstract
point of view and prove under general assumptions a structure theorem for the
set of (tempered) invariant measures.

####  3.1.3 Tempered probability measures

Given a duality function, we define the $D$-transform of a probability measure
$\mu$ on the configuration space $\Omega$ by

$\widehat{\mu}(\xi)=\int D(\xi,\eta)d\mu(\eta),$ (3.4)

where we implicitly assume that for all $\xi\in\Omega_{f}$, $D(\xi,\cdot)$ is
$\mu$-integrable.

######  DEFINITION 3.1.

We then say that a measure $\mu$ is tempered if

  1. 1.

$\mu$ satisfies a uniform moment condition, i.e., for all $n\in\mathbb{N}$

$c_{n}:=\sup_{|\xi|\leq n}\int D(\xi,\eta)d\mu(\eta)<\infty.$ (3.5)

  2. 2.

$\mu$ is determined by its $D$-transform, i.e., if
$\widehat{\mu}=\widehat{\nu}$ then $\mu=\nu$.

  3. 3.

The vector space

${\mathcal{D}}=\text{vct}\\{D(\xi,\cdot):\xi\in\Omega_{f}\\}$

is dense in $L^{2}(\mu)$.

Notice that by the assumptions on the duality functions, the condition (3.5)
can be expressed equivalently by the requirement that all moments of the
occupation variables are finite uniformly in $x$, i.e., for all
$n\in\mathbb{N}$

$\sup_{x\in G}\int\eta_{x}^{n}d\mu(\eta)<\infty.$

Using the Hölder inequality, we then also obtain that under (3.5) we have that
for all $n\in\mathbb{N}$

$\sup_{\xi,\xi^{\prime}:|\xi|\leq n,|\xi^{\prime}|\leq n}\int
D(\xi,\eta)D(\xi^{\prime},\eta)d\mu(\eta)<\infty.$ (3.6)

The condition that the $D$-transform determines the measure uniquely is
implied by a growth condition on $c_{n}$ which implies that the measure $\mu$
is uniquely determined by its multivariate moments. In these settings, the
condition that ${\mathcal{D}}$ is dense is also natural. In the setting of
processes of exclusion type, i.e., $E=\\{1,\ldots,\alpha\\}$, the condition of
density of ${\mathcal{D}}$ is natural and follows from the Stone Weierstrass
theorem, i.e., ${\mathcal{D}}$ is uniformly dense in the set of continuous
functions ${\mathscr{C}}(\Omega)$.

####  3.1.4 Assumptions on the dual process

For the dual process $\\{\xi(t):t\geq 0\\}$, we assume that it is not positive
recurrent and irreducible on the sets
$\Omega_{n}=\\{\xi:\sum_{x}\xi_{x}=n\\}$. Moreover we assume that eventually
the process $\\{\xi(t):t\geq 0\\}$ started at $\xi$ with $|\xi|=n$, spreads
out over the infinite set $\Omega_{n}$. This is expressed via the condition
that for all $\xi^{\prime}\in\Omega_{f}$

$\lim_{t\to\infty}\widehat{\mathbb{P}}_{\xi}(\xi(t)\perp\xi^{\prime})=1,$
(3.7)

where we denote $\xi\perp\xi^{\prime}$ the event that the supports of $\xi$
and $\xi^{\prime}$ are disjoint. In words, (3.7) means that the probability
that the configuration at time $t$ has non-zero occupation at fixed sites
tends to zero as $t\to\infty$.

####  3.1.5 Ergodic measures

We denote by ${\mathscr{I}}$ the set of invariant probability measures of the
process $\\{\eta(t):t\geq 0\\}$ and by ${\mathscr{T}}$ the set of tempered
probability measures on $\Omega$. Both ${\mathscr{I}}$ and ${\mathscr{T}}$ are
convex sets.

We are the interested in characterizing the ergodic measures which belong to
${\mathscr{T}}$. We recall that we call a measure $\mu\in{\mathscr{I}}$
ergodic if $S(t)f=f$ implies $f=\int fd\mu$ almost surely. The set of ergodic
measures coincides with ${\mathscr{I}}_{e}$, the set of extreme points of
${\mathscr{I}}$. Ergodicity is implied by mixing which is the property that
for all $f,g\in L^{2}(\mu)$

$\lim_{t\to\infty}cov_{\mu}(f,S(t)g)=\lim_{t\to\infty}\int\left(f-\int
f\mathop{}\\!\mathrm{d}\mu\right)\left(S(t)g-\int
g\mathop{}\\!\mathrm{d}\mu\right)=0.$ (3.8)

By bilinearity of the covariance and the fact that, for $\mu\in{\mathscr{I}}$,
$S(t)$ is a contraction in $L^{2}(\mu)$, it suffices to show (3.8) for a set
of functions $f,g\in W$, where $W$ is such that the vectorspace spanned by $W$
is a dense subspace in $L^{2}(\mu)$.

###  3.2 Successful coupling

We say that the dual process admits a successful coupling if for all
$n\in\mathbb{N}$, $\xi,\xi^{\prime}\in\Omega_{n}$ there exists a coupling
$\\{(\xi^{(1)}(t),\xi^{(2)}(t)):t\geq 0\\}$ of the processes $\\{\xi(t):t\geq
0\\}$ starting from $\xi$ and $\xi^{\prime}$ such that the following stopping
time

$\tau_{\xi,\xi^{\prime}}=\inf\\{T>0:\xi^{(1)}(t)=\xi^{(2)}(t)\ \forall t\geq
T\\}$

is almost surely finite. We call this stopping time the coupling time. For
this paper, we will make use of the following consequence of a successful
coupling,

$\lim_{t\to\infty}\widehat{\mathbb{P}}_{\xi,\xi^{\prime}}(\xi^{(1)}(t)\not=\xi^{(2)}(t))=0,$
(3.9)

where $\widehat{\mathbb{P}}_{\xi,\xi^{\prime}}$ is the path space measure of
$\\{(\xi^{(1)}(t),\xi^{(2)}(t)):t\geq 0\\}$ starting from $\xi$ and
$\xi^{\prime}$.

###  3.3 Characterization of tempered invariant measures

The following theorem has two parts: the first parts is well-known and appears
in various context, e.g. [11] chapter 2, chapter 8. We give its proof in this
general context mainly for the sake of completeness. The second part is
inspired by [9] in the context of the inclusion process.

######  THEOREM 3.1.

  1. 1.

If there exists a succesful coupling for the dual process, then for every
tempered invariant measure $\mu$ there exists a function
$f:\mathbb{N}\to[0,\infty)$ such that for all $\xi\in\Omega_{n}$,

$\widehat{\mu}(\xi)=f(n).$ (3.10)

  2. 2.

If $\mu$ is a probability measure on $\Omega$ which is tempered, invariant and
ergodic then $f(n)=f(1)^{n}$. As a consequence, $\mu$ is a product measure.

PROOF. To prove item 1, by duality and the assumption that $\mu$ is tempered,
we can use Fubini’s theorem combined with duality to compute

$\displaystyle\widehat{\mathbb{E}}_{\xi}\widehat{\mu}(\xi(t))=\int\widehat{\mathbb{E}}_{\xi}(D(\xi(t),\eta))\mathop{}\\!\mathrm{d}\mu(\eta)=\int\mathbb{E}_{\eta}D(\xi,\eta(t))d\mu(\eta)=\int
D(\xi,\eta)d\mu(\eta)=\widehat{\mu}(\xi).$ (3.11)

Here in the last equality we used the invariance of $\mu$. We conclude that
$\widehat{\mu}$ is a harmonic function, which is bounded on each $\Omega_{n}$
by the assumption that $\mu$ is tempered.

Therefore, using the assumed existence of a successful coupling, by (3.9)
together with dominated convergence, we obtain for
$\xi,\xi^{\prime}\in\Omega_{n}$ the following

$\displaystyle\widehat{\mu}(\xi)$ $\displaystyle=$
$\displaystyle\widehat{\mathbb{E}}_{\xi,\xi^{\prime}}\widehat{\mu}(\xi^{(1)}(t))$
(3.12) $\displaystyle=$
$\displaystyle\widehat{\mathbb{E}}_{\xi,\xi^{\prime}}\widehat{\mu}(\xi^{(1)}(t))I(\xi^{(1)}(t))=\xi^{(2)}(t)))+o(1)$
$\displaystyle=$
$\displaystyle\widehat{\mathbb{E}}_{\xi,\xi^{\prime}}\widehat{\mu}(\xi^{(2)}(t))I(\xi^{(1)}(t))=\xi^{(2)}(t)))+o(1)$
$\displaystyle=$
$\displaystyle\widehat{\mathbb{E}}_{\xi^{\prime}}(\widehat{\mu}(\xi(t)))+o(1)=\widehat{\mu}(\xi^{\prime})+o(1),$

where $o(1)\to 0$ as $t\to\infty$. This gives that $\widehat{\mu}$ is constant
on $\Omega_{n}$, i.e., $\widehat{\mu}(\xi)=f(n)$ for some
$f:\mathbb{N}\to\mathbb{R}$.

To prove item 2, we start by using the ergodicity combined with duality to
write

$\displaystyle\int D(\xi,\eta)\mathop{}\\!\mathrm{d}\mu(\eta)\int
D(\xi^{\prime},\eta)\mathop{}\\!\mathrm{d}\mu(\eta)$ (3.13) $\displaystyle=$
$\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\int
D(\xi,\eta)S(t)D(\xi^{\prime},\cdot)(\eta)\mathop{}\\!\mathrm{d}\mu(\eta)dt$
$\displaystyle=$ $\displaystyle\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}\int
D(\xi,\eta)\widehat{\mathbb{E}}_{\xi^{\prime}}D(\xi(t).\cdot)(\eta)\mathop{}\\!\mathrm{d}\mu(\eta)dt,$

Here in the first step we used that
$\frac{1}{T}\int_{0}^{T}S(t)D(\xi^{\prime},\cdot)(\eta)dt\to\int
D(\xi^{\prime},\eta)d\mu(\eta)$ holds almost surely and in $L^{1}(\mu)$, and
in the second step we used duality. Now let $\xi\in\Omega_{n},\
\xi^{\prime}\in\Omega_{m}$ be given. By item 1, we have
$f(n)=\widehat{\mu}(\xi),\ f(m)=\widehat{\mu}(\xi^{\prime})$. By the
homogeneous factorization of $D$, we have for $\xi\in\Omega_{n},\
\xi^{\prime}\in\Omega_{m},\ \xi\perp\xi^{\prime}$

$D(\xi,\eta)D(\xi^{\prime},\eta)=D(\xi+\xi^{\prime},\eta),$

and therefore, if $\xi\perp\xi^{\prime}$, we have that

$\int D(\xi,\eta)D(\xi^{\prime},\eta)\mathop{}\\!\mathrm{d}\mu=f(n+m).$

Now combine (3.13) and the assumption (3.7) with the temperedness of the
measure $\mu$ to conclude

$\displaystyle f(n)f(m)$ $\displaystyle=$ $\displaystyle\int
D(\xi,\eta)\mathop{}\\!\mathrm{d}\mu(\eta)\int
D(\xi^{\prime},\eta)\mathop{}\\!\mathrm{d}\mu(\eta)$ (3.14) $\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\int
D(\xi,\eta)\widehat{\mathbb{E}}_{\xi^{\prime}}D(\xi(t),\cdot)(\eta)\mathop{}\\!\mathrm{d}\mu(\eta)+o(1)$
$\displaystyle=$ $\displaystyle\frac{1}{T}\int_{0}^{T}\int
D(\xi,\eta)\widehat{\mathbb{E}}_{\xi^{\prime}}D(\xi(t),\cdot)(\eta)I(\xi(t)\perp\xi)\mathop{}\\!\mathrm{d}\mu(\eta)+o(1)$
$\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\int\widehat{\mathbb{E}}_{\xi^{\prime}}D(\xi+\xi(t),\cdot)(\eta)I(\xi(t)\perp\xi)\mathop{}\\!\mathrm{d}\mu(\eta)+o(1)$
$\displaystyle=$
$\displaystyle\frac{1}{T}\int_{0}^{T}\mathbb{E}_{\xi^{\prime}}\left(I(\xi(t)\perp\xi)f(n+m)\right)+o(1)$
$\displaystyle=$ $\displaystyle f(n+m)+o(1),$

where $o(1)\to 0$ as $T\to\infty$ via (3.6), (3.5). This proves that for all
$\xi\in\Omega_{n},\xi^{\prime}\in\Omega_{m}$ we have

$\widehat{\mu}(\xi+\xi^{\prime})=\widehat{\mu}(\xi)\widehat{\mu}(\xi^{\prime}),$

which gives $f(n)=f(1)^{n}$. This implies that for all $x_{1},\ldots,x_{n}\in
G$, $k_{1},\ldots,k_{n}\in\mathbb{N}$,

$\int\prod_{i=1}^{n}d(k_{i},\eta_{x_{i}})\mathop{}\\!\mathrm{d}\mu(\eta)=f(1)^{k_{1}+\ldots+k_{n}}=\prod_{i=1}^{n}\int
d(k_{i},\eta)\mathop{}\\!\mathrm{d}\mu(\eta),$

which implies that $\mu$ is a product measure. $\;\qed$

In the next theorem we prove that invariant tempered product measures are
ergodic. This, combined with Theorem 3.1, completes the characterization of
the set of tempered ergodic measures.

We introduce

$K:=\left\\{\int D(\delta_{x},\eta)d\mu(\eta):\mu\ \text{is an invariant
tempered product measure}\right\\}.$

######  THEOREM 3.2.

  1. 1.

If $\mu$ is an invariant tempered product measure then it is ergodic.

  2. 2.

If there exists a successful coupling for the dual process, then the only
tempered invariant measures which are ergodic are the product measures
$\mu_{\theta}$ for which $\widehat{\mu}_{\theta}(\xi)=\theta^{|\xi|}$ with
$\theta\in K$.

  3. 3.

If there exists a successful coupling for the dual process, then

$({\mathscr{T}}\cap{\mathscr{I}})_{e}={\mathscr{T}}\cap{\mathscr{I}}_{e}=\\{\mu_{\theta}:\theta\in
K\\}.$

PROOF. For item 1, as indicated in the section where we defined mixing, it
suffices to show that

$\lim_{t\to\infty}\int
D(\xi,\eta)\mathbb{E}_{\eta}D(\xi^{\prime},\eta(t))\mathop{}\\!\mathrm{d}\mu(\eta)=\widehat{\mu}(\xi)\widehat{\mu}(\xi^{\prime})$
(3.15)

because by assumption the vectorspace spanned by the $D(\xi,\cdot)$ is dense
in $L^{2}(\mu)$.

Using duality, the assumption (3.7), the product character of the measure
$\mu$ as well as the assumed temperedness of $\mu$ (cf. (3.6)), and denoting
$o(1)$ for a term which converges to zero as $t\to\infty$, we write

$\displaystyle\int
D(\xi,\eta)\mathbb{E}_{\eta}D(\xi^{\prime},\eta(t))\mathop{}\\!\mathrm{d}\mu(\eta)$
$\displaystyle=$ $\displaystyle\int
D(\xi,\eta)\widehat{\mathbb{E}}_{\xi^{\prime}}D(\xi(t),\eta)\mathop{}\\!\mathrm{d}\mu(\eta)$
(3.16) $\displaystyle=$ $\displaystyle\widehat{\mathbb{E}}_{\xi^{\prime}}\int
D(\xi,\eta)D(\xi(t),\eta)I(\xi(t)\perp\xi)\mathop{}\\!\mathrm{d}\mu(\eta)+o(1)$
$\displaystyle=$ $\displaystyle\left(\int
D(\xi,\eta)\mathop{}\\!\mathrm{d}\mu(\eta)\int\widehat{\mathbb{E}}_{\xi^{\prime}}D(\xi(t),\eta)\mathop{}\\!\mathrm{d}\mu(\eta)\right)+o(1)$
$\displaystyle=$ $\displaystyle\int
D(\xi,\eta)\mathop{}\\!\mathrm{d}\mu(\eta)\int\mathbb{E}_{\eta}(D(\xi^{\prime},\eta(t)))\mathop{}\\!\mathrm{d}\mu(\eta)+o(1)$
$\displaystyle=$ $\displaystyle\int
D(\xi,\eta)\mathop{}\\!\mathrm{d}\mu(\eta)\int
D(\xi^{\prime},\eta)\mathop{}\\!\mathrm{d}\mu(\eta)+o(1).$

Item 2 follows immediately from item 1 and item 2 of Theorem 3.1. To prove
item 3, we only have to prove that

$({\mathscr{T}}\cap{\mathscr{I}})_{e}={\mathscr{T}}\cap{\mathscr{I}}_{e}.$

The implication “$\mu\in{\mathscr{T}}\cap{\mathscr{I}}_{e}$ implies
$\mu\in({\mathscr{T}}\cap{\mathscr{I}})_{e}$” is obvious. To prove the other
implication, start from $\mu\in({\mathscr{T}}\cap{\mathscr{I}})_{e}$ and
assume that we have

$\mu=\lambda\nu_{1}+(1-\lambda)\nu_{2},$

with $\nu_{1},\nu_{2}\in{\mathscr{I}}$ and $0<\lambda<1$. Then we have,
because $\mu\in{\mathscr{T}}$, that $\nu_{1},\nu_{2}\in{\mathscr{T}}$, and
therefore, $\nu_{1},\nu_{2}\in{\mathscr{T}}\cap{\mathscr{I}}$. But then, using
that $\mu\in({\mathscr{T}}\cap{\mathscr{I}})_{e}$ we have
$\mu=\nu_{1}=\nu_{2}$, therefore we conclude that $\mu\in{\mathscr{I}}_{e}$.
$\;\qed$

##  4 Existence of a Successful Coupling

We can now state the main result of our paper, i.e., the characterization of
the tempered ergodic measures for the three models of Section 2.1.

######  THEOREM 4.1.

For all $s\in\\{-1,0,1\\}$, the measures $\mu_{\rho,s}$ defined in Proposition
2.2 are the only tempered ergodic measures of the process generated by
$\mathscr{L}_{s}$.

By Theorem 3.2 we only need to show the existence of a successful coupling for
the dual processes of our models. To prove this result we proceed as follows:

First, we consider independent simple symmetric random walkers on $V$ where
the jump rate depends on the layer (this can be seen as the RTP with
$\lambda=0$ in (2.3)), which will be needed for the proof of our models.
Second, we deal with interacting particles, distinguishing the transient and
recurrent cases for both models. Finally, we prove the existence of a
successful coupling for a general RTP, distinguishing the cases where there
are random walk jumps and the case where there are only active jumps.

###  4.1 Successful coupling of multi-layer symmetric IRW

Let $\textbf{Y}^{(1)}(t)$ and $\textbf{Y}^{(2)}(t)$ be two configurations of
simple symmetric independent random walkers on the multi-layer state space
$\mathbb{N}^{V}$.

######  PROPOSITION 4.1.

There exists a successful coupling for
$(\textbf{\emph{Y}}^{(1)}(t),\textbf{\emph{Y}}^{(2)}(t))$.

PROOF. Since the particles move independently, by the Ornstein coupling we
only have to show that there exists a successful coupling of two independent
random walkers $(Y^{(1)}(t),\sigma^{(1)}(t))$ and
$(Y^{(2)}(t),\sigma^{(2)}(t))$.

First we define the stopping time

$\varsigma=\inf{\\{t\geq 0:\sigma^{(1)}(t)=\sigma^{(2)}(t)\\}}.$ (4.1)

After the stopping time $\varsigma$, we let the two random walkers copy each
others internal state jumps, i.e., we define the processes
$(\tilde{Y}^{(i)}(t),\tilde{\sigma}^{(i)}(t))$ for $i=1,2$ such that
$(\tilde{Y}^{(i)}(t),\tilde{\sigma}^{(i)}(t))=(Y^{(i)}(t),\sigma^{(i)}(t))$
for $t\leq\varsigma$ and $\tilde{\sigma}^{(i)}(t+\varsigma)=\tilde{\sigma}(t)$
for $t\geq 0$.

Now let $\\{e_{1},e_{2},...,e_{d}\\}$ be the standard basis vectors of
$\mathbb{Z}^{d}$, then we can write

$\tilde{Y}^{(1)}(\varsigma+t)-\tilde{Y}^{(2)}(\varsigma+t)=a_{1}(t)e_{1}+a_{2}(t)e_{2}+...+a_{d}(t)e_{d}$

Here every $a_{k}(t)$ is a simple symmetric random walk on $\mathbb{Z}$ with
(time-dependent) transition rates $\pi_{\tilde{\sigma}(t)}(e_{k})>0$. Now
define the stopping times

$\tau_{a_{k}}:=\inf\\{t\geq 0:a_{k}(t)=0\\}.$

It is clear that every $\tau_{a_{k}}$ is a.s. finite. After time
$\tau_{a_{k}}$ we let the processes
$(\tilde{Y}^{(1)}(t),\tilde{\sigma}^{(1)}(t))$ and
$(\tilde{Y}^{(2)}(t),\tilde{\sigma}^{(2)}(t))$ copy each others jumps in the
direction of $e_{k}$, i.e. $a_{k}(t)=0$ for all $t\geq\tau_{a_{k}}$. The proof
is now finished from the observation that for the coupling time $\tau$ we have
that $\tau=\max_{1\leq k\leq d}\tau_{a_{k}}$. $\;\qed$

###  4.2 Successful coupling of multi-layer SEP

Let
$\textbf{X}^{(1)}(t):=\left(X_{1}^{(1)}(t),X_{2}^{(1)}(t),...,X_{n}^{(1)}(t)\right)$
and
$\textbf{X}^{(2)}(t)=\left(X^{(2)}_{1}(t),X^{(2)}_{2}(t),...,X^{(2)}_{n}(t)\right)$
be two finite configurations of multi-layer SEP particles. We split the proof
of the successful coupling up in two parts, namely the transient case and the
recurrent case.

####  4.2.1 Transient case

Assume $d\geq 3$, then the random walk $\pi_{\sigma}(\cdot)$ is transient on
$\mathbb{Z}^{d}$ for every $\sigma\in S$. Let $\textbf{Y}(t)$ be an IRW
processes on $\Omega$ with finitely many particles. Since the transition rates
are transient, for any $R>1$ and any starting position
$\textbf{y}=(y_{1},y_{2},...,y_{n})$ such that $||y_{i}-y_{j}||_{1}>R$ for all
$i\neq j$,with positive probability $p(R)$ the particles in $\textbf{Y}(t)$
starting from y will never have _collisions_. Here a collision means that
there is a $t>0$ such that two particles $(Y_{1}(t),\sigma_{1}(t))$ and
$(Y_{2}(t),\sigma_{2}(t))$ in the configuration $\textbf{Y}(t)$ are at
neighboring positions of each other, i.e. we either have

$||Y_{1}(t)-Y_{2}(t)||_{1}=1\ \ \ \text{and}\ \ \
\sigma_{1}(t)=\sigma_{2}(t),$

or

$\hskip 21.33955ptY_{1}(t)=Y_{2}(t)\ \ \ \ \ \ \ \ \ \ \ \ \ \text{and}\ \ \
c(\sigma_{1}(t),\sigma_{2}(t))>0.$

It follows that if there are no collisions, then the multi-layer SEP particles
move the same as multi-layer IRW particles.

We now let the configurations $\textbf{X}_{1}(t)$ and $\textbf{X}_{2}(t)$ move
for some time $T>0$. We denote by $R(T)$ the minimal distance between two
particles in the same configuration at time $T$, i.e.

$R(T):=\min_{i\neq
j}\\{||X_{i}^{(1)}(t)-X_{j}^{(1)}(t)||_{1},||X^{(2)}_{i}(t)-X^{(2)}_{j}(t)||_{1}\\}$

After time $T$, we start the coupling attempt by letting the SEP particles
copy the jumps of IRW particles starting from $\textbf{X}^{(1)}(T)$ and
$\textbf{X}^{(2)}(T)$. By Proposition 4.1, this attempt is successful with
probability larger than $p(R(T))$. This proof is now finished by noting that
in the transient case, we have that $R(T)\to\infty$ as $T\to\infty$ and
$p(R)\to 1$ as $R\to\infty$.

####  4.2.2 Recurrent case

For the case of $d\leq 2$, where every $\pi_{\sigma}(\cdot)$ is recurrent, we
will define the multi-layer SEP process on the _ladder graph_ (see e.g. [6]),
i.e., we define the state space $\Omega_{-1}^{\prime}:=\\{0,1\\}^{V\times A}$
with $A=\\{1,2,...,\alpha\\}$. This space can be seen as the space where on
every site $v\in V$ there is a _ladder_ with $\alpha$ steps, and every
particle chooses a step of this ladder if it moves to a new site. We can
easily go back from a configuration $\eta^{\prime}\in\Omega_{-1}^{\prime}$ to
a configuration in $\eta\in\Omega_{-1}$ by setting

$\eta(x,\sigma)=\sum_{i=1}^{\alpha}\eta^{\prime}(x,\sigma,i),\ \ \ \ \text{for
all $(x,\sigma)\in V$}.$ (4.2)

We now define the process on $\Omega_{-1}^{\prime}$ through the generator

$\mathscr{L}_{-1}^{\prime}f(\eta^{\prime})=\sum_{j,k=1}^{\alpha}\sum_{v,w\in
V}\frac{p(v,w)}{\alpha}\eta^{\prime}_{(v,j)}\left(\alpha-\eta^{\prime}_{(w,k)}\right)\nabla_{(v,j),(w,k)}f(\eta^{\prime}),$

i.e., it is the simple symmetric exclusion process on $V\times A$ where
particles choose a step on the ladder $A$ uniformly. It is easy to see that
$\mathscr{L}_{-1}^{\prime}$ on $\Omega_{-1}^{\prime}$ corresponds to the
generator $\mathscr{L}_{-1}$ on $\Omega_{-1}$ through $\eqref{ladder}$. The
successful coupling of the multi-layer SEP now follows from the successful
coupling of the simple symmetric exclusion process on $\Omega_{-1}^{\prime}$.
Since the set $V\times A$ is countable, this result is already known, for
example in [11].

###  4.3 Successful coupling of multi-layer SIP

The successful coupling of SIP on $\mathbb{Z}^{d}$ has already been shown by
Kuoch and Redig in [9] and can be extended to our framework of multi-layer
particles. In the transient case, this proof uses the same principle as the
proof of a successful coupling for multi-layer SEP above. The proof of the
recurrent case actually uses a similar approach as in the transient case, in
which it lets the particles spread out over time and afterwards makes a
coupling attempt. The probability that this coupling attempt is successful has
non-zero probability. If the attempt fails, i.e., there is a collision, a new
coupling attempt is made. Since these coupling attempts have non-zero
probability of success and are independent, there will be a successful
coupling eventually. For more details on both proofs, see [9].

###  4.4 Successful coupling of $\widehat{\textbf{RTP}}$

For the $\widehat{\text{RTP}}$ process we will also look at two cases, namely
the case with random walk jumps of particles, i.e., $\kappa>0$ in (2.3), and
without random walk jumps. For the case of $\kappa>0$, we will see that the
successful coupling of $\widehat{\text{RTP}}$ is a corollary of Proposition
4.1 by copying the active and internal jumps of the process. If $\kappa=0$,
then we will need that the additional assumption that
$vct\\{\mathscr{R}(v)\\}=\mathbb{Z}^{d}$, with $\mathscr{R}(v)$ the range of
the velocity function $v:S\to\mathbb{Z}^{d}$. With this assumption we are able
to use a similar argument as in the proof of Proposition 4.1 to prove the
existence of a successful coupling.

#### Successful coupling of $\widehat{\text{RTP}}$ with $\kappa>0$

Let $(X^{(1)}(t),\sigma^{(1)}(t))$ and $(X^{(2)}(t),\sigma^{(2)}(t))$ be two
$\widehat{\text{RTP}}$($\kappa,\lambda$) processes in $\mathbb{Z}^{d}$. Then
we can write this as

$\begin{split}&(X^{(1)}(t),\sigma^{(1)}(t))=(Y^{(1)}(t)+Z^{(1)}(t),\sigma^{(1)}(t)),\\\
&(X^{(2)}(t),\sigma^{(2)}(t))=(Y^{(2)}(t)+Z^{(2)}(t),\sigma^{(2)}(t)).\end{split}$

$Y^{(i)}(t)$ are random walks starting from $X^{(i)}(0)$, $Z^{(i)}(t)$ are the
active jumps starting from $0$, and $\sigma^{(i)}(t)$ are the internal states.

We define the stopping time $\varsigma$ as in (4.1) and the processes
$(\tilde{X}^{(i)}(t),\tilde{\sigma}^{(i)}(t))$ for $i=1,2$ as
$(\tilde{X}^{(i)}(t),\tilde{\sigma}^{(i)}(t))=(X^{(i)}(t),\sigma^{(i)}(t))$
for $t\leq\varsigma$, and

$(\tilde{X}^{(i)}(\varsigma+t),\tilde{\sigma}^{(i)}(\varsigma+t))=(\tilde{Y}^{(i)}(t)+\tilde{Z}(t),\tilde{\sigma}(t))$

where $\tilde{Y}^{(i)}(t)$ is again a random walk starting from
$X^{(i)}(\varsigma)$, $\tilde{Z}(t)$ are again the active jumps starting from
0, and $\tilde{\sigma}(t)$ is the internal state starting from
$\sigma^{(1)}(\varsigma)=\sigma^{(2)}(\varsigma)$. Note that the difference of
the positions of the two processes is now equal to

$\begin{split}\tilde{X}^{(1)}(\varsigma+t)&-\tilde{X}^{(2)}(\varsigma+t)=\tilde{Y}^{(1)}(t)-\tilde{Y}^{(2)}(t).\\\
\end{split}$

The result now follows from Proposition 4.1.

#### Successful coupling of $\widehat{\text{RTP}}$ with $\kappa=0$ and
$\lambda>0$

If there is no random walk, to ensure that the two processes are able to find
each other we will need to make a further assumption on the set of internal
states $S$ and the function $v:S\to\mathbb{Z}^{d}$, namely that the range of
$v$ spans the whole of $\mathbb{Z}^{d}$, i.e.,

$vct\\{\mathscr{R}(v)\\}=\mathbb{Z}^{d}.$ (4.3)

Then, just as in the previous case, for two $\widehat{\text{RTP}}$ processes
$(X^{(1)}(t),\sigma^{(1)}(t))$ and $(X^{(2)}(t),\sigma^{(2)}(t))$ we define
the stopping time $\varsigma$ as in (4.1), and set up the processes
$(\tilde{X}^{(i)}(t),\tilde{\sigma}^{(i)}(t))$ for $i=1,2$ such that
$(\tilde{X}^{(i)}(t),\tilde{\sigma}^{(i)}(t))=(X^{(i)}(t),\sigma^{(i)}(t))$
for $t\leq\varsigma$ and $\sigma^{(i)}(\varsigma+t)=\tilde{\sigma}(t)$ for
$t>0$.

By 4.3 we can now write

$\tilde{X}^{(1)}(\varsigma+t)-\tilde{X}^{(2)}(\varsigma+t)=b_{1}(t)v(\sigma_{1})+b_{2}(t)v(\sigma_{2})+...+b_{m}(t)v(\sigma_{m})$

for $m=|S|$, $\sigma_{k}\in S$ and $b_{k}(t)\in\mathbb{Z}$ for all $k$ and
$t\geq 0$. For every $k$, the couple $(b_{k}(t),\tilde{\sigma}(t))$ is a
random walk on $\mathbb{Z}\times S$ with the following dynamics:

  * -

If $\tilde{\sigma}(t)=\sigma_{k}$, $b_{k}(t)$ is a simple symmetric random
walker on $\mathbb{Z}$ with rate $\lambda$.

  * -

If $\tilde{\sigma}(t)\neq\sigma_{k}$, $b_{k}(t)$ is immobile.

Since $S$ is finite and the transition rates $c(\sigma,\sigma^{\prime})$ are
irreducible, these random walks are recurrent as their discrete counterparts
are recurrent. This implies that the stopping times

$\tau_{b_{k}}:=\inf\\{t\geq 0:b_{k}(t)=0\\}$ (4.4)

are almost surely finite. After every time $\tau_{b_{k}}$, we let the process
$\tilde{X}^{(2)}(\varsigma+t)$ copy the jumps of
$\tilde{X}^{(1)}(\varsigma+t)$ in the direction of $v(\sigma_{k})$. For the
coupling time $\tau$, we then again have that $\tau=\max_{1\leq k\leq
m}\tau_{b_{k}}$.

######  REMARK 4.1.

We are able to extend the results of this section to the case where we take
$S$ countable. We would need the additional assumption that the transition
rates $c(\sigma,\sigma^{\prime})$ are positive recurrent, which ensures that
we return to any $\sigma\in S$ in almost surely finite time. For the
interacting particles we would then distinguish between the cases where the
transition rates $p(v,w)$ in (2.1) are transient and recurrent (note that the
latter need not be the case where $d\leq 2$). For the $\widehat{\text{RTP}}$,
the positive recurrence of $c(\sigma,\sigma^{\prime})$ ensures that every
stopping time $\tau_{b_{k}}$ in (4.4) is almost surely finite.

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