# Feedback Loops in Opinion Dynamics of Agent-Based Models with Multiplicative
Noise

Nataša Djurdjevac<EMAIL_ADDRESS>Zuse Institute Berlin, Berlin,
Germany  Jonas Köppl  Zuse Institute Berlin, Berlin, Germany  Weierstrass
Institute for Applied Analysis and Stochastics Berlin, Germany  Ana Djurdjevac
Freie Universität Berlin, Institut für Mathematik und Informatik, Berlin,
Germany

###### Abstract

We introduce an agent-based model for co-evolving opinion and social dynamics,
under the influence of multiplicative noise. In this model, every agent is
characterized by a position in a social space and a continuous opinion state
variable. Agents’ movements are governed by positions and opinions of other
agents and similarly, the opinion dynamics is influenced by agents’ spatial
proximity and their opinion similarity. Using numerical simulations and formal
analysis, we study this feedback loop between opinion dynamics and mobility of
agents in a social space. We investigate the behavior of this ABM in different
regimes and explore the influence of various factors on appearance of emerging
phenomena such as group formation and opinion consensus. We study the
empirical distribution and in the limit of infinite number of agents we derive
a corresponding reduced model given by a partial differential equation (PDE).
Finally, using numerical examples we show that a resulting PDE model is a good
approximation of the original ABM.

Keywords: opinion dynamics, feedback loop, agent-based modeling,
multiplicative noise, group formation, mean-field limit, empirical
distribution, stochastic partial differential equations.

##  1 Introduction

Opinion dynamics is one of the most important processes of our society, as our
opinions influence not only our individual actions and behavior; but can also
shape collective dynamics governing societal change and social movement.
Complex interaction patterns between individuals and coupled social mechanisms
in different environments are shown to be the crucial drivers of opinion
dynamics [1]. With the introduction of online social media, the way people
interact and share their opinions has drastically changed. For example,
physical proximity is now not anymore a constrain for communication, everybody
can engage in information transmission and express their opinions to a large
number of people in different social, political and cultural environment.
Additionally, large amounts of data became available about how people
influence and are getting influenced in their opinions [2], which provided new
insights into the social mechanisms and emerging phenomena such as formation
of echo chambers and opinion consensus.

During the last decades, extensive research has been carried out in order to
understand how people shape their opinions in their social space, see recent
reviews [3, 4]. Governed by an increasing amount of available large-scale
social data and fast computational advances, the topic of opinion dynamics
gathered an interdisciplinary research community [3, 5]. Existing work ranges
from the studies on (1) model-driven approaches that produce formal models for
opinion dynamics that can be analysed using theories from mathematics and
statistical physics to (2) data-driven approaches that are used to explore
empirical data using knowledge from social sciences. Using computer
simulations and computational analysis, opinion dynamics models can be used as
a tool for understanding social mechanisms, uncovering social interaction
patterns and exploring influences of various factors on e.g. group formation
and opinion consensus [6, 7, 3, 8]. Furthermore, mathematical description of
models is a starting point that enables the use of analytical tools. In such a
way we can get theoretical predictions of the models, such as: long time
behaviour, limiting behaviour of the system when the number of agents tends to
infinity (macroscale) and description of the fluctuations in the case when the
number of agents is very big but still finite (mesoscale). In addition, this
type of analysis is a basis for developing rigorous numerical analysis which
implies error estimates that should be expected by numerical computations and
which determine the choice of parameters in the model that should be used in
the experiments. However, most existing models are rather simple and rarely
connect to empirical studies and available real-world data [9]. In order to
close this gap between model- and data-driven approaches, new formal models
need to be introduced that can better represent real-world social systems and
capture complex mechanisms that govern how people shape their opinions.

The largest group of formal models for studying opinion dynamics are agent-
based-models (ABMs), where a process of opinion formation takes place through
interactions between individual agents. One example of such ABMs are Voter
models [10], that describe opinion change between agents with discrete opinion
states, where agents interaction dynamics is defined through an underlying
social network. In the DeGroot model [11], agents have continuous-valued
opinions that are formed as the average opinions of all other agents. Further
mathematical literature mostly focuses on bounded confidence models [12],
where the dynamics of opinions depends only on interactions among agents that
have similar opinions, without assuming an underlying interaction network, but
rather all-to-all possible interactions. Different extensions of these models
have been considered in order to account for more realistic scenarios. For
example, introducing stochastic effects [13, 14, 15, 16, 6, 17], complex
interaction mechanisms [18, 19] and different types of agents (e.g. stubborn
agents, influencers, campaigners) [20, 21]. Additionally, in the context of
complex social networks, the co-evolution of the opinion and the network
dynamics has been studied, where the changes in the network structure
influence the opinion dynamics and vice versa. It has been shown that this co-
evolution process in network models governs the appearance of emerging
structures, e.g. echo chambers [22]. However, most of the existing ABMs for
opinion dynamics do not include the dynamics of agents in a social space,
despite that this process determines agents’ interaction patterns. Extending
on the models for epidemic spreading [23] and cultural dissemination [24, 25],
recently the so-called mobile agents [26, 7] were introduced to account for
the feedback between the spatial movement of agents and the social contagion
dynamics. While simulation results on co-evolving dynamics (both for network
models and ABMs) provided many useful insights into the social mechanisms
behind opinion formation, theoretical considerations of such models are still
largely missing.
In this article, we introduce a mathematical ABM, that includes feedback loops
in opinion dynamics of agents moving in a social space, influencing and being
influenced in their opinions. The focus of this manuscript is on the non-
trivial two-way interaction between the agents’ movement in a social space and
their opinion dynamics. More precisely, agents’ spatial movements can induce
changes in opinion states over time and additionally, opinion dynamics can
influence the spatial position of agents and opinion states of agents in their
vicinity. This feedback loop between spatial and opinion changes is at the
core of the systems co-evolving dynamics. We consider that both spatial and
opinion dynamics are governed by stochastic dynamics with multiplicative
noise, which generalizes the case of constant, additive noise that is usually
considered [14, 15, 16, 6, 17]. We explore the impact the feedback has on the
behavior of the system and in particular on the grouping of agents in opinion
and/or social space. From the mathematical point of view, our ABM can be seen
as an interacting particle system. In particular, on the microscopic level
(level of agents) we formulate our model as a system of coupled stochastic
differential equations (SDEs) with multiplicative noise. From the application
point of view, the interest is to have weak regularity assumptions on the
drift and diffusion coefficients. Next, we study the corresponding limiting
equation in the case when the number of agents tends to infinity, i.e. the so
called McKean-Vlasov equation. Furthermore, we show the well-posedness results
for the limiting system, which is a very popular and challenging topic in the
field of stochastic analysis. Nowadays, there is extensive literature about
these results, the standard results are [27, 28, 29] and see the references
therein. Another interesting class of results that is discussed in the
literature and that we also consider, is the so-called propagation of chaos,
meaning that one wants to prove the convergence of the microscopic model to
McKean-Vlasov SDEs. The challenge in these proofs lies in the weak assumptions
on the coefficients. Since the topic of this article is not the theoretical
investigation of the weak regularity assumption, in order to illustrate our
message we concentrate on the simple case of Lipschitz bounded coefficients
with multiplicative noise case. Furthermore, due to the high computational
cost of ABM simulations when the number of agents is large, we suggest the
standard model reduction approach that considers instead the empirical density
rather than each agent individually. We derive the formal equations of the
empirical density and its so called hydrodynamic limit. These results are in
the spirit of the standard, so called Dean-Kawasaki equation [30]. In the
setting of the social dynamics, this model reduction for the uncoupled system
has already been considered in [31]. Using a numerical example, we illustrate
the expected behaviour of the system on the macroscopic scale, that is given
by the partial differential equation.

The article is organized as follows. Our agent-based model for opinion
dynamics with feedback loops is introduced and studied through numerical
simulations in Section 2. Next, we develop a theoretical framework for
studying the system at the macroscopic level by an mean-field approach in
Section 3 and we present the well-posedness result of the McKean-Vlasov SDE
system with Lipschitz coefficients and the convergence results of propagation
of chaos. In Section 4 we present formal derivation of the equation that
describes the dynamics of the empirical measure and its hydrodyanamic limit.
We illustrate the limiting behaviour of the system on the macroscopic scale,
by a numerical example. Finally, we derive our conclusions and possible future
directions in Section 5.

##  2 Model Description

We consider a closed system of $N$ interacting agents and agents’ co-evolving
opinion and social dynamics. At time $t\in[0,T]$, every agent $k,\
k=1,\dots,N$ has a position state $X^{k}_{t}\in\mathbb{R}^{d}$ and an opinion
state $\Theta^{k}_{t}\in\mathbb{R}$. The position state $X^{k}_{t}$ of an
agent is a point in an abstract social space, such that a distance between two
agents refers to their social proximity that is described by their social
similarity. In real-world social systems, information about a position in
social space may be inferred from e.g. online social media. The opinion
$\Theta^{k}_{t}$ of an agent $k$ is considered to be a continuous variable.
For more generality, this model can be extended to incorporate several opinion
entities, such that $\Theta^{k}_{t}\in\mathbb{R}^{m}$. However, for technical
simplicity in this paper we will assume that $m=1$. The state of the system at
time $t\geq 0$ for the set of $N$ agents is given by

$Z_{t}=(X_{t},\Theta_{t})\in(\mathbb{R}^{d})^{N}\times\mathbb{R}^{N},$

where the $k$-th row of the systems’ state corresponds to the state
$Z^{k}_{t}=(X^{k}_{t},\Theta^{k}_{t})$ of the $k-$th agent. All agents follow
the same rules that describe how their positions and opinions change. More
precisely, agents move in a social space governed by the position of other
agents and their opinions. Similarly, the opinion states of agents are
influenced by both agents’ spatial proximity and their opinions. This feedback
loop between spatial and opinion changes determines the systems adaptive
dynamics. Additionally, to be able to account for external influences on
agents and the sometimes seemingly random nature of human interactions, we
model this system via a coupled system of stochastic differential equations
with multiplicative noise of the form

$\begin{cases}dX_{t}&=\tilde{U}(X_{t},\Theta_{t})dt+\sigma_{sp}(X_{t},\Theta_{t})dB^{sp}_{t},\\\
\
d\Theta_{t}&=\tilde{V}(X_{t},\Theta_{t})dt+\sigma_{op}(X_{t},\Theta_{t})dB^{op}_{t},\end{cases}$
(1)

where:

  * •

$\tilde{U}:(R^{d})^{N}\times\mathbb{R}^{N}\to(\mathbb{R}^{d})^{N}$ is a
spatial interaction map that models how the positions and opinions of the
agents influence the spatial movement of the agents,

  * •

$\tilde{V}:(R^{d})^{N}\times\mathbb{R}^{N}\to(\mathbb{R}^{d})^{N}$ is an
opinion interaction map that models how the positions and opinions of the
agents influence the opinion states of the agents,

  * •

$B^{sp}$ and $B^{op}$ are independent Brownian motions starting in $0$,

  * •

$\sigma_{sp}(X_{t},\Theta_{t}),\sigma_{op}(X_{t},\Theta_{t})$ are diffusion
coefficients for spatial and opinion dynamics respectively.

Dynamics given by the SDEs in (1) is rather abstract and in this form it
doesn’t provide much intuition on how it can be adapted to known social
mechanisms coming from real-world systems. Thus, in the following we will
focus on how complex interaction patterns and stochastic influences can be
enforced in our model.

###  2.1 Pairwise interactions

We start by exploring the simplest type of interaction dynamics, namely
pairwise (or 2-body) interactions that only take into account interactions
between pairs of agents. More precisely, we consider the case where the
interaction maps $\tilde{U},\tilde{V}$ are linear functions of simpler
interaction maps $U,V$, i.e. we define
$\tilde{U}=(\tilde{U}_{1},...,\tilde{U}_{N}):(R^{d})^{N}\times\mathbb{R}^{N}\to(\mathbb{R}^{d})^{N}$
with

$\displaystyle\tilde{U}_{k}(X,\Theta)=\frac{1}{N}\sum_{j=1}^{N}U(X^{k},X^{j},\Theta^{k},\Theta^{j}),$
(2)

for some pair-interaction map
$U:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}^{d}$
and analogously for $\tilde{V}$. In this model, agents shape their opinions
based on the mean opinion of other agents through pairwise interactions, which
is the setting of many of the classical models for opinion dynamics [11, 32,
12]. Thus, the dynamics of the $k$-th agent are given by

$\displaystyle\begin{cases}dX^{k}_{t}=\frac{1}{N}\sum_{j=1}^{N}U(X^{k}_{t},X^{j}_{t},\Theta^{k}_{t},\Theta^{j}_{t})dt+\sigma_{sp}(X_{t},\Theta_{t})dB^{sp,k}_{t},\\\
\
d\Theta^{k}_{t}=\frac{1}{N}\sum_{j=1}^{N}V(X^{k}_{t},X^{j}_{t},\Theta^{k}_{t},\Theta^{j}_{t})dt+\sigma_{op}(X_{t},\Theta_{t})dB^{op,k}_{t}.\end{cases}$
(3)

Note that in order to simplify the notation, we do not write $X_{t}^{k,N}$,
$\Theta_{t}^{k,N}$, but instead $X^{k}_{t}$, $\Theta_{t}^{k}$ respectively. As
a more concrete example one can consider the following model that can be seen
as an extension of a classical model by DeGroot [11], in a sense that it does
not assume interactions between all agents, but instead assumes that two
agents can only interact if their positions in a social space are closer than
a certain interaction radius $R$, like in the bounded confidence models. The
reason for such modeling decision is that agents that are further away in a
social space, i.e. are having a low social similarity, may have conflicting
attitudes and social norms, and thus lack motivation to interact with each
other. An opinion interaction map that models this idea is

$\displaystyle V(x_{1},x_{2},\theta_{1},\theta_{2}):=\alpha\cdot
1_{[0,R]}\left(\left\lVert
x_{1}-x_{2}\right\rVert\right)\cdot(\theta_{2}-\theta_{1}),$ (4)

where $\alpha$ is the opinion strength parameter and
$\left\lVert\cdot\right\rVert$ refers to the Euclidean distance. In our model,
$\alpha$ regulates the strength of the social influence on the agent’s
opinion, i.e. the higher the $\alpha$ the more influence the pairwise opinion
difference has on an agent’s opinion. As given by (4), this model formalizes
the idea that two agents can only interact if their positions in a social
space are close enough and if they interact, then their opinions become more
similar. Since we do not distinguish between individual agent types, we assume
that $\alpha$ is a constant, i.e. it is equal for all agents in the system. In
the literature there are different variations of this classical interaction
dynamics [33, 19]. As discussed above, our model extends these by introducing
the movements of agents in a social space and its feedback loop with the
opinion dynamics. An example of such dynamics is given by the spatial
interaction map

$\displaystyle U(x_{1},x_{2},\theta_{1},\theta_{2}):=\beta\cdot
1_{[0,R]}\left(\left\lVert
x_{1}-x_{2}\right\rVert\right)\cdot\text{sgn}(\theta_{1}\cdot\theta_{2})\cdot(x_{2}-x_{1}),$
(5)

where $\beta$ denotes the spatial strength parameter regulating how much
agents compromise when updating their positions. The third term in (5)
introduces a direction of agents’ spatial motion based on their opinions, i.e.
agents can either attract or repel each other, depending on whether their
opinions are similar or different to each other. Thus, movement of an agent in
a social space is determined by its spatial closeness and opinion similarity
with other agents in the system.

###  2.2 Multi-body interactions

Most existing models of opinion dynamics only take pairwise interactions into
account, ignoring any higher order interaction dynamics. While this simplifies
mathematical considerations significantly, this is a very rough approximation
of real life interactions and does not allow to model group effects such as
peer pressure, see e.g. [34, 35]. Recently there has been an increasing
interest in going beyond pairwise interactions to create more realistic models
for social dynamics, e.g. [36]. Our model can be extended with such effects by
including the spatial interaction dynamics as introduced in the previous
section by (5) and considering multi-body interactions for the opinion
dynamics proposed in [36]. In particular, the effect of peer pressure within
groups of agents in spatial proximity can be modeled by the opinion
interaction map

$\displaystyle
V(x_{1},x_{2},x_{3},\theta_{1},\theta_{2},\theta_{3}):=\alpha\cdot\left(\prod_{i,j=1}^{3}1_{[0,R]}(\left\lVert
x_{i}-x_{j}\right\rVert)\right)s(\left\lvert\theta_{2}-\theta_{3}\right\rvert)[(x_{2}-x_{1})+(x_{3}-x_{1})],$
(6)

where $s:[0,\infty)\to\mathbb{R}$ is a non-increasing positive-definite
function, e.g. $s(x)=\exp(\lambda x)$, for some $\lambda<0$. This means, that
the influence of agents $2$ and $3$ on the opinion of agent $1$ is stronger if
they have similar opinions. Different from the classical DeGroot model, in a
case when multi-body interactions are defined on hypergraphs, shifts in the
average opinion of the system have been observed [36]. Similar results are to
be expected in our model and will be the topic of our future work. Here, in
order to keep our analytical results more trackable, we will focus on the case
of pairwise interactions.

###  2.3 Stochastic influence: Multiplicative noise

Many of the classical models of opinion dynamics are of a deterministic nature
[11, 12, 32]. Historically, the first stochastic versions of these models have
always used additive noise, i.e. stochastic noise with constant (in time,
status and over the population) strength [37, 38]. Having a constant noise
coefficient also implies that the driving noise of agents $i$ and $j$ are
independent for all $i,j\in\left\\{1,\dots,N\right\\}$. This makes the model
mathematically easier, but such a modelling choice is questionable from a
social sciences perspective. The usual role of the noise is to account for
external influences, that are not already incorporated into the model through
agents interactions, such as randomly occurring environmental changes or some
other significant events. However, these events influence all (or at least
most) agents, introducing non-trivial correlation between the individual
noises. The choice of additive noise also doesn’t account for more complex
social mechanisms observed in real world systems [7]. Namely, the strength of
the noise can additionally depend on how homogeneous the opinions of agent’s
peers are. In a highly polarised group of individuals, one might be more
susceptible to random influences than in a group where everyone has roughly
the same opinion.
Recently, a few models considered such complex mechanisms by introducing a
multiplicative noise, e.g. in the context of opinion dynamics [7, 39], animal
movement [40] and flocking of a Cucker-Smale system [41].

Extending on the ideas from [7], one concrete example of a multiplicative
noise for our model is defined by

$\displaystyle\sigma_{op}^{i}(X_{t},\Theta_{t})=\min_{j:\left\lVert
X_{t}^{i}-X_{t}^{j}\right\rVert\leq
R}\left\lvert\Theta_{t}^{i}-\Theta_{t}^{j}\right\rvert\cdot\text{Id},i=1,\ldots,N.$
(7)

Thus, the noise in this model is characterized by a similarity bias, i.e. the
closer the opinions of agents’ peers are to the opinion of that agent, the
smaller the random fluctuations. Similarly, agents have a higher probability
to move away from agents with very different opinions. These effects have been
shown to yield fast formation of stable spatial clusters in which a local
consensus is reached [7].

###  2.4 Numerical simulations of the ABM

In this section, we show the main properties of our proposed model for opinion
dynamics with feedback loops given by (3), (4) and (5). Using numerical
simulations, we study how different parameters influence grouping of agents
into social and opinion clusters. Additionally, we demonstrate a difference
between additive and multiplicative noise and how this affects the stability
of clusters and opinion distribution within clusters.

(a) $\sigma=0.01$

(b) $\sigma=0.05$

(c) $\sigma=0.15$

Figure 1: Snapshots from numerical simulations at final time $T=2.5$ for
different influences of additive noise: (a) $\sigma=0.01$, (b) $\sigma=0.05$
and (c) $\sigma=0.15$. Position of agents indicated their position in a social
space. Color of agents denotes their opinions according to the color-bar.
Other parameters are fixed to $R=0.015$ and $\alpha=\beta=20$.

(a) $\sigma=0.01$

(b) $\sigma=0.05$

(c) $\sigma=0.15$

Figure 2: Opinion trajectories of agents’ over time-period $[0,2.5]$ for
different influences of additive noise: (a) $\sigma=0.01$, (b) $\sigma=0.05$
and (c) $\sigma=0.15$. Other parameters are fixed to $R=0.015$ and
$\alpha=\beta=20$.

(a) $\sigma=0.01$

(b) $\sigma=0.05$

(c) $\sigma=0.15$

Figure 3: Distribution of agents’ opinions at final time $T=2.5$ for different
influences of additive noise: (a) $\sigma=0.01$, (b) $\sigma=0.05$ and (c)
$\sigma=0.15$. Other parameters are fixed to $R=0.15$ and $\alpha=\beta=20$.

We run stochastic simulations for $N=100$ agents, where at $t=0$ all agents
are placed uniformly at random inside $[-0.25,0.25]^{2}$ and their initial
opinions are distributed uniformly in the interval $[-1,1]$. We run each
simulation for $250$ time-steps, i.e. until $T=2.5$ where $\Delta t=0.01$. In
every time-step, movement of agents and their opinions are obtained using a
Euler–Maruyama scheme [42, 43].
First, we consider the case of additive noise, where we assume that
$\sigma:=\sigma_{op}=\sigma_{sp}$ and we distinguish between different noise
strengths, namely $\sigma=0.01$, $\sigma=0.05$ and $\sigma=0.15$. We fix the
interaction radius $R=0.15$ and include strong opinion influence $\alpha=20$
and strong spatial influence $\beta=20$. In Figure 1 we plot simulation
snapshots at the final time $T=2.5$ for these different values of $\sigma$,
where agents’ positions correspond to their position in a social space and
agents’ colours indicate their opinion. We observe a strong influence of noise
on the cluster formation process. Namely, the higher the $\sigma$, the denser
the inter-cluster connections are, such that for $\sigma=0.15$ there is no
clear separation into different clusters neither in social nor in opinion
space. This can be seen in Figure 2 of opinion trajectories of individual
agents. Additionally, stronger noise in the system leads to more diversified
opinion distributions when reaching consensus, as can be observed in the final
distribution of agents’ opinions in Figure 3. For small values of noise, i.e.
$\sigma=0.01$, the system reaches a stable state and we see several spatially
separated clusters of agents that are stable and within each cluster agents
have similar opinions. This behavior has been observed in previous studies of
network models [22, 44].

(a) $\alpha=\beta=50$

(b) $\alpha=\beta=5$

(c) $\alpha=50,\beta=5$

Figure 4: Snapshots from numerical simulations at final time $T=2.5$ for
different influences of opinion and spatial strength: (a) $\alpha=\beta=50$,
(b) $\alpha=\beta=5$, (c) $\alpha=50,\beta=5$. Position of agents indicated
their position in a social space. Color of agents denotes their opinions
according to the color-bar. Other parameters are fixed to $R=0.15$ and
$\sigma=0.05$.

(a) $\alpha=50$

(b) $\alpha=5$

(c) $\alpha=50,\beta=5$

Figure 5: Distribution of agents’ opinions at final time $T=2.5$ for different
influences of opinion and spatial strength: (a) $\alpha=\beta=50$, (b)
$\alpha=\beta=5$. Position of agents indicated their position in a social
space. Color of agents denotes their opinions according to the color-bar.
Other parameters are fixed to $R=0.15$ and $\sigma=0.05$.

Next, we study the influence of opinion strength $\alpha$ and spatial strength
$\beta$ on the cluster formation process. Additional to results shown in
Figure1(b) for $\alpha=\beta=20$, in Figure 4 we show the snapshots after
$250$ time-steps for $\alpha=\beta=50$, $\alpha=\beta=5$ and
$\alpha=50,\beta=5$. Large values of $\alpha,\beta$ mean that agents are
strongly influenced by their peers, such that they compromise heavily towards
their neighbours when updating their opinions and positions. This leads to
faster formation of stable clusters where within the clusters agents have
similar opinions, see Figure 4 and Figure 5. Because of this effect, in
standard bounded confidence models, these parameters are often called
‘convergence parameters’ as they affect the speed of convergence [45].
Comparing the three scenarios in this experiment and in particular the cases
when $\alpha=\beta$ to $\alpha\neq\beta$, we see that the spatial strength is
important for formation of spatial clusters, but also in the case when spatial
strength is small $\beta=5$ and opinion strength is large $\alpha=50$, because
of the feedback loop agents tend to form large, loose groups. This effect
emphasises the importance of the feedback loop in this system.

Figure 6: Results of one simulation of the ABM with multiplicative noise, for
$\alpha=\beta=20$ and $R=0.15$. Left: Snapshot of the dynamics at $T=2.5$.
Middle: Opinion trajectories during $[0,2.5]$. Right: Distribution of agents’
opinions at $T=2.5$. Position of agents indicated their position in a social
space. Color of agents denotes their opinions according to the color-bar.
Other parameters are fixed to $R=0.015$ and $\sigma=0.05$.

Finally, we examine the effects of multiplicative noise on the dynamics of our
ABM given by (3), (4) and (5). To this end, we consider multiplicative noise
$\sigma_{op}=\sigma_{sp}$, as introduced in (7). In Figure 6 we see a fast
formation of $8$ spatially well separated clusters. In each of these clusters
local consensus is reached, as discussed in Section 2.3. These clusters are
stable in their spatial position and opinion distribution. In comparison to
the case of additive noise, see Figure 1, clusters obtained through the
influence of multiplicative noise are constructed faster and are more stable.

##  3 Theoretical analysis: Coupled mean-field limit

In this section we consider the theoretical setting that describes the
feedback loop dynamics of the system and its limit when $N\to\infty$. In
particular, we start by briefly motivating the mean-field equation, and then
we state its well-posedness and the convergence of ABM to this mean-field
equation.

From the theoretical point of view, one of the main challenges of these
results lies in the regularity assumptions of drift and diffusion
coefficients. The standard result is considering Lipschitz coefficients and
additive noise, as presented in [27]. On the one hand, one can consider the
dynamics just for one agent with singular interactions and additive noise,
such as in [29] and its extensions to multiplicative noise case as in [46].
However, as noted in [47], these results can not be trivially extended for $N$
particle systems. Thus, different techniques need to be used to obtain results
for the singular interactions, such as mixed $L^{p}$ drifts. The literature on
the weaker assumptions on drift and diffusion than the standard global
Lipschitz assumption is tremendous as it is a very popular topic, see [48, 49,
50, 51, 52].

However, in order to illustrate the main message of this article that is the
consideration of the feedback loop dynamics with the multiplicative noise, it
is enough to consider the standard Lipschitz setting. Since the case of the
multiplicative noise in this setting can not be found in an easily accessible
form, for the completeness of results, we state in the appendix the proofs of
the well-posedness and convergence results, without relying on the
sophisticated analysis results. This result can be seen as an extension of the
standard result in [27].

In order to simplify the notation, we will write $\int$ for
$\int_{\mathbb{R}^{d}\times\mathbb{R}}=\int_{\mathbb{R}^{d+1}}$ and we define
$\mathcal{C}:=C([0,T];\mathbb{R}^{d+1})$. Hence, for the rest of the article,
we will assume that potentials $U$ and $V$ are real valued Lipschitz functions
on $\mathbb{R}^{2d}\times\mathbb{R}^{2}$. More precisely, we will always
assume the following on the regularity of initial data:

###### Assumption 1

Let $(B^{sp},B^{op})$ be a Brownian motion in $\mathbb{R}^{d}\times\mathbb{R}$
with respect to a filtration $\mathcal{F}=(\mathcal{F}_{t})_{t\geq 0}$, the
initial conditions $X_{0}$ and $\Theta_{0}$ are $\mathcal{F}_{0}$-measurable
and square-integrable, and the maps

$\displaystyle
U:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}^{d},\quad
V:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R},$
(8)
$\displaystyle\sigma_{sp}:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}^{d\times
d},\quad\sigma_{op}:\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$
(9)

are bounded and Lipschitz continuous.

###  3.1 Motivation for the limiting equations

In this section we will present a general motivation on how one can derive the
so-called mean-field limit of coupled SDEs on $[0,T]$ for a fixed number
$N\in\mathbb{N}$ of interacting agents which is a special case of our ABM (3).
The mean-field limit is the coupled system of SDEs that describes the averaged
dynamics of the system when the number of agents tends to infinity.

In particular, from now on the coupled SDE system that we will analyse is
given by

$\displaystyle\begin{cases}dX^{k,N}_{t}&=\frac{1}{N}\sum_{j=1}^{N}U(X^{k}_{t},X^{j}_{t},\Theta^{k}_{t},\Theta^{j}_{t})dt+\frac{1}{N}\sum_{j=1}^{N}\sigma_{sp}(X^{k}_{t},X^{j}_{t},\Theta^{k}_{t},\Theta^{j}_{t})dB^{sp,k}_{t},\\\
\
d\Theta^{k,N}_{t}&=\frac{1}{N}\sum_{j=1}^{N}V(X^{k}_{t},X^{j}_{t},\Theta^{k}_{t},\Theta^{j}_{t})dt+\frac{1}{N}\sum_{j=1}^{N}\sigma_{op}(X^{k}_{t},X^{j}_{t},\Theta^{k}_{t},\Theta^{j}_{t})dB^{op,k}_{t},\end{cases}$
(10)

for $k=1,...,N$, an independent family $(B^{sp,k},B^{op,k})_{k\in\mathbb{N}}$
of $(d+1)$-dimensional Brownian motions.

Let us define the empirical density by

$\mu_{t}^{N}:=\frac{1}{N}\sum_{j=1}^{N}\delta_{(X^{j,N}_{t},\Theta^{j,N}_{t})}$
(11)

with $\delta$ being the Dirac distribution. Now the previous system can
alternatively be written as

$\displaystyle\begin{cases}dX^{k,N}_{t}&=\int_{\mathbb{R}^{d}\times\mathbb{R}}U(X^{k}_{t},y,\Theta^{k}_{t},\theta)\mu^{N}_{t}(dyd\theta)dt+\int_{\mathbb{R}^{d}\times\mathbb{R}}\sigma_{sp}(X^{k}_{t},y,\Theta^{k}_{t},\theta)\mu^{N}_{t}(dyd\theta)dB^{sp,k}_{t},\\\
\
d\Theta^{k,N}_{t}&=\int_{\mathbb{R}^{d}\times\mathbb{R}}V(X^{k}_{t},y,\Theta^{k}_{t},\theta)\mu^{N}_{t}(dyd\theta)dt+\int_{\mathbb{R}^{d}\times\mathbb{R}}\sigma_{op}(X^{k}_{t},y,\Theta^{k}_{t},\theta)\mu^{N}_{t}(dyd\theta)dB^{op,k}_{t},\\\
\
\mu_{t}^{N}&=\frac{1}{N}\sum_{j=1}^{N}\delta_{(X^{j,N}_{t},\Theta^{j,N}_{t})}.\end{cases}$
(12)

Utilizing the law of large numbers we will show that the mean-field limit of
the system is of the form

$\displaystyle\begin{cases}d\overline{X}_{t}&=\left(\int_{\mathbb{R}^{d}\times\mathbb{R}}U(\overline{X}_{t},y,\overline{\Theta}_{t},\theta)\mu_{t}(dyd\theta)\right)dt+\left(\int_{\mathbb{R}^{d}\times\mathbb{R}}\sigma_{sp}(\overline{X}_{t},y,\overline{\Theta}_{t},\theta)\mu_{t}(dyd\theta)\right)dB^{sp}_{t},\\\
\
d\overline{\Theta}_{t}&=\left(\int_{\mathbb{R}^{d}\times\mathbb{R}}V(\overline{X}_{t},y,\overline{\Theta}_{t},\theta)\mu_{t}(dyd\theta)\right)dt+\left(\int_{\mathbb{R}^{d}\times\mathbb{R}}\sigma_{op}(\overline{X}_{t},y,\overline{\Theta}_{t},\theta)\mu_{t}(dyd\theta)\right)dB^{op}_{t},\\\
\ \overline{\mu}_{t}&=\text{Law}(\overline{X}_{t},\overline{\Theta}_{t}),\\\ \
\overline{X}_{t=0}&=X_{0},\,\overline{\Theta}_{t=0}=\Theta_{0},\end{cases}$
(13)

for suitable initial values $X_{0},\Theta_{0}$ and a $(d+1)$-dimensional
Brownian motion $(B^{sp}_{t},B^{op}_{t})$.

###### Remark 1

One can write the system more compactly by summarizing the state of each agent
into a single $\mathbb{R}^{d+1}$-valued random variable $Y=(X,\Theta)$. In
order to do this, we define the combined interaction map to be

$\displaystyle
W:\mathbb{R}^{d}\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d+1},\quad(x_{1},\theta_{1},x_{2},\theta_{2})\mapsto(U(x_{1},x_{2},\theta_{1},\theta_{2}),V(x_{1},x_{2},\theta_{1},\theta_{2})).$

With this notation we can write the system (13) as

$\displaystyle\begin{cases}dY_{t}&=\left(\int_{\mathbb{R}^{d+1}}W(Y_{t},y,\theta)\mu_{t}(d(y,\theta))\right)dt+\left(\int_{\mathbb{R}^{d+1}}\tilde{\sigma}(Y_{t},y,\theta)\mu_{t}(d(y,\theta))\right)d\tilde{B}_{t},\\\
\ \mu_{t}&=\text{Law}(Y_{t}),\end{cases}$ (14)

where $\tilde{B}$ is a $(d+1)$-dimensional standard Brownian motion and
$\tilde{\sigma}$ is a $(d+1)\times(d+1)$ diagonal matrix with
$\text{diag}(\tilde{\sigma_{sp}(X_{t},\Theta_{t})})=(\sigma_{sp}(X_{t},\Theta_{t}),...,\sigma_{sp}(X_{t},\Theta_{t}),\sigma_{op}(X_{t},\Theta_{t}))$.
In the case of additive noise, if $U,V$ are assumed to be Lipschitz
continuous, then $W$ is also a Lipschitz function and one can apply the
classical existence and uniqueness and convergence results from [27] directly
to $(\ref{reduced-system})$. With the focus on the different interpretation of
processes $X_{t}$ and $\Theta_{t}$ from the application point of view, we will
keep the separate notion which enables us to directly see the limiting
equations for position and opinion.

###  3.2 Well-posedness result of the coupled mean-field SDE

Here, we will study the well-posedness of the limiting stochastic differential
equations given by (13).

The proof is based on the classical results on mean-field theory (see e.g. for
additive noise [27]) and the classical existence and uniqueness theory for
SDEs via the standard fixed-point argument.

In order to set up our fixed point argument we, let us define

$\displaystyle\mathcal{P}^{2}$
$\displaystyle:=\mathcal{P}^{2}(C([0,T];\mathbb{R}^{d+1}))$
$\displaystyle:=\left\\{\mu:\mu\text{ is a probability measure on
}C([0,T];\mathbb{R}^{d+1})\text{ s.t. }\int{\left\lVert
x\right\rVert_{\infty}^{2}\mu(dx)<\infty}.\right\\}.$

Furthermore, we need to equip this space with a notion of distance, that turns
$\mathcal{P}^{2}$ into a complete metric space. To make use of a common tool
for this type of proof, namely Gronwall’s inequality, we need to consider a
whole family of distances (which are not necessarily metrics due to a lack of
definiteness). To be precise we define the truncated 2-Wasserstein distance
for $t\in[0,T]$ by

$\displaystyle
D_{t}^{2}(\mu_{1},\mu_{2}):=\inf_{\mu}\int_{\mathcal{C}\times\mathcal{C}}\left(\sup_{s\leq
t}(\left\lVert
X_{s}(\omega_{1})-X_{s}(\omega_{2})\right\rVert^{2}+\left\lvert\Theta_{s}(\omega_{1})-\Theta_{s}(\omega_{2})\right\rvert^{2})\right)d\mu(\omega_{1},\omega_{2}),$
(15)

where we take the infimum over all couplings $\mu$ of $\mu_{1}$ and $\mu_{2}$
and $X_{s}(\cdot),\Theta_{s}(\cdot)$ are the projections onto the time $s$
marginal of the $d$-dimensional component and the $1$-dimensional component
respectively. Note that for $t=T$, i.e. if we don’t truncate, we obtain the
standard $2$-Wasserstein metric $\mathcal{W}_{2}^{2}(\cdot,\cdot)$ on
$\mathcal{P}^{2}$.

###### Remark 2

Observe that the Lipschitz Assumption 1 on the coefficients also implies, that
the induced maps defined by

$\displaystyle\hat{U}:\mathbb{R}^{d}\times\mathbb{R}\times\mathcal{P}^{2}(\mathbb{R}^{d+1})\to\mathbb{R}^{d},\quad(x,\theta,\mu)\mapsto\int_{\mathbb{R}^{d+1}}U(x,y,\theta,\eta)\mu(dyd\theta),$
(16)

and analogously $\hat{V},\hat{\sigma}_{sp},\hat{\sigma}_{op}$, satisfy a
Lipschitz-type inequality (2) with respect to the product metric, when we
equip $\mathcal{P}^{2}$ with the $2$-Wasserstein metric. Indeed, for
$(x_{1},\theta_{1},\mu_{1}),(x_{2},\theta_{2},\mu_{2})$ we have by Jensen’s
inequality

$\displaystyle\left\lVert\hat{U}(x_{1},\theta_{1},\mu_{1})\\!\\!-\\!\\!\hat{U}(x_{2},\theta_{2},\mu_{2})\right\rVert^{2}\\!\\!\leq\\!\\!\int\left\lVert
U(x_{1},y_{1},\theta_{1},\eta_{1})\\!-\\!U(x_{2},y_{2},\theta_{2},\eta_{2})\right\rVert^{2}\pi(d(y_{1},\theta_{1})d(y_{2},\theta_{2})),$

where $\pi$ is an arbitrary coupling of $\mu_{1},\mu_{2}$. We can now use the
Lipschitz assumption on $U$ to get

$\displaystyle\int\left\lVert
U(x_{1},y_{1},\theta_{1},\eta_{1})-U(x_{2},y_{2},\theta_{2},\eta_{2})\right\rVert^{2}\pi(d(y_{1},\theta_{1})d(y_{2},\theta_{2}))$
$\displaystyle\ \leq$ $\displaystyle 2L^{2}\left\lVert
x_{1}-x_{2}\right\rVert+2L^{2}\left\lVert\theta_{1}-\theta_{2}\right\rVert+2L^{2}\int\left(\left\lVert
y_{1}-y_{2}\right\rVert^{2}+\left\lVert\eta_{1}-\eta_{2}\right\rVert^{2}\right)\pi(d(y_{1},\theta_{1})d(y_{2},\theta_{2})).$

Now we can take the infimum over all couplings $\pi$ of $\mu_{1}$ and
$\mu_{2}$ on the right hand side to obtain

$\displaystyle\left\lVert\int_{\mathbb{R}^{d+1}}U(x_{1},y_{1},\theta_{1},\eta_{1})\mu_{1}(dy_{1}d\eta_{1})-\int_{\mathbb{R}^{d+1}}U(x_{2},y_{2},\theta_{2},\eta_{2})\mu_{2}(dy_{2}d\eta_{2})\right\rVert^{2}$
$\displaystyle\ \leq$ $\displaystyle 2L^{2}(\left\lVert
x_{1}-x_{2}\right\rVert^{2}+\left\lVert\theta_{1}-\theta_{2}\right\rVert^{2}+\mathcal{W}^{2}_{2}(\mu_{1},\mu_{2})).$

As already announced, to show the well-posedness of (13), we rely on Banach’s
classical fixed-point theorem. Therefore we need to make sure that the metric
space $(\mathcal{P}^{2},D^{2}_{T})$ is sufficiently regular.

###### Lemma 1

$(\mathcal{P}^{2},D^{2}_{T})$ is a complete metric space.

The proof of this lemma is standard and is given in Appendix A.1.

In addition, for the proof of the next theorem we need the following a priori
estimate.

###### Lemma 2

Assume that the coefficients $U,V,\sigma_{op},\sigma_{sp}$ are bounded by some
positive constant $K>0$ and that the initial values $X_{0},\Theta_{0}$ satisfy
$\mathbb{E}[\left\lVert X_{0}\right\rVert^{2}]<\infty$ and
$\mathbb{E}[\left\lvert\Theta_{0}\right\rvert^{2}]<\infty$. Then for any $T>0$
there exists a constant $C=C(T,K)>0$ such that every solution $(X,\Theta)$ to
(13) satisfies

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\lVert
X_{t}\right\rVert^{2}+\left\lvert\Theta_{t}\right\rvert^{2}\right]\leq
C\left(\mathbb{E}\left[\left\lVert
X_{0}\right\rVert^{2}+\left\lvert\Theta_{0}\right\rvert^{2}\right]+K^{2}(T^{2}+T)\right),$

In particular, it holds that $Law(X,\Theta)\in\mathcal{P}^{2}$.

The proof of this lemma is classic and it is based on the Burkholder-Davis-
Gundy inequality. For the completeness we wrote the proof in Appendix A.1. Now
we can state the well-posedness result:

###### Theorem 1

Let $X_{0}$ and $\Theta_{0}$ be $\mathbb{R}^{d}$-valued, respectively
$\mathbb{R}$-valued, random variables with finite second moment i.e.

$\displaystyle\mathbb{E}\left[\left\lVert
X_{0}\right\rVert^{2}\right]<\infty,\quad\mathbb{E}\left[\left\lvert\Theta_{0}\right\rvert^{2}\right]<\infty.$

Under Assumption 1, there exists a unique (pathwise and in law) solution to
equation $\eqref{lim_system}$.

As already announced, the proof of the existence is based on the fixed point
argument and it can be found in a more general setting. Nevertheless, for the
completeness we sketch in the Appendix 1.1 the basic idea of the proof in our
setting that is simpler than in the general setting, and hence easier
accessible.

###  3.3 Convergence of the microscopic model to the mean-field equation

In this section we will prove that the the system of coupled SDEs for fixed
$N\in\mathbb{N}$ given by (10) indeed converges to the mean-field limit, i.e.
to show the so called propagation of chaos. For $N\in\mathbb{N}$ we denote the
(measure-valued) empirical process of this system by
$\mu^{N}=(\mu^{N}_{t})_{t\geq 0}$, i.e. for $t\geq 0$ we set

$\displaystyle\mu_{t}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{(X^{i,N}_{t},\Theta^{i,N}_{t})}.$
(17)

For fixed $i$ we consider the process
$(\overline{X}^{i}_{t},\overline{\Theta}^{i}_{t})_{t\geq 0}$ that solves

$\displaystyle\begin{cases}d\overline{X}^{i}_{t}&=\left(\int_{\mathbb{R}^{d+1}}U(\overline{X}^{i}_{t},y,\overline{\Theta}^{i}_{t},\theta)\mu_{t}(dy\
d\theta)\right)dt+\left(\int_{\mathbb{R}^{d+1}}\sigma_{sp}(\overline{X}^{i}_{t},y,\overline{\Theta}^{i}_{t},\theta)\mu_{t}(dy\
d\theta)\right)dB^{i,sp}_{t},\\\ \
d\overline{\Theta}^{i}_{t}&=\left(\int_{\mathbb{R}^{d+1}}V(\overline{X}^{i}_{t},y,\overline{\Theta}^{i}_{t},\theta)\mu_{t}(dy\
d\theta)\right)dt+\left(\int_{\mathbb{R}^{d+1}}\sigma_{op}(\overline{X}^{i}_{t},y,\overline{\Theta}^{i}_{t},\theta)\mu_{t}(dy\
d\theta)\right)dB^{i,op}_{t},\\\ \
{\mu}_{t}&=\text{Law}(\overline{X}_{t}^{i},\overline{\Theta}_{t}^{i}).\end{cases}$

The existence of $(\overline{X}^{i}_{t},\overline{\Theta}^{i}_{t})_{t\geq 0}$
follows from Theorem 1 and ${\mu}=({\mu}_{t})_{t\geq 0}$ does not depend on
$i$. From now on we will denote the law of
$(\overline{X}^{i}_{t},\overline{\Theta}^{i}_{t})_{t\geq 0}$ by ${\mu}$.

Next, we will study the case when $N$ tends to $\infty$. The key idea for the
proof is to use the law of large numbers (LLN) for empirical measures
$\overline{\mu}^{N}$ of i.i.d. copies of the process
$(\overline{X}^{i},\overline{\Theta}^{i})$, cf. [53, Theorem 11.4.1]. We will
see that by a uniform integrability argument, LLN also implies that

$\displaystyle\mathbb{E}\left[D^{2}_{T}({\mu},\overline{\mu}^{N})\right]\to
0\quad\text{as }N\to\infty.$

The rest of the needed estimates are then the same as in the proof of Theorem
1 and are mainly done for the purpose of showing that

$\displaystyle\mathbb{E}\left[D_{T}^{2}({\mu},\mu^{N})\right]\leq
C\mathbb{E}\left[D^{2}_{T}({\mu},\overline{\mu}^{N})\right]$

for some constant $C>0$.

###### Theorem 2

For any $i\in\mathbb{N}$ we have

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq t}\left\lVert
X^{i,N}_{s}-\overline{X}^{i}_{s}\right\rVert^{2}+\left\lvert\Theta^{i,N}_{s}-\overline{\Theta}^{i}_{s}\right\rvert^{2}\right]\to
0\quad\text{as }N\to\infty.$

Moreover, we have

$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[D_{T}^{2}({\mu},\mu^{N})\right]=0.$
(18)

The details of the proof are presented in Appendix 1.2.

##  4 Characterization of the empirical measure and its limit

In this section we derive the formal stochastic partial differential equation
(SPDE) for the empirical measure $\mu^{N}$, that is the so-called
Dean–Kawasaki type equation [30, 54] with multiplicative noise and we derive
the McKean-Vlasov type PDE for the limiting measure $\mu$. Recall that if
$X=(X_{t})_{t\geq 0}$ is a Markov process with generator $\mathcal{L}$, then
its law $\mu=(\mu_{t})_{t\geq 0}$ is a (weak) solution to the linear PDE

$\displaystyle\partial_{t}\mu=\mathcal{L}^{*}\mu,$ (19)

also known as the Kolmogorov forward equation. A similar equation still holds
true for solutions $(X,\Theta,\mu)$of the type of equation (13) and empirical
measure for (10). However, the PDE that we will derive will be non-linear.

###  4.1 Derivation of the PDE for the law of the coupled mean-field SDEs

We consider a system of coupled mean-field SDEs given by (13) and we want to
derive equation for the hydrodynamic limit
$\mu_{t}=\text{Law}(X_{t},\Theta_{t})$.

Let $\phi:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a smooth and
compactly supported function. Then by Itô’s formula we have

$\displaystyle d\phi(X_{t},\Theta_{t})=$ $\displaystyle\\!\\!\\!\left(\int
U(X_{t},y,\Theta_{t},\theta)\cdot\nabla_{x}\phi(X_{t},\Theta_{t})\mu_{t}(dyd\theta)\right)dt+\left(\int
V(X_{t},y,\Theta_{t},\theta)\frac{\partial}{\partial\theta}\phi(X_{t},\Theta_{t})\mu_{t}(dyd\theta)\right)dt$
$\displaystyle\ +$
$\displaystyle\frac{1}{2}\Bigg{(}\\!\\!\sum_{i,j=1}^{d}\left(\int\sigma_{sp}\\!(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\\!\cdot\\!\left(\\!\int\\!\sigma_{sp}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\right)\\!\\!^{\top}\right)_{i,j}\\!\\!\\!\\!\times\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}\phi(X_{t},\Theta_{t})\\!\\!\Bigg{)}dt$ $\displaystyle\ +$
$\displaystyle\frac{1}{2}\left(\left(\int\sigma_{op}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\right)^{2}\frac{\partial^{2}}{\partial\theta^{2}}\phi(X_{t},\Theta_{t})\right)dt$
$\displaystyle\ +$
$\displaystyle\left((\nabla_{x}\phi(X_{t},\Theta_{t}))\\!\\!^{\top}\cdot\int\sigma_{sp}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta))\right)dB^{sp}_{t}$
$\displaystyle\ +$
$\displaystyle\left(\int\sigma_{op}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\cdot\frac{\partial}{\partial\theta}\phi(X_{t},\Theta_{t})\right)dB^{op}_{t}.$

Taking the expectation the martingale part vanishes, differentiating in $t$
and recalling that $\mu_{t}=\text{Law}(X_{t},\Theta_{t})$ we can rewrite the
previous equation as

$\displaystyle\frac{d}{dt}\int\phi(z,\eta)\mu_{t}(dzd\eta)=$
$\displaystyle\int\bigg{[}\left(\int
U(z,y,\eta,\theta)\cdot\nabla_{z}\phi(z,\eta)\mu_{t}(dyd\theta)\right)+\left(\int
V(z,y,\eta,\theta)\frac{\partial}{\partial\eta}\phi(z,\eta)\mu_{t}(dyd\theta)\right)$
$\displaystyle\ \quad+$
$\displaystyle\frac{1}{2}\Bigg{(}\sum_{i,j=1}^{d}\left(\int\sigma_{sp}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\cdot\left(\\!\\!\int\sigma_{sp}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\right)^{\top}\right)_{i,j}$
$\displaystyle\quad\times\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}\phi(X_{t},\Theta_{t})\Bigg{)}+\frac{1}{2}\left(\left(\int\sigma_{op}(X_{t},y,\Theta_{t},\theta)\mu_{t}(dyd\theta)\right)^{2}\frac{\partial^{2}}{\partial\theta^{2}}\phi(X_{t},\Theta_{t})\right)\bigg{]}\mu_{t}(dzd\eta).$

Assuming that $\mu_{t}$ is absolutely continuous with respect to Lebesgue
measure and that its density $\mu_{t}(z,\eta)$ is sufficiently regular, we can
apply Fubini’s theorem and integration by parts. Since $\phi$ is an arbitrary
sufficiently regular test-function, we see that $\mu=(\mu_{t})_{t\geq 0}$ is a
distribution-valued solution of the PDE

$\displaystyle\partial_{t}\mu_{t}(z,\eta)=$
$\displaystyle-\text{div}_{z}\left(\mu_{t}(z,\eta)\cdot
U(z,\eta,\mu_{t})\right)-\text{div}_{\eta}\left(\mu_{t}(z,\eta)\cdot
V(z,\eta,\mu_{t})\right)$ (20)
$\displaystyle+\frac{1}{2}\sum_{i,j=1}^{d}\frac{\partial^{2}}{\partial
z_{i}\partial
z_{j}}\left(\mu_{t}(z,\eta)\cdot\sigma_{sp}(z,\eta,\mu_{t})\cdot\sigma_{sp}(z,\eta,\mu_{t})^{\top}\right)_{i,j}$
$\displaystyle\ +$
$\displaystyle\frac{1}{2}\frac{\partial^{2}}{\partial\eta^{2}}\left(\mu_{t}(z,\eta)\sigma_{op}(z,\eta,\mu_{t})^{2}\right),$

where we use the shorthand notation

$\displaystyle U(z,\eta,\mu_{t})$
$\displaystyle:=\int_{\mathbb{R}^{d}\times\mathbb{R}}U(z,y,\eta,\theta)\mu_{t}(dyd\theta),$
(21) $\displaystyle\ V(z,\eta,\mu_{t})$
$\displaystyle:=\int_{\mathbb{R}^{d}\times\mathbb{R}}V(z,y,\eta,\theta)\mu_{t}(dyd\theta),$
(22) $\displaystyle\ \sigma_{space}(z,\eta,\mu_{t})$
$\displaystyle:=\int_{\mathbb{R}^{d}\times\mathbb{R}}\sigma_{space}(z,y,\eta,\theta)\mu_{t}(dyd\theta),$
(23) $\displaystyle\ \sigma_{op}(z,\eta,\mu_{t})$
$\displaystyle:=\int_{\mathbb{R}^{d}\times\mathbb{R}}\sigma_{op}(z,y,\eta,\theta)\mu_{t}(dyd\theta).$
(24)

###### Remark 3

Note that in the case of the additive noise this equation becomes

$\displaystyle\partial_{t}\mu_{t}(z,\eta)=$
$\displaystyle-\text{div}_{z}\Big{(}U(z,\eta,\mu_{t})\mu_{t}(z,\eta)\Big{)}-\text{div}_{\eta}\Big{(}V(z,\eta,\mu_{t})\mu_{t}(z,\eta)\Big{)}$
$\displaystyle+\frac{1}{2}\sigma_{1}^{2}\Delta_{z}\mu_{t}(z,\eta)+\frac{1}{2}\sigma_{2}^{2}\Delta_{\eta}\mu_{t}(z,\eta),$

which coincides with the standard result from [27]. Hence, as expected the
only difference are the coefficients of the second order part of the
differential operator on the right hand side, which are either constants or
functions.

###  4.2 SPDE description for the empirical measure $\mu^{N}$

The goal is to derive a formal SPDE for the empirical measure $\mu^{N}$, i.e.
the analogue to the Dean-Kawasaki equation, but with multiplicative noise.
More precise, for $N\in\mathbb{N}$ let $(X^{N}_{t},\Theta_{t}^{N})_{t\geq 0}$
be such that its components $(X^{k,N}_{t},\Theta_{t}^{k,N})_{t\geq 0}$ solve
the system (10).

In the following, we want to (formally) derive the SPDE that is solved by the
empirical measure $\mu^{N}=(\mu^{N}_{t})_{t\geq 0}$ defined by (11).

Let $\phi:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}$ be a smooth and
compactly supported function. By definition of the empirical measure we have

$\displaystyle\int_{\mathbb{R}^{d}\times\mathbb{R}}\phi(z,\eta)\mu_{t}^{N}(dzd\eta)=\frac{1}{N}\sum_{i=1}^{N}\phi(X^{i,N}_{t},\Theta^{i,N}_{t}).$

Hence, we will first calculate each of the summands individually. For fixed
$i=1,\dots,N$ we denote by $B^{sp,i}=(B^{sp,i,1},\dots,B^{sp,i,d})$ the
components of the driving Brownian motion of the $i$-th agent. By Itô’s
formula we have

$\displaystyle d\phi(X_{t}^{i,N},\Theta_{t}^{i,N})$
$\displaystyle=\int\mu^{i,N}_{t}(dzd\eta)\Bigg{[}\bigg{(}\int
U(z,\eta,\mu_{t}^{N})\cdot\nabla_{x}\phi(z,\eta)+V(z,\eta,\mu^{N}_{t})\frac{\partial}{\partial\theta}\phi(z,\eta)\bigg{)}dt$
$\displaystyle\ +$
$\displaystyle\frac{1}{2}\left(\sum_{i,j=1}^{d}\left(\sigma_{sp}(z,\eta,\mu_{t}^{N})\cdot\sigma_{sp}(z,\eta,\mu_{t}^{N})^{\top}\right)_{i,j}\frac{\partial^{2}}{\partial
z_{i}\partial z_{j}}\phi(z,\eta)\right)dt$ $\displaystyle\ +$
$\displaystyle\frac{1}{2}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})^{2}\frac{\partial^{2}}{\partial\eta^{2}}\phi(z,\eta)\right)dt$
$\displaystyle\ +$
$\displaystyle\left((\nabla_{z}\phi(z,\eta))^{\top}\sigma_{sp}(z,\eta,\mu_{t}^{N})\right)dB^{sp,i}_{t}+\left(\sigma_{op}(z,\eta,\mu_{t}^{N})\frac{\partial}{\partial\theta}\phi(z,\eta)\right)dB^{op,i}_{t}\Bigg{]},$

where we again used the shorthand notation introduced in (21)-(24).

Assuming that $\mu^{i,N}_{t}$ is absolutely continuous with respect to
Lebesgue measure with a sufficiently smooth density $\mu_{t}^{i,N}(z,\eta)$
and applying integration by parts with respect to the Lebesgue integral, we
obtain

$\displaystyle d\phi(X_{t}^{i,N},$ $\displaystyle\Theta_{t}^{i,N})=$
$\displaystyle\int
dzd\eta\phi(z,\eta)\Bigg{[}-\text{div}_{z}\left[U(z,\eta,\mu^{N}_{t})\mu_{t}^{i,N}(z,\eta)\right]dt-\text{div}_{\eta}\left[V(z,\eta,\mu^{N}_{t})\mu^{i,N}_{t}(z,\eta)\right]dt$
$\displaystyle\ +$
$\displaystyle\frac{1}{2}\left(\sum_{i,j=1}^{N}\frac{\partial^{2}}{\partial
z_{i}\partial
z_{j}}\left(\left(\sigma_{sp}(z,\eta,\mu^{N}_{t})\cdot\sigma_{sp}(z,\eta,\mu^{N}_{t})^{\top}\right)_{i,j}\mu^{i,N}_{t}(z,\eta)\right)\right)dt$
$\displaystyle\ +$
$\displaystyle\frac{1}{2}\frac{\partial^{2}}{\partial\eta^{2}}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})^{2}\mu^{i,N}(z,\eta)\right)dt-\sum_{k,l=1}^{d}\frac{\partial}{\partial
z_{k}}\left(\sigma_{sp}^{l,k}(z,\eta,\mu^{N}_{t})\mu^{i,N}(z,\eta)dB^{sp,i,k}_{t}\right)$
$\displaystyle\ -$
$\displaystyle\frac{\partial}{\partial\eta}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})\mu^{i,N}(z,\eta)dB^{op,i}_{t}\right)\Bigg{]}.$

Utilizing that for each $i=1,\dots,N$ we have

$\displaystyle\frac{d}{dt}\phi(X^{i,N},\Theta^{i,N})=\int_{\mathbb{R}^{d+1}}dzd\eta\phi(z,\eta)\left[\frac{\partial}{\partial
t}\mu^{i,N}_{t}(z,\eta)\right],$

and after the summation over $i$ we conclude that the empirical measure
$(\mu^{N}_{t})_{t\geq 0}$ solves the SPDE

$\displaystyle\partial_{t}\mu^{N}_{t}=$
$\displaystyle-\text{div}_{z}\left[U(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)\right]-\text{div}_{\eta}\left[V(z,\eta,\mu^{N}_{t})\mu^{N}_{t}\right]$
$\displaystyle+\frac{1}{2}\sum_{i,j=1}^{N}\left(\frac{\partial^{2}}{\partial
z_{i}\partial
z_{j}}\left(\sigma_{sp}(z,\eta,\mu^{N}_{t})\cdot\sigma_{sp}(z,\eta,\mu^{N}_{t})^{\top}\right)_{i,j}\mu^{N}_{t}(z,\eta)\right)$
$\displaystyle+\frac{1}{2}\frac{\partial^{2}}{\partial\eta^{2}}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})^{2}\mu^{N}(z,\eta)\right)$
$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{q}\sum_{l=1}^{d}\frac{\partial}{\partial
z_{k}}\left(\sigma_{sp}^{l,k}(z,\eta,\mu^{N}_{t})\mu^{i,N}_{t}(z,\eta)dB^{sp,i,k}_{t}\right)$
$\displaystyle-\frac{1}{N}\sum_{i=1}^{N}\frac{\partial}{\partial\eta}\left(\sigma_{op}(z,\eta)\mu^{i,N}_{t}(z,\eta)dB^{op,i}_{t}\right).$

However, this is not yet a closed equation for the empirical measure, because
it still depends on the individual trajectories through the noise term. As in
the case of additive noise [30], we will first calculate the covariance of
this noise term and then replace it by a statistically identical term. For
this, let’s denote the two noise terms by

$\displaystyle\xi^{sp}(t,z,\eta)$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{d}\sum_{l=1}^{d}\frac{\partial}{\partial
z_{k}}\left(\sigma_{sp}^{l,k}(z,\eta,\mu^{N}_{t})\mu^{i,N}_{t}(z,\eta)dB^{sp,i,k}_{t}\right),$
$\displaystyle\ \xi^{op}(t,z,\eta)$
$\displaystyle:=\frac{1}{N}\sum_{i=1}^{N}\frac{\partial}{\partial\eta}\left(\sigma_{op}(z,\eta)\mu^{i,N}_{t}(z,\eta)dB^{op,i}_{t}\right).$

Then, the covariances are given by

$\displaystyle\mathbb{E}\left[\xi^{sp}(t,z,\eta)\xi^{sp}(s,y,\theta)\right]$
$\displaystyle=\delta(t-s)\frac{1}{N^{2}}\sum_{i=1}^{N}\sum_{k=1}^{d}\sum_{l,g=1}^{d}\frac{\partial}{z_{k}}\left(\sigma_{sp}^{l,k}(z,\eta,\mu^{N}_{t})\mu^{i,N}_{t}(z,\eta)\right)$
$\displaystyle\hskip
113.81102pt\times\frac{\partial}{x_{k}}\left(\sigma_{sp}^{g,k}(x,\theta,\mu^{N}_{s})\mu^{i,N}_{s}(x,\theta)\right),$
$\displaystyle\ \mathbb{E}\left[\xi^{sp}(t,z,\eta)\xi^{op}(s,x,\theta)\right]$
$\displaystyle=0,$ $\displaystyle\
\mathbb{E}\left[\xi^{op}(t,z,\eta)\xi^{op}(s,x,\theta)\right]$
$\displaystyle=\delta(t-s)\frac{1}{N^{2}}\sum_{i=1}^{N}\frac{\partial}{\partial\eta}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})\mu^{i,N}_{t}(z,\eta)\right)$
$\displaystyle\hskip
85.35826pt\times\frac{\partial}{\partial\theta}\left(\sigma_{op}(x,\theta,\mu^{N}_{s})\mu^{i,N}_{s}(x,\theta)\right).$

First note that by the definition of $\mu^{i,N}_{t}$ as a Dirac delta
distribution, we have for each $i=1,\dots,N$

$\displaystyle\mu^{i,N}_{t}(z,\eta)\mu^{i,N}_{t}(x,\theta)=\delta(z-x)\delta(\eta-\theta)\mu^{i,N}(z,\eta)=\delta(z-x)\delta(\eta-\theta)\mu^{i,N}(x,\theta).$

Therefore we can rewrite the non-trivial covariances as

$\displaystyle\mathbb{E}\left[\xi^{sp}(t,z,\eta)\xi^{sp}(s,y,\theta)\right]$
$\displaystyle=\delta(t-s)\frac{1}{N}\sum_{i=1}^{N}\sum_{k=1}^{d}\sum_{l,g=1}^{d}\frac{\partial}{\partial
z_{k}}\frac{\partial}{\partial
x_{k}}\left(\delta(\eta-\theta)\delta(z-x)\sigma^{l,k}_{sp}(z,\eta,\mu^{N}_{t})\sigma_{sp}^{g,k}(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)\right),$

and

$\displaystyle\mathbb{E}\left[\xi^{op}(t,z,\eta)\xi^{op}(s,x,\theta)\right]$
$\displaystyle=\delta(t-s)\frac{1}{N}\frac{\partial}{\partial\eta}\frac{\partial}{\partial\theta}\left(\delta(z-x)\delta(\eta-\theta)\sigma_{op}(z,\eta,\mu^{N}_{t})^{2}\mu^{N}_{t}(z,\eta)\right).$

Furthermore, let $\zeta$ be a $(d+1)$-dimensional Gaussian random field
indexed by $\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathbb{R}$ with
covariance given by

$\displaystyle\mathbb{E}\left[\zeta^{i}(t,z,\eta)\zeta^{j}(s,x,\theta)\right]=\delta^{i,j}\delta(t-s)\delta(z-x)\delta(\eta-\theta).$

We define two noise fields $\xi^{sp^{\prime}},\xi^{op^{\prime}}$ via

$\displaystyle\xi^{sp^{\prime}}(t,z,\eta)$
$\displaystyle:=\frac{1}{\sqrt{N}}\sum_{k=1}^{d}\sum_{l,g=1}^{d}\frac{\partial}{\partial
z_{k}}\frac{\partial}{\partial
x_{k}}\left(\sigma^{l,k}_{sp}(z,\eta,\mu^{N}_{t})\sigma_{sp}^{g,k}(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)^{\frac{1}{2}}\zeta(t,z,\eta)\right)$
$\displaystyle\ \xi^{op^{\prime}}(t,z,\eta)$
$\displaystyle:=\frac{1}{\sqrt{N}}\frac{\partial}{\partial\eta}\frac{\partial}{\partial\theta}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)^{\frac{1}{2}}\zeta(t,z,\eta)\
\right).$

Note that these noise fields are statistically equivalent to
$\xi^{sp},\xi^{op}$. Altogether, we conclude that the dynamics of the
empirical measure $\mu^{N}$ of the feedback-loop dynamics with multiplicative
noise given by (1) is described by the following formal SPDE

$\displaystyle\partial_{t}\mu^{N}_{t}=$
$\displaystyle-\text{div}_{z}\left[U(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)\right]-\text{div}_{\eta}\left[V(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)\right]$
(25)
$\displaystyle+\frac{1}{2}\sum_{i,j=1}^{N}\left(\frac{\partial^{2}}{\partial
z_{i}\partial
z_{j}}\left(\sigma_{sp}(z,\eta,\mu^{N}_{t})\cdot\sigma_{sp}(z,\eta,\mu^{N}_{t})^{\top}\right)_{i,j}\mu^{N}_{t}(z,\eta)\right)$
$\displaystyle+\frac{1}{2}\frac{\partial^{2}}{\partial\eta^{2}}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})^{2}\mu^{N}(z,\eta)\right)$
$\displaystyle-\frac{1}{\sqrt{N}}\sum_{k=1}^{d}\sum_{l,g=1}^{d}\frac{\partial}{\partial
z_{k}}\frac{\partial}{\partial
x_{k}}\left(\sigma^{l,k}_{sp}(z,\eta,\mu^{N}_{t})\sigma_{sp}^{g,k}(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)^{\frac{1}{2}}\zeta^{k}(t,z,\eta)\right)$
$\displaystyle-\frac{1}{\sqrt{N}}\frac{\partial}{\partial\eta}\frac{\partial}{\partial\theta}\left(\sigma_{op}(z,\eta,\mu^{N}_{t})\mu^{N}_{t}(z,\eta)^{\frac{1}{2}}\zeta^{d+1}(t,z,\eta)\
\right).$

###### Remark 4

The previous equation (25) is the generalization of the standard Dean-Kawasaki
equation [30] to the dynamics with the multiplicative case. In particular, in
the case of the additive noise this SPDE becomes

$\displaystyle\partial_{t}\mu^{N}_{t}(x,\theta)=$
$\displaystyle-\text{div}_{x}\left[\mu^{N}_{t}(x,\theta)\cdot\int
U(x,y,\theta,\eta)\mu^{N}_{t}(y,\eta)dyd\eta\right]$
$\displaystyle-\text{div}_{\theta}\left[\mu^{N}_{t}(x,\theta)\cdot\int
V(x,y,\theta,\eta)\mu^{N}_{t}(y,\eta)dyd\eta\right]$
$\displaystyle+\frac{1}{2}\sigma_{1}^{2}\Delta_{x}\mu_{t}^{N}(x,\theta)+\frac{1}{2}\sigma_{2}^{2}\Delta_{\theta}\mu^{N}_{t}(x,\theta)$
$\displaystyle-\frac{1}{N}\sigma_{1}\text{div}_{x}\left[\mu^{N}_{t}(x,\theta)^{\frac{1}{2}}\cdot\zeta(t,x,\theta)\right]-\frac{1}{N}\sigma_{2}\text{div}_{\theta}\left[\mu^{N}_{t}(x,\theta)^{\frac{1}{2}}\cdot\zeta(t,x,\theta)\right]$

which is the standard type of Dean-Kawasaki equation. Concerning the SPDE
formulations of an ABM, observe that the diffusion part that appears in the
dynamics of the empirical measure (see [31]) is exactly of this type and the
case with multiplicative noise is its generalization.

###  4.3 Numerical experiment

We will here illustrate the behaviour of the feedback loop system at the
macroscopic level, i.e. when the number of agents tend to infinity. As we
showed in the previous Section, this dynamics is described by the PDE (20). We
will simulate it using finite difference method with one spatial dimension and
one opinion dimension. The considered domain is $[-2,2]^{2}$ with the grid
size $0.05$ and the time interval is $[0,1]$ with time step $dt=0.0001$. We
use no-flux boundary conditions. The initial conditions are randomly chosen
four clusters with normal distribution. Other parameters are chosen in a
similar way as in the experiments made in Section 2.4. In particular, the
potentials $U$ and $V$ are given by (4) and (5) respectively, with the
additional scaling parameter that is taken to be $0.5$ and represents scaled
space/opinion interaction strength. Note that these potentials do not satisfy
our regularity Assumption (1), however, as already explained, from the
literature is expected that the PDE equation for the empirical density has the
same form as (20). As in the ABM, the interaction radius is taken to be $0.15$
and we consider the additive noise with strength $0.01$ for both space and
opinion dynamics. In Figure 7 we show the empirical density of agents from the
numerical discretization of the equation (20) at the initial time $t=0$,
intermediate time $t=0.5$ and final time $t=1$. We observe that the behavior
of the ABM shown in 2.4 agrees with the emerging dynamics of the PDE model.
Namely, the agents empirical density show the cluster formation that is in
agreement with the ABM and reflects the feedback loop dynamics of the system.
Note that the diffusive behaviour depends on the choice of the scaled
space/opinion strength parameter. In particular, it is expected to have
stronger clustering effects with the increasing of space/opinion strength
parameter. More detailed investigation of this, and the effect of the boundary
conditions will be the topic for the future research.

(a) $t=0$

(b) $t=0.5$

(c) $t=1$

Figure 7: Empirical density of agents in mean-field limit given by equation
(20) at initial time $t=0$, intermediate time $t=0.5$ and final time $t=1$.

##  5 Conclusion

Literature on opinion dynamics is very rich and diverse, spanning from
empirical approaches, that rely on real-world data, to mathematical models,
that mostly consider simple social rules and allow for rigorous analysis.
However, there is a large gap between data- and model- driven approaches, that
asks for introducing new formal models that can better represent complex
social mechanisms governing how people shape and share their opinions. As a
step towards closing this gap, in this paper we introduced an agent-based
model for studying the feedback loop between opinion and social dynamics. This
co-evolving dynamics governs how agents position in a social space (e.g.
online social media) is influencing and is being influenced by other agents
opinions. Additionally, unlike most existing models for opinion dynamics that
consider only deterministic dynamics or additive noise, in order to account
for more realistic scenarios, we introduce the influence of a multiplicative
noise. In order to explore how these different factors influence the
appearance of emerging phenomena in the system, we tested our ABM model on
several toy examples. In particular, we simulated the model for different
parameter choices and we showed how these govern grouping of agents into
spatial clusters, within which agents hold similar opinions. Our experiments
have shown that cluster formation with respect to agents’ social and/or
opinion states is strongly influenced by the feedback loop. Further
investigations of this model and possible effects the feedback loop could have
in real-world social systems, will be the topic of future research.
Additionally, we formulated the feedback loop model in a rigorous mathematical
framework and considered its behaviour in the case when the number of agents
tends to infinity. Although the well-posedness propagation of chaos results
have been proved for these type of systems in various weak regularity
assumptions scenarios, we stated the proofs of these results in the case of
the Lipschitz assumption that are easier to access and present the extension
of the work presented in [27]. Finally, we derived formal equation for the
empirical density of the ABM that is in the spirit of the so-called Dean-
Kawasaki equation, and the equation of its hydrodynamic limit. Motivated by
applications, considering the feedback loop dynamics with more singular
interaction potentials and its numerical analysis would be a natural next step
and will be considered in the future.

## Acknowledgments

The authors would like to thank Christof Schütte, Nicolas Perkowski and Luzie
Helfmann for insightful discussions on ABM, McKean-Vlasov SDE and mean-field
limit. We acknowledge the support of Deutsche Forschungsgemeinschaft (DFG)
under the Germany’s Excellence Strategy - The Berlin Mathematics Research
Center MATH+ (EXC-2046/1, project ID 390685689). A. Dj. has been partially
funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114
”Scaling Cascades in Complex Systems”, Project Number 235221301, Project C10.

##  6 Appendix

### Appendix A.1.

######  proof 1 (Proof of Lemma 1)

This follows from Theorem [55, Theorem 6.18] and the fact that

$\displaystyle d((x,\theta),(y,\eta)):=\sup_{s\leq
T}\left\lVert(x_{s},\theta_{s})-(y_{s},\eta_{s})\right\rVert,\quad(x,\theta),(y,\eta)\in
C([0,T];\mathbb{R}^{d+1}),$

where
$\left\lVert(x_{s},\theta_{s})\right\rVert:=\left(\sum_{k=1}^{d}\left\lvert
x_{s}^{k}\right\rvert^{2}+\left\lvert\theta_{s}\right\rvert^{2}\right)^{2},$
defines a metric on $C([0,T];\mathbb{R}^{d+1})$.

######  proof 2 (Proof of Lemma 2)

For $n\in\mathbb{N}$ consider the stopping time

$\displaystyle\tau_{n}:=\min\left\\{\inf\left\\{t\geq 0:\left\lvert
X_{t}\right\rvert\geq n\right\\},\inf\left\\{t\geq
0:\left\lvert\Theta_{t}\right\rvert\geq n\right\\}\right\\}.$

Then for the stopped process $(X_{t}^{\tau_{n}},\Theta_{t}^{\tau_{n}})_{t\geq
0}$ it holds that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\lVert
X_{t}^{\tau_{n}}\right\rVert^{2}+\left\lvert\Theta_{t}^{\tau_{n}}\right\rvert^{2}\right]\leq\mathbb{E}\left[\left\lVert
X_{0}\right\rVert^{2}+\left\lvert\Theta_{0}\right\rvert^{2}\right]+2n^{2}.$

Via Cauchy-Schwarz and the Burkholder-Davis-Gundy inequality one can now
derive the upper bound

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left\lVert
X_{t}^{\tau_{n}}\right\rVert^{2}+\left\lvert\Theta_{t}^{\tau_{n}}\right\rvert^{2}\right]\leq
C\left(\mathbb{E}\left[\left\lVert
X_{0}\right\rVert^{2}+\left\lvert\Theta_{0}\right\rvert^{2}\right]+K^{2}(T^{2}+T)\right),$

where the constant $C$ only depends on $T$ and $K$. The right hand side of
this inequality does not depend on $n$, so we can conclude that the unstopped
process $X$ also satisfies this inequality by letting $n$ tend to infinity.

######  proof 3 (Proof of Theorem 1)

As already announced, the proof of the existence is based on the fixed point
argument. Let us define a map $\Phi:\mathcal{P}^{2}\to\mathcal{P}^{2}$ by
mapping $\mu\in\mathcal{P}^{2}$ to the law of the solution of the coupled
system of SDEs

$\displaystyle\begin{cases}dX^{\mu}_{t}&=\left(\int_{\mathcal{C}}U(X^{\mu}_{t},y,\Theta^{\mu}_{t},\theta)\mu(dyd\theta)\right)dt+\left(\int_{\mathcal{C}}\sigma_{sp}(X_{t}^{\mu},y,\Theta_{t}^{\mu},\theta)\mu(dyd\theta)\right)dB^{sp}_{t},\\\
\
d\Theta^{\mu}_{t}&=\left(\int_{\mathcal{C}}V(X^{\mu}_{t},y,\Theta^{\mu}_{t},\theta)\mu(dyd\theta)\right)dt+\left(\int_{\mathcal{C}}\sigma_{op}(X_{t}^{\mu},y,\Theta_{t}^{\mu},\theta)\mu(dyd\theta)\right)dB^{op}_{t},\\\
\ X^{\mu}_{t=0}&=X_{0},\\\ \ \Theta^{\mu}_{t=0}&=\Theta_{0}.\end{cases}$

The existence and uniqueness of a solution to this system follows from the
fact that for fixed $\mu$ the drift and diffusion coefficient are Lipschitz
continuous and bounded. Moreover, from standard a priori estimates for
classical SDEs it follows that $\Phi(\mu)\in\mathcal{P}^{2}$.

It remains to show, that $\Phi$ is indeed a contraction on $\mathcal{P}^{2}$.
To see this, fix two probability measures $\mu^{1},\mu^{2}\in\mathcal{P}^{2}$
and let $(X^{1},\Theta^{1}),(X^{2},\Theta^{2})$ be the corresponding
solutions, i.e. for $i=1,2$ we have

$\displaystyle\begin{cases}dX^{i}_{t}&=\left(\int_{\mathcal{C}}U(X^{i}_{t},y,\Theta^{i}_{t},\theta)\mu^{i}(dyd\theta)\right)dt+\left(\int_{\mathcal{C}}\sigma_{sp}(X_{t}^{i},y,\Theta_{t}^{i},\theta)\mu^{i}(dyd\theta)\right)dB^{sp}_{t},\\\
\
d\Theta^{i}_{t}&=\left(\int_{\mathcal{C}}V(X^{i}_{t},y,\Theta^{i}_{t},\theta)\mu^{i}(dyd\theta)\right)dt+\left(\int_{\mathcal{C}}\sigma_{op}(X_{t}^{i},y,\Theta_{t}^{i},\theta)\mu^{i}(dyd\theta)\right)dB^{op}_{t}.\end{cases}$

To estimate $D^{2}_{T}(\Phi(\mu^{1}),\Phi(\mu^{2}))$, we will use that the law
of $(X^{1},\Theta^{1},X^{2},\Theta^{2})$ is actually a coupling of the two
probability measures $\Phi(\mu^{1})$ and $\Phi(\mu^{2})$. Therefore, by
definition of the $2$-Wasserstein distance as the infimum over all couplings
we have

$D^{2}_{T}(\Phi(\mu^{1}),\Phi(\mu^{2}))\leq\mathbb{E}\left[\sup_{s\leq
T}\left\lVert
X^{1}_{s}-X^{2}_{s}\right\rVert^{2}+\left\lvert\Theta^{1}_{s}-\Theta^{2}_{s}\right\rvert^{2}\right].$

We will now set up a Gronwall-type argument to obtain an upper bound of the
right-hand side of this inequality in terms of the $2$-Wasserstein distance of
$\mu^{1}$ and $\mu^{2}$. As a first step, we use Hölder’s and Young’s
inequality to see that for fixed $s\leq T$ we have

$\displaystyle|$ $\displaystyle X_{s}^{1}-X^{2}_{s}|^{2}\leq
2s\int_{0}^{s}\left\lvert\int_{\mathcal{C}}\\!\\!U(X^{1}_{r},y_{r},\Theta^{1}_{r},\theta_{r})\mu^{1}(dyd\theta)-\int_{\mathcal{C}}\\!\\!U(X^{2}_{r},y_{r},\Theta^{2}_{r},\theta_{r})\mu^{2}(dyd\theta)\right\rvert^{2}ds$
$\displaystyle+$ $\displaystyle
2\left\lvert\int_{0}^{s}\left(\int_{\mathcal{C}}\sigma_{sp}(X_{r}^{1},y_{r},\Theta_{r}^{1},\theta_{r})\mu^{1}(dyd\theta)-\int_{\mathcal{C}}\sigma_{sp}(X_{r}^{2},y_{r},\Theta_{r}^{2},\theta_{r})\mu^{2}(dyd\theta)\right)dB^{sp}_{r}\right\rvert^{2}$

where $y_{r},\theta_{r}$ are the projections onto the time $r$ coordinate of
the $d$-dimensional spatial component and the $1$-dimensional opinion
component respectively. We obtain the analogoue estimate for
$\left\lvert\Theta_{s}^{1}-\Theta_{s}^{2}\right\rvert^{2}$.

Utilizing then Doob’s $L^{2}-$inequality to estimate the stochastic part and
Lipschitz-type inequality (2), enable us to apply Gronwall’s lemma which
implies that for all $t\in[0,T]$ we have

$\displaystyle\mathbb{E}\left[\sup_{s\leq t}\left\lVert
X^{1}_{s}-X^{2}_{s}\right\rVert^{2}+\left\lvert\Theta^{1}_{s}-\Theta^{2}_{s}\right\rvert^{2}\right]\leq
C\int_{0}^{t}\mathcal{W}^{2}_{2,\mathbb{R}^{d+1}}(\mu^{1}_{r},\mu^{2}_{r})dr,$
(26)

where the constant $C$ is given explicitly by

$\displaystyle C:=2(8+2T)L^{2}\exp(2(8+2T)L^{2})$

and $\mathcal{W}_{2,\mathbb{R}^{d+1}}$ is the usual $2$-Wasserstein distance
on $\mathcal{P}^{2}(\mathbb{R}^{d+1})$. Next observe that for two measures
$\mu,\nu\in\mathcal{P}^{2}((C([0,T];\mathbb{R}^{d+1}))$ and fixed $r\in[0,T]$
we have

$\displaystyle\mathcal{W}^{2}_{2,\mathbb{R}^{d+1}}(\mu_{r},\nu_{r})\leq
D_{r}^{2}(\mu,\nu).$ (27)

Applying $(\ref{projection-inequality})$ to (26) gives us

$\displaystyle\mathbb{E}\left[\sup_{s\leq t}\left\lVert
X^{1}_{s}-X^{2}_{s}\right\rVert^{2}+\left\lvert\Theta^{1}_{s}-\Theta^{2}_{s}\right\rvert\right]\leq
C\int_{0}^{t}D^{2}_{r}(\mu^{1},\mu^{2})dr.$

Furthermore, since the law of $(X^{1},\Theta^{1},X^{2},\Theta^{2})$ is a
coupling of $\Phi(\mu^{1}),\Phi(\mu^{2})$, the definition of the
$2$-Wasserstein distance (15) implies

$\displaystyle
D^{2}_{T}(\Phi(\mu^{1}),\Phi(\mu^{2}))\leq\mathbb{E}\left[\sup_{s\leq
T}\left\lVert
X^{1}_{s}-X^{2}_{s}\right\rVert^{2}+\left\lvert\Theta^{1}_{s}-\Theta^{2}_{s}\right\rvert\right]\leq
C\int_{0}^{T}D^{2}_{r}(\mu^{1}_{r},\mu^{2}_{r})dr.$

After establishing this inequality, the well-posedness follows from the usual
application of Banach’s fixed point theorem. By Lemma 2 this shows that the
solution is unique in law. Now if we let $\mu$ be the unique law of the
solution, and let it be the drift and diffusion coefficient of (13), then we
have a standard SDE with bounded and Lipschitz continuous drift and diffusion
coefficients. For such systems it is well known, that pathwise uniqueness
holds. Therefore we can conclude that the solution to (13) is not only unique
in law but also pathwise unique.

### Appendix A.2.

######  proof 4 (Proof of Theorem 2)

For fixed $t\in[0,T]$ we have by Hölder’s inequality

$\displaystyle\left\lVert X^{i,N}_{t}-\overline{X}^{i}_{t}\right\rVert^{2}$
$\displaystyle\ \leq$ $\displaystyle
2t\int_{0}^{t}\left\lVert\int_{\mathcal{C}}U(X^{i,N}_{s},y,\Theta^{i,N},\theta)\nu_{N}(dyd\theta)-\int_{\mathcal{C}}U(\overline{X}^{i}_{s},y,\overline{\Theta^{i,N}},\theta)\mu(dyd\theta)\right\rVert^{2}ds$
$\displaystyle\ +$ $\displaystyle
2\left\lVert\int_{0}^{t}\left(\int_{\mathcal{C}}\sigma_{sp}(X^{i,N}_{s},y,\Theta^{i,N},\theta)\nu_{N}(dyd\theta)-\int_{\mathcal{C}}\sigma_{sp}(\overline{X}^{i}_{s},y,\overline{\Theta^{i,N}},\theta)\mu(dyd\theta)\right)dB^{sp}_{s}\right\rVert^{2}.$

We obtain the analogue estimate for
$\left\lvert\Theta^{i,N}_{t}-\overline{\Theta}_{t}^{i}\right\rvert^{2}$.

Similarly as before, taking the supremum of previous estimates and integrating
with respect to $\mathbb{P}$ and Doob’s $L^{2}$-inequality and the Lipschitz
type inequality (2) we obtain

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq t}\left\lVert
X^{i,N}_{s}-\overline{X}^{i}_{s}\right\rVert^{2}+\left\lvert\Theta^{i,N}_{s}-\overline{\Theta}^{i}_{s}\right\rvert^{2}\right]$
$\displaystyle\ \leq$ $\displaystyle
2(2t+8)L^{2}\mathbb{E}\left[\int_{0}^{t}\left\lVert
X^{i,N}_{s}-\overline{X}^{i}_{s}\right\rVert^{2}+\left\lvert\Theta^{i,N}_{s}-\overline{\Theta}^{i}_{s}\right\rvert^{2}+\mathcal{W}_{2,\mathbb{R}^{d+1}}^{2}(\mu^{N}_{s},\mu_{s})ds.\right],$

where $\mathcal{W}_{2,\mathbb{R}^{d+1}}(\cdot,\cdot)$ is the usual
$2$-Wasserstein metric on the set of probability measures on
$\mathbb{R}^{d+1}$ with finite second moment. The application of Gronwall’s
inequality and the analogue arguments as in the proof of Theorem 1, yield the
estimate

$\displaystyle\mathbb{E}\left[\sup_{0\leq s\leq t}\left\lVert
X^{i,N}_{s}-\overline{X}^{i}_{s}\right\rVert^{2}+\left\lvert\Theta^{i,N}_{s}-\overline{\Theta}^{i}_{s}\right\rvert^{2}\right]\leq
C\mathbb{E}\left[\int_{0}^{t}D_{s}^{2}(\mu,\mu^{N})ds\right].$

By construction we know that
$(\overline{X}^{i},\overline{\Theta}^{i})_{i\in\mathbb{N}}$ is an $i.i.d.$
family with $\text{Law}(\overline{X}^{i},\overline{\Theta}^{i})=\mu$ for all
$i\in\mathbb{N}$. We define the empirical measure of the
$(\overline{X}^{i},\overline{\Theta}^{i})$’s by

$\displaystyle\overline{\mu}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{(\overline{X}^{i},\overline{\Theta}^{i})}.$
(28)

Note that the random measure
$\frac{1}{N}\sum_{i=1}^{N}\delta_{(X^{i,N},\Theta^{i,N},\overline{X}^{i},\overline{\Theta}^{i})}$
is a random coupling of the random measures $\mu^{N}$ and
$\overline{\mu}^{N}$. Therefore the definition of the truncated
$2$-Wasserstein distance gives us the pathwise inequality

$\displaystyle
D_{t}^{2}(\mu^{N},\overline{\mu}^{N})\leq\frac{1}{N}\sum_{i=1}^{N}\sup_{0\leq
s\leq t}\left(\left\lVert
X^{i,N}_{s}-\overline{X}^{i}_{s}\right\rVert^{2}+\left\lvert\Theta^{i,N}_{s}-\overline{\Theta}^{i}_{s}\right\rvert^{2}\right).$
(29)

Hence, by our previous considerations we have

$\displaystyle\mathbb{E}\left[D_{t}^{2}(\mu^{N},\overline{\mu}^{N})\right]\leq
C\mathbb{E}\left[\int_{0}^{t}D_{s}^{2}(\mu,\mu^{N})ds\right].$ (30)

Moreover, combining this with the triangle inequality for the truncated
$2$-Wasserstein distance (15), we obtain

$\displaystyle\mathbb{E}\left[D_{t}^{2}(\mu,\mu^{N})\right]$
$\displaystyle\leq
2\mathbb{E}\left[D_{t}^{2}(\mu,\overline{\mu}^{N})\right]+2C\mathbb{E}\left[\int_{0}^{t}D_{s}^{2}(\mu,\mu^{N})ds\right].$

Now we can apply Gronwall’s inequality and take $t=T$ to get

$\displaystyle\mathbb{E}\left[D^{2}_{T}(\mu,\mu^{N})\right]\leq
2e^{2CT}\mathbb{E}\left[D_{T}^{2}(\mu,\overline{\mu}^{N})\right],$

where the right hand side tends to $0$ as $N\to\infty$ by the LLN for
empirical measures of i.i.d. random variables, cf. [53, Theorem 11.4.1].
Since this LLN only gives us the almost sure convergence of the random
measures $\mu^{N}\to\mu$, it remains to show that the family
$(D_{T}^{2}(\mu,\mu^{N}))_{N\in\mathbb{N}}$ is uniformly integrable.

This can be shown by observing that

$\displaystyle D_{T}^{2}(\mu,\overline{\mu}^{N})\leq
2D^{2}_{T}(\mu,\delta_{0})+2\mathcal{W}^{2}_{2}(\delta_{0},\overline{\mu}^{N})\leq
2D^{2}_{T}(\mu,\delta_{0})+2\frac{1}{N}\sum_{j=1}^{N}\sup_{s\leq
T}\left\lVert(\overline{X}^{j}_{s},\overline{\Theta}^{j}_{s})\right\rVert^{2},$
(31)

where $\delta_{0}$ is the Dirac zero measure on $C([0,T];\mathbb{R}^{d+1})$.
The right hand side of this inequality is uniformly integrable, since the
first term does not depend on $N$ and the family
$(\frac{1}{N}\sum_{j=1}^{N}\sup_{s\leq
T}\left\lVert(\overline{X}^{j}_{s},\overline{\Theta}^{j}_{s})\right\rVert^{2})_{N\in\mathbb{N}}$
is uniformly integrable as the empirical average of i.i.d. copies of an
integrable random variable. Then the uniform integrability of the empirical
averages follows directly from the definition.

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