Modelling interactions between active and passive agents moving through heterogeneous environments
We study the dynamics of interacting agents from two distinct inter-mixed populations: One population includes active agents that follow a predetermined velocity field, while the second population contains exclusively passive agents, i.e. agents that have no preferred direction of motion. The orientation of their local velocity is affected by repulsive interactions with the neighboring agents and environment. We present two models that allow for a qualitative analysis of these mixed systems. We show that the residence times of this type of systems containing mixed populations is strongly affected by the interplay between these two populations. After showing our modeling and simulation results, we conclude with a couple of mathematical aspects concerning the well-posedness of our models.
Key words: Crowd dynamics; lattice gas model; fire and smoke dynamics; particle methods; heterogeneous domains.
PACS: 02.70.Uu, 07.05.Tp, 05.06.-k.
MSC 2010: 65Z05, 82C80, 91E30.
Unlike fluid flows, pedestrian flows are rarely uniform. Hence, their motion is difficult to predict accurately. The main source of non-uniformity stems from the fact that pedestrian flows are “thinking flows”, i.e., both agent-agent interactions and agent-structure interactions are always active and are much more complex than the standard Van der Waals-like (attraction-repulsion) interactions which govern to a large extent the molecular description of fluids and gases. In this framework, we consider a particular type of non-uniformity. Looking at a heterogeneous environment (e.g. a complex office building), we consider our target pedestrian flow to contain the dynamics of interacting agents from two distinct populations:
active agents, knowing where to go (they are aware of a predetermined optimal velocity field leading towards the exits),
passive agents, randomly exploring the environment (they have no information about the exit routes, but base their motion on interaction with other agents).
We are particularly interested in investigating what mechanisms can be responsible for the minimization of the residence time of the pedestrians when an emergency evacuation situation has occurred, for instance, due to the unexpected occurrence of a fire that produces a significant amount of smoke. Our standing assumption is that the use of a purely macroscopic crowd model, which encodes the motion of a uniform flow, is prone to underestimate the residence time and does not properly capture crowd interaction.
In this chapter, we present conceptually different crowd dynamics models that describe the joint evolution of such passive and active agents. One of the models employs systems of nonlinear stochastic differential equations of motion one-way coupled with the diffusive-convective dynamics of the smoke, while an other model is a lattice-gas-type approach based on a Monte Carlo stochastic dynamics. Both models give estimates of the residence time of the particles as well as of the local occupancy (local pedestrian densities). When treating such scenarios, the complexity of the work is high. One of the difficulties is the handling of agent-structure interaction. It is worth noting that even if all agents were active and their wanted path is known a priori, if their number is sufficiently high, given a certain prescribed internal geometry of the facility, the agent-agent and agent-structure interactions normally lead to clogging or to the faster-is-slower effect; see e.g. Zuriguel2014 () and Garcimartin2015 (). Another difficulty is to handle the presence of the fire, and consequently, of the smoke and of the increased discomfort the agents feel. We refer the reader to OmarMSC () for one possible way of treating the presence of obstacles and to Omar2017 () for hints on how to introduce the fire physics in the evolution equations describing the dynamics of the crowd. In this framework, we focus exclusively on the effect of knowledge of the geometry on the actual dynamics of the agents.
After reviewing a number of relevant related contributions, we proceed with the description of two closely-related modeling scenarios where the type of models previously mentioned apply. Then we solve the models numerically and illustrate the typical behavior of the output: positions, residence times, discomfort values, etc. We also discuss a few basic aspects concerning the mathematical well-posedness of one crowd model related to Model 1. We close the chapter with a discussion section where we also include hints towards further potential contributions in this context. The results reported here should be seen as preliminary. More efforts are currently invested to develop these research directions.
2 Related contributions
Escape evacuation and social human behaviour are closely connected. In an emergency situation, building occupants require information about the surrounding environment and social interactions in order to evacuate successfully. The experiments in Horiuchi1986 () can serve as a typical example for the relevance of distinguishing between two groups of occupants: regular users of the building and those less familiar with it.
In the model presented in Chu2014 (), the building occupants are modelled as agents who decide their evacuation actions on the basis of their infrastructural knowledge and their interactions with the social groups and the neighboring crowd therein. The authors showed that both familiar agents with the geometry building and social influence can dramatically impact on egress performance. As a multi-agent evacuation simulation tool, the ESCAPES system (presented in Tsai2011 ()) describes a realistic spread of knowledge to model two types of different knowledge: exit knowledge together with event knowledge. The conclusions made based on these models are supported by experimental findings such as those reported in ronchi17 (), where an evacuation was performed and the exit choice of participants was investigated, as well as the effect of the evacuation geometry.
Commonly, agent-based crowd models are based on developing individual trajectories. Yet for dense crowds, additional dynamics come into play. This has been observed in, for instance, corbetta14 (), where the interaction in dense crowds has been measured and analyzed, obtaining statistics for aggregate dynamics. These macroscopic properties have been observed from a theoretical perspective as well in e.g. luding07 (). One way to bridge the gap between models for regular and dense crowds is to use models defined on different spatial scales, giving rise to a so-called multiscale model. In cristiani11 (), a multiscale model is proposed in terms of a granular flow formulation to display both microscopic as well as macroscopic crowd behaviour. For an investigation of handling contacts in such flows of granular matters applied to crowds, we refer the reader to the work of Maury and co-authors, compare Faure2015 (). All these papers assume that the exits are visible. For study cases when the walking environment is not visible due to the lack of light, we refer the reader to Cialela (). There the main question is whether the grouping of the agents (involving higher coordination costs and information overload) has a chance to favorize an eventually quicker evacuation. From a different perspective, interesting connections to crisis management issues are made in Bellomo2016_1 () and references cited therein.
3 Agent-based dynamics (Model 1)
In this section we introduce an agent-based model in a continuous two-dimensional multiple connected region , containing obstacles with a fixed location, a fire that produces smoke, and an exit. represents the environment in which the crowd is present and tries to find the fastest way to the exit, avoiding any obstacles and the fire. The crowd is represented by the two aforementioned groups, active and passive agents. At time , the crowd starts to evacuate from . In the rest of this section, refers to the geometry displayed in Figure 1.
Active agents have a perfect knowledge of the environment and the locations of the obstacles, but are not aware of the location of the fire prior to experiencing sensory cues. Passive agents have no information on the environment and follow their neighbours to reach the exit. A similar model as the one described below has been presented in Omar2017 ().
Active and passive agents are seen as members from the sets and , respectively. The dynamics governing their motion are described in the following sections.
3.1 Active agents
The motion of the active agents is governed by a potential field model proposed by Hughes in hughes02 () and adapted in treuille06 (). It functions similarly to a floor field function, its counterpart in lattice models presented in e.g. tan15 () and cao2014 (). We modify the potential field model to account for the presence of obstacles and the effects of fire and smoke.
The potential field agrees with the principle of minimization of effort, serving as a dynamic generalized distance transform. Let be an arbitrary point selected in . We introduce a marginal cost field , defined as
The marginal cost field represents the effort of moving through a certain location and consists of a base level of constant walking effort , information on the geometry and the obstacles , and information on the fire source . Here, takes value 1 if the agent is aware of the location of the fire and 0 otherwise.
Let be a path going from point to point . Then the effort of walking on the path can be expressed as
At the beginning of the simulation, is 0 for all agents. When an active agent experiences a significant increase in temperature because of his proximity to the location of the fire, is set to 1 and changes, and as a result, the fire is avoided.
Let be the set of all inaccessible locations in the geometry (i.e. those parts of covered by obstacles). Then for all , the geometry information (i.e. the obstacle cost field) can be expressed as
where is a parameter of the order of the size of the agents. The obstacle cost makes sure that obstacle locations are inaccessible, and adds a tiny layer of repulsion around each obstacle to ensure the basic fact that agents do not run into walls.
The preferred path for an agent with location and motion target is determined as
where we minimize over the set of all possible motion paths from to . In this framework, the active agents are aware of all exits, and the optimal path is made available by means of the potential function , a solution to the equation
where denotes the standard Euclidean norm. Passive agents do not have access to the optimal paths.
Figure 3 and Figure 3 display the potential field and the corresponding paths for our case study. Figure 5 and Figure 5 display the adaption active agents make as soon as they become aware of the fire locations and take an alternative route out.
Let denote the position of active agent at time . We express their motion within the geometry by
where represents the initial configuration of the active agents and represents a predefined walking speed. In (3), represents a given discomfort term that influences agent interactions at the macroscopic scale. The discomfort measures how much agents locally have to deviate from their ideal velocity. We are on purpose vague concerning this macroscopic discomfort. In a follow-up publication, we will tackle a multiscale non-uniform crowd model where will be part of the solution to a macro-micro flow problem.
3.2 Passive agents
Since we assume in this context that passive agents are unfamiliar with their environment, it is reasonable to postulate that to obtain information, they rely solely on neighbouring agents. This is a modelling assumption which has been confirmed for e.g. primates in meunier06 (). This idea has already been applied in other crowd dynamics models (cf. e.g. helbing00 ()).
To model this strategy, we choose to apply a Cucker-Smale-like model which averages the velocity of nearby agents (an idea introduced originally in cucker07 ()). A Brownian term is added to this swarming-like model to represent disorienting and chaotic effects which inherently appear while moving through an unknown environment.
We denote the positions and velocities of agent from population as and , and positions and velocities from member of the complete set as and , respectively.
We express the motion of passive agents in the following way
In (4), are weight factors, decreasing as a function of distance, defined as
In (5), is the sight radius in the agents’ location, affected by smoke level (see Section 3.3). It should also be noted that we do not take into account those walls that block the transfer of information between agents, since they are ignored in (5). However, in the simulations described in the next section, the size of the walls generally exceeds the size of the interaction radius. The term is simply an a priori known normalization factor depending of the smoke level; one can take just for simplicity. In the context of (5), the gradient in the discomfort level111Here, we assume that the discomfort is perceptible, known. bounds asymptotically the speed of the passive agents.
Note that, based on (4), passive agents follow a set of coupled second-order differential equations (a social force-like model), while following (3), the active agents are expected to respect a set of coupled first-order differential equations (a social velocity-like model). We believe that the ’social inertia’ is much higher in the case of passive agents, so we keep the classical Langevin structure of the balance of forces, while for the active agents we choose an overdamped version.
Another important observation is that in this model, passive agents do not know which of the other agents are active, and which are passive themselves; they follow others indiscriminately.
3.3 Smoke effects
In addition to repelling the agents, the fire produces smoke which propagates in and reduces the visual acuity of the agents. The creation and propagation of the smoke is modelled as a diffusion-dominated reaction-advection-diffusion process.
The smoke density , is assumed to respect the following equation
where represents the smoke diffusivity, determined by the environment, is the outer normal vector to , is a given drift corresponding to, for instance, ventilation systems or indoor airflow, while encodes the shape and intensity of the fire, viz.
In our context, is the molecular diffusion coefficient for the smoke and a slight space dependence in is allowed. At a later stage, maybe eventually also an -dependence of can be foreseen, if one would replace (6) by an averaged version where the free motion paths and the geometry are perceived as some sort of “homogenized” porous medium.
Figure 6 illustrates a snapshot of the smoke density for our case study.
4 Results Model 1: Agent-based dynamics
This section contains our numerical results obtained using the agent-based dynamics described in the previous section.
The results are run in crowd simulation prototyping application Mercurial (mercurial ()). This is an open-source framework developed in Python and Fortran to simulate hybrid crowd representations as the one described in Section 3. It provides both agent-based- and continuum-level visualizations and supports the design of arbitrary two-dimensional geometries. More details on the structure and implementation of Mercurial are found in OmarMSC ().
Figure 1 shows the geometry of our case study. It has a fairly simple structure to ensure the exit can be reached even without environment knowledge. However, the placement of the obstacles is such that zones of congestion easily occur and paths to the exit will necessarily have to be curved.
The simulation was run twice with 1000 agents, varying the ratio between active and passive agents. The first run (Case 1), of which a snapshot is presented in Figure 8, contains a total of 800 active agents and 200 passive agents. The second run (Case 2), illustrated with a snapshot in Figure 10, contains a total of 200 active agents and 800 passive agents.
Figure 8 and Figure 10 illustrate a coherent discomfort field in an ongoing simulation due to a large number of agents with conflicting directions. The corresponding agents configuration (i.e. their spatial distribution) is displayed in Figure 8 and Figure 10. Visible is that close to the fire a lot of discomfort is generated. The main cause for this congestion is the conflict between active agents who have identified the location of the fire and want to move in different directions and active agents which are still unaware and want to exit the geometry through that particular corridor. This reminisces of the panic zone that occurs in crowd disasters close to the origin of the panic. Notice how this zone is much more present in Case 1 than in Case 2, due to the lack of active agents in Case 2. While the passive agents take a lot longer to reach the exit, their following-dominated behaviour amounts to less discomfort in doing so.
It would be interesting to have a partial differential equation describing at least approximately the macroscopic space-time evolution of such discomfort field available. Also, such an object would be very useful from a practical point of view – it would allow a fast detection of zones of high discomfort, which could be helpful in taking management decisions to reduce the potential of risks and accidents.
In Case 1 (Figure 8), we observe that all agents belong to a collective moving towards the exit, regardless of population.
As one would expect, Case 2 (Figure 10) displays less order than Case 1. Most agents move in smaller groups, either guided by active agents or randomly moving throughout the geometry.
Figure 12 and Figure 12 depict the agents leaving the environment as a function of time. In Figure 12 we observe three stages: the first stage (from to ) corresponds to the group of active agents that exit the geometry without any obstructions, guiding most of the passive agents while doing so. The second stage (from to ) has virtually no agents that reach the exit; all the remaining agents are trapped in the high discomfort panic-like zone close to the fire. The third stage shows the final active agents have escaped the panic zone, reaching the exit.
Figure 12 displays a similar first stage, but because the discomfort zones are a lot less intensive, there is no pronounced second and third stage.
Notice how after the bulk of the active agents have left the geometry, the egress of the passive agents has reduced to a random walk.
These observations are supported by Figure 14 and Figure 14, where the cumulative discomfort for each location in is displayed. Case 1 (Figure 14) shows significantly higher discomfort both near the fire and where the geometry narrows itself. Case 2 (Figure 14) shows a much higher usage of the space in the geometry, i.e. agents walking in locations that do not belong to any shortest path. However, the high discomfort zones are an order of magnitude lower than in Case 1, due to the ‘flexibility’ of passive agents.
Concluding, simulations support the following observations. Differences in environment knowledge can have a significant impact on several aspects on the dynamics of crowds in e.g. evacuations. While it is true that additional knowledge decreases evacuation time, the autonomy of active agents can cause problems when their information turns out to be incorrect. When steering passive agents it is significantly more difficult to maintain order in the evacuation, but the fact that they can be guided can relieve discomfort and reduce congestion.
5 Lattice gas dynamics (Model 2)
The second model we shall tackle here is a lattice gas model. Namely, we consider a Simple Exclusion Process (SEP) kipnis1998 () on a two-dimensional lattice :
According to the basic tenets of the SEP dynamics, there can be only one particle per site, and particles jump independently towards one of the nearest neighbor sites on the lattice, provided that the arrival site is empty.
We shall hereafter assume that the system is closed, namely particles may not hop outwards from any of the boundary sites of , except from a subset of lattice sites , called the “exit door”: any particle located in and hopping upwards is annihilated. Note that particles may just leave the system through the exit door: no inward flux of particles is considered in this model. The numerical investigation of the lattice gas model aims, indeed, to shed light on the characteristic time scales characterizing the particle evacuation from the system.
As in the case of Model 1, we distinguish between two species of particles, namely aware or active particles, and, respectively, unaware or passive particles. For simplicity of the notation, we shall refer to them as particles “” and “” in this section. While the species performs a symmetric simple exclusion dynamics on the lattice, particles of the species experience both a horizontal and a vertical drift, denoted below as and , that enhance the rates at which particles of such species hop towards the exit door. The microscopic dynamics is defined as follows. Call and the occupation number on the site (which is either or ) of the species, respectively, and . Given two nearest neighbor sites , , such that the bond joining to is entirely contained in , we define the hopping rate from to of a particle of the species (no matter if the jump occurs along a horizontal or a vertical bond) as:
To define the corresponding hopping rate for particles of the species we shall distinguish between vertical and horizontal bonds. For the vertical bonds, i.e. when , we set:
for bonds directed upwards and downwards, respectively. The microscopic dynamics of the species , ruled by Eq. (10), highlights the intrinsic bias of the species to move upwards, i.e. towards the exit, and prevents any redundant vertical motion in the opposite direction. Moreover, (10) also includes a drift term that marks the tendency of the species to reach the exit door with a rate that is higher than the unitary rate defining the unbiased dynamics of the species , see Eq. (9). For the horizontal bonds, i.e. when , we shall first consider the case , for which we set:
The presence of a horizontal drift term , in Eq. (11) reminds us that particles of the species , unlike particles of the species , move preferably towards the right (resp. the left), when the horizontal coordinate of the departure site is (resp. ). Instead, when and , we set:
Equation (12) says that, when , particles of the species may only hop upwards, namely they point directly towards the exit door without wandering along the horizontal direction.
The rates associated to those bonds joining any boundary site of , that is not part of the exit door, to any external site, are all set equal to 0. Finally, the rates associated to the vertical bonds joining a site with a site , are defined as follows:
The proposed lattice gas model also accounts for the presence of fixed obstacles inside the domain, that correspond to a subset of lattice sites that are inaccessible, represented by the black spots shown in Figure 15. To complete the description of the microscopic dynamics we shall, hence, also set equal to zero all the rates associated to those bonds joining two sites, one (or even both) of them belonging to .
The study of the evacuation of particles shall be pursued by considering, for each species, the behavior of the number of particles and the particle current (through the exit door) as a function of time. The particle current is defined as
where and denote, respectively, the number of particles of a given species at the initial time and at the time .
6 Results Model 2: Lattice gas dynamics
This section contains our preliminary numerical results obtained using Model 2.
The dynamics was implemented by running a set of Kinetic Monte Carlo (KMC) simulations (Landau:2005 ()). KMC methods are notoriously suited to describe transient phenomena, in which physical time plays a crucial role in the microscopic evolution (Bortz (); Voter ()). Note that, denoting by the number of time steps considered in the KMC simulation, the physical time , considered in (14), is obtained as , where each (corresponding to the time elapsed between two consecutive particle jumps on the lattice) is an exponentially distributed random variable with a parameter given by the sum of all the rates associated to the lattice bonds, defined in Section 5, see refs. CM (); CCM () for details.
The results of the KMC simulations for this model are portrayed in Figure 15 and Figure 16. In Figure 15 we show the microscopic configurations of the particles and at different times, namely from the initial configuration (top left panel) until the time when the evacuation of particle is essentially completed (cf. the bottom right panel, corresponding to some time steps of the dynamics).
In the left panel of Figure 16, we show the effect of varying the width of the exit door as well as the drifts and on the total number of particles and as a function of time. Clearly, by increasing the width of the exit door, particles of both species evacuate with a higher pace. The left panel of Figure 16 also highlights an interesting effect that is obtained by varying and : an increase of the drift terms induces a higher evacuation rate for particles of the species , and also, consequently, for particles of the species , which have access to a larger number of empty sites on the lattice. In the right panel of Figure 16 we show the behavior of the particle current, defined in (14), for the two species and as a function of time, for fixed values of the parameters , and . The higher evacuation rate observed for the aware particles stems directly from the definition of the rates for the two species and , given in section 5.
7 Mathematical aspects of social dynamics in mixed populations
In this section, we discuss the solvability of a social dynamics model of mixed populations, resembling an overdamped version of Model 1. Note that Model 2 is well-posed by construction. Here, interesting questions would be pointing towards the rigorous derivation of the corresponding hydrodynamic limit equations DMP (), and/or the numerical evaluation of non-equilibrium collective effects (e.g. the inclusion of a reaction mechanism within the microscopic dynamics, allowing particles to switch from one species to the other, or the presence of long-range interactions between particles), but these aspects are not in our focus for the moment.
This section contains a couple of technical preliminaries needed to state the evolution problem in a functional analytic framework. We use standard methods to handle the well-posedness of a coupled set of SDEs for the agents dynamics, also linked to a linear parabolic equation describing the motion of the smoke.
7.1 Technical preliminaries, notation and assumptions
We consider a two dimensional domain, which we refer to as . This domain presents the geometry of the evacuation scenario. In addition, as a building geometry, parts of the domain are filled with obstacles ( and ) denoted by and the fire denoted by . Moreover, the domain has exits denoted by . Let for and be , or at least satisfying the exterior sphere condition. A typical example of such is depicted in Figure 17.
The space , , is endowed with the norm
while for the space we consider the norm
Our analysis of the stochastic differential equations (SDEs) describing the evolution of our populations follows the line of reasoning from Flandoli95 () and Prato14 () (the compactness method of SPDEs and martingale solutions). We refer to Evans2013 () and Pavliotis2014 () for the basic concepts and usual notations.
Let be a continuous-time stochastic process. We define the family of laws
This is a family of probability distribution of .
Recall the classical Ascoli-Arzelà theorem:
A family of functions is relatively compact (in uniformly topology) if
for every , the set is bounded.
for every and there is such that
whenever for all .
We introduce the definition of Hölder seminorms, for as
and the supremum norm as
Using Ascoli-Arzelà theorem, starting from the facts:
there is such that for all .
for some , there is an such that for all ,
we infer that the set
is relatively compact in .
For , and , the space is defined as the set of all such that
This space is endowed with the norm
Moreover, we know that if , then
and . Relying on the Ascoli-Arzelà theorem, we have the following situation:
for some and with , there is such that for all .
If i’ and ii” hold, then the set
is relatively compact in , if .
In this framework, we require the following assumptions:
(see also section 7.3).
(bounded maximal discomfort).
The smoke matrix diffusion coefficient satisfies the uniform ellipticity condition, i.e. there exists positive constants such that
The smoke interface exchange coefficient on the boundary of our domain is such that there exist positive constants satisfying
Changing the functional framework will naturally lead to a reconsideration of these assumptions.
First-order social agents dynamics
We focus on the interaction between two groups of pedestrians, one familiar (active agents) and one unfamiliar (passive agents, visitors) with the geometry. To keep the presentation simple, we decide to tackle here the case when both active and passive agents follow a first-order dynamics (overdamped Langevin equations). To this end, we modify the dynamics of the passive agents, deviating this way from Model 1.
Let denote the position of the agent from group at time . The crowd dynamics in group A is expressed by the first-order differential equation encoding optimal environment knowledge within the domain , viz.
where is a discomfort threshold proportional to the overall population size, say , with and is the local discomfort (realization of social pressure) so that
In (22), is the Dirac (point) measure and is a ball center x with small enough radius such that . Hence, the discomfort represents a finite measure on the bounded set . In addition, we assume the following structural relation between the smoke extinction and the walking speed:
where are given positive numbers. Note that every member of this group wants to follow the motion path explicitly given by (with the potential function solving the Eikonal equation), which minimizes the distance between particle positions and the exit location .
As mentioned before, concerning the second population, since the agents do not know the geometry, they must rely on the information from their neighbour. The unfamiliarity with the local environment is expressed here by means of a Brownian motion term . Moreover, the passive agents like to be stay away the fire – for this to happen we use a repulsion term pinpointing to the location of the fire source . Hence, the dynamics is described as a stochastic differential equation as follows
Here is the constant diffusion coefficient matrix, while , and is a weight factor decreasing as a function of distance. They are defined as
In (24), is the sight radius in the evacuees location. Since is in general not differentiable everywhere (cf. e.g. (51)), in order to be able to take the gradient of , we consider from the start a mollified , say . Furthermore, note that can in principle also depend on the space position. This way the random effects can be skipped in the regions where the geometry is not available for walking. It is worth noting that we have many ways to express how the active agents sense the fire. We choose here to introduce the fire location as a region to be avoided and impose it in the definition domain of the Eikonal equation. It is worth comparing this model for the evolution of the passive agents and the one prescribed in Model 1. Notice here the following important aspects: not only the dynamics is over-damped, but also the expression of the social velocity is slightly adapted to avoid an implicit definition.
Our evolution system consists of an ODE (21) coupled to an SDE (23). Therefore, due to the randomness incorporated in the SDE (23), the ODE becomes an SDE after coupling. So, we can consider (21) and (23) as a SDE system. Note that this system is one-way coupled with the reaction-diffusion-drift equation describing the smoke evolution.
Furthermore, we set
and initial datum
In this section, we use the compactness method for proving the existence of solutions; we follow the arguments by G. Da Prato and J. Zabczyk () (cf. Prato14 (), Section ) and a result of F. Flandoli (1995) (cf. Flandoli95 ()) for martingale solutions. The starting point of this argument is based on considering a sequence of solutions of the following stochastic differential equation
To ensure the applicability of the compactness argument, we need the following structural assumptions:
be a consequence of continuous functions and uniformly Lipschitz in .
be equi-bounded .
It is not difficult to see that in our case, Assumptions () and () are fulfilled. By from Remark 1, we have Lipschitz in . Moreover, by the Assumption , we obtain is Lipschitz for . On the other hand, the term is a finite measure on bounded sets – it is automatically Lipschitz. These considerations lead to the fact that is Lipschitz in . In addition, by together with taking (as a mollified ) implies that is uniformly Lipschitz in . By the formula (24), the weight factors are Lipschitz in . Thus, inherites the Lipschitz property. Clearly, from these arguments, we obtain not only that and are Lipschitz, but also that these functions are equibounded and . Hence, we have satisfying both assumptions () and ().
The compactness argument proceeds as follows. We begin with solutions of the system (21) and (23), describing in (31). The construction of these solutions can be investigated on a probability space with a filtration and a Brownian motion . Next, let be the laws of which is defined cf. (15). Then, by using Prokhorov’s theorem, we show that the sequence of laws is weakly convergent to in . Then, by using the “Skorohod representation Theorem”, the weak convergence is in a new probability space with a new stochastic process, for a new filtration. This leads to some arguments for weak convergence results of two stochastic processes in two different probability spaces that we need to use to obtain the existence of our SDE system. Finally, we prove the uniqueness of solutions to our system.
Let us start with handling the tightness of the laws through the following lemma.
Assume () and () hold. The family of is tight in
In order to prove the tightness, let us recall the following compact sets (as in the preliminaries section 7.1):
Now, we will show that for a given , there are such that
This means that
A sufficient condition is
Now, we consider the first one . Using Markov’s inequality (cf. Jacod2004 (), Corollary 5.1), we get
Since bounded, then we have
Taking the expectation, we have the following estimate
Hence, for , we can choose such that .
From now on, we consider the second inequality in (32). This reads
Let us introduce another class of compact sets now in the Sobolev space (which for suitable exponents lies in ). Additionally, we recall the relatively compact sets in (20) such that
A sufficient condition for to be relatively compact in the underlying space is . Having this in mind, we wish to prove that there exist and with together with the property: given , there is such that
for every .
Using Markov’s inequality, we obtain
For , we have
Let us introduce some further notation. For a vector , we set . At this moment, we consider the following expression: