Stochastic partial differential equation models for highway traffic

UNIVERSITY OF CALIFORNIASanta Barbara Stochastic Partial Differential Equation Models for Highway Traffic A Dissertation submitted in partial satisfaction of the requirement for the degr

UNIVERSITY OF CALIFORNIA

Santa Barbara

Stochastic Partial Differential Equation Models for Highway Traffic

A Dissertation submitted in partial satisfaction of the requirement for the degree

of Doctor of Philosophy in Mathematics

by

Gunnar Gunnarsson

Committee in charge:

Professor Guillaume Bonnet, Committee Chairman

Professor Michael Crandall, Committee Chairman

Professor Bj¨orn Birnir

September 2006

UMI Number: 3233741

3233741 2006

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by ProQuest Information and Learning Company

The dissertation of Gunnar Gunnarsson is approved

Bj¨orn Birnir

Michael Crandall, Committee Chairman

Guillaume Bonnet, Committee Chairman

September 2006

Stochastic Partial Differential Equation Models for Highway Traffic

Copyright c

byGunnar Gunnarsson

To my family

Academically, I would like to especially thank professor Guillaume Bonnetfor his endless support and help while I often found myself lost in the world ofstochastics I am endebted to professor Michael Crandall for offering his help, at

a crucial time, when I switched fields of research and to professor Bj¨orn Birnir forgetting me in touch with Guillaume and for his guidance and assistance in manyother matters

Lastly, I thank the wonderful staff and faculty at the department of matics as well as my fellow graduate students My stay here, would certainly nothave been this nice if it were not for the good times I spent in Medina Teel’s office,

mathe-or with Bill Lyons in our shared office mathe-or on the balcony Also, I might have pletely lost touch with the world of soccer, were it not for professor Darren Long.Finally, I thank professor Daryl Cooper for selflessly taking over my teaching load

com-in time of need Thank you all

Vita of Gunnar Gunnarsson

Education

PhD Mathematics, Sept 2006, University of California, Santa Barbara

MA Mathematics, March 2002, University of California, Santa Barbara

BSc Mathematics, May 2000, University of Iceland, Reykjavik

Work Experience

Lecturer and Teaching Assistant, 2000 - 2006, UC, Santa Barbara, USA

Researcher, 2002, Decode Genetics, Reykjavik, Iceland

Programmer, 1998 - 2000, K¨ogun hf., Reykjavik, Iceland

Programmer, 1999, RISC, Research Inst of Symbolic Computing, Linz, AustriaTeaching Assistant, 2001 - 2003, University of Iceland, Reykjavik, IcelandAwards and Fellowships

Graduate Division Dissertation Fellowship, Fall 2005

UCSB Affiliates Dissertation Fellowship, Fall 2005

Memorial Fund of Helga Jonsdottir, Fall 2004

Teaching Award, Spring 2004

Thor Thors Fellowship, Spring 2001

Raymond L Wilder Award, Spring 2001

Fulbright Fellowship, June 2000

Stochastic Partial Differential Equation Models for Highway Traffic

byGunnar Gunnarsson

We introduce a new model for multi-lane highway traffic, based onstochastic partial differential equations We prove that the model iswell-posed; has one and only one solution We prove the existenceconstructively and thus derive a numerical scheme to compute thesolution

We examine measured traffic data and introduce a new method andalgorithm to estimate the fundamental diagram, an integral part ofalmost every macroscopic highway traffic model

1.1 Introduction 1

1.2 Notation 3

1.3 Models 4

1.3.1 Macroscopic Models 4

1.3.2 Microscopic Models 13

2 SPDE Model 18 2.1 Introduction 18

2.2 The SPDE Model 23

2.3 Discrete System 26

2.4 Convergence to the SPDE System 35

2.4.1 Mild Formulation 36

2.4.2 Tightness 39

2.4.3 Martingale Problem Representation 45

2.5 Uniqueness 49

2.6 Conclusions and Remarks 54

3.1 Data Analysis 56

3.1.1 Viewing the Data 57

3.1.2 Fundamental Diagram 60

3.1.3 Lane Shifting 64

3.2 Numerical Simulations 65

Bibliography 67 A Relative Compactness Criteria 71 A.1 Arzela-Ascoli 71

A.2 Relative Compactness in L2γ 72

List of Figures

1.1 Conservation of mass 4

1.2 Stop and go traffic 6

1.3 Riemann initial data 9

1.4 What would you do? 9

1.5 Direction of travel 10

2.1 Lane numbering 19

2.2 An example of a non-convex fundamental diagram 26

2.3 Discretization 29

2.4 Definition of the xn and jn 34

3.1 Bird’s eye view (cars driving across page) 57

3.2 Density 58

3.3 Density of one vehicle 59

3.4 ρ as a function of time and space 60

3.5 Density ranges 61

3.6 Finding the linear trend 62

3.7 Inverse of estimated slopes 63

3.8 Estimate of fundamental diagram 64

3.9 Densities around lane switches (the diagonal is only for show) 65

3.10 Density on one of the lanes 66

them in real life And finally, the importance of being able to effectively evacuatelarge populated areas has recently been discussed in the media in the US Thoseare just few examples of the utility of traffic modeling.

Different disciplines may consider traffic problems from different perspectives

In economics, for example, one could see the problem at the scale of a highwaysystem, and could try to control the overall traffic flow by introducing toll roads[28] In civil engineering, one might focus on developing methods of measuringthe traffic or designing roads for more efficient traffic flow A physicist wouldperhaps treat the traffic as a system of interacting particles and then apply thetools of statistical mechanics to derive equations for the large scale behaviour ofthe traffic flow This approach gives rise to kinetic macroscopic models which

we discuss further below Computer scientists might treat each car as a dot in agrid, whose behaviour is determined by the position of the dots around it Such

a system of dots is called a cellular automata Of course one can also attempt toaccurately model the small scale behaviour of individual drivers We will call allmodels that treat individual vehicles, microscopic Such models are also called car-following models since one postulates an equation, describing the behaviour of adriver based on information about the vehicle in front Finally, a mathematician’srole in the greater scheme of things could be to prove the well-posedness of models

or to study some of their qualitative properties

In table 1.1 we list some of the different kinds of models, and give historicalreferences In the sections that follow we take a closer look at the entries in thetable that are labeled with an ∗

Scale Type of equation Historical references

ODEs∗ Treiber et al [27]

Microscopic Delay equations Newell [18]

Cellular automata Nagel et al [25]

First order PDE∗ Lighthill, Whitham [30]

Macroscopic Second order PDE∗ Aw, Rascle [3], Helbing [12],

Zhang [31]

Table 1.1: Different traffic models and related references

We will use the following notation throughout this document

• We denote by (t, x), time and space, t ∈ [0, ∞), x ∈ I ⊆ R If units arerequired we will measure time in seconds and length in feet since those arethe units of the data we work with in Chapter 3

• We denote by ρ(x, t) ∈ [0, 1] the renormalized density of cars, i.e ρ sents the number of cars per mile, divided by the maximum number of carsper mile This maximum is of course somewhat arbitrary since the maxi-mum number of cars per mile is not well defined, it depends on the lengthsand types of vehicles

repre-• The flux of cars past a point x at time t will be denoted by q(x, t) (#cars/sec)

• The traffic flow velocity will be denoted by v We have, by definition,v(x, t) = q/ρ when ρ 6= 0, 0 otherwise

When using the above quantities we are considering traffic as a kind of anisotropicfluid

q(x0, t) q(x1, t)

Figure 1.1: Conservation of mass

Assuming for now that ρ and q are continuous differentiable quantities we get,see Figure 1.1,

ddt

Z x 1

x 0

ρ(x, t)dx = q(x0, t) − q(x1, t), (1.3.1)i.e the change in mass is only due to flux through the endpoints Now dividing

by x1− x0 and letting x0 → x1 gives

ρt(x, t) + qx(x, t) = 0 (1.3.2)

In the context of highway traffic, this step it is not as easy to justify as it is

in fluid dynamics, where the ratio of the size of the particles to the size of thedomain of interest is a lot smaller

Simplest model

The single equation (1.3.2) has two unknowns, so to close the system we needanother equation The simplest assumption is that q = Qeq(ρ), e.g Qeq(ρ) =ρ(1 − ρ)2, leading to the equation

where Qeq is the so called equilibrium flux The graph of ρ 7→ Qeq(ρ) is called thefundamental diagram and is an essential part of macroscopic modeling of highwaytraffic This equation can then be solved if we are given the initial distribution

of cars, i.e ρ(x, 0) = f (x) This model is usually called the Richards (LWR) model, after the authors that first introduced it [24], [16].The behaviour of the solution of this model can be deduced using the method

Lighthill-Whitham-of characteristics In this case the characteristics are straight lines Generally thesolution forms a shock and one resorts to weak solutions called entropy solutions.For more on this equation, its solutions and shock fitting see [30]

The assumption q = Q(ρ) is somewhat dubious It comes from assuming thatthe behaviour of drivers depends only on the density of cars around them Onecould for example that drivers also respond to the rate of change of density.Another drawback of this model is its inability to predict stop-and-go traffic.This situation in large cities is where drivers experience one traffic jam afteranother and travel almost freely between them In the (x, ρ) plane this would bedescribed by large amplitude waves, see Figure 1.2 The reason for this inability

is that the solution of equation (1.3.3) satisfies a maximum principle:

ρ(x, t) ∈ [infy ρ(y, 0), sup

yρ(y, 0)]

Free travel

Jams ρ

x

Figure 1.2: Stop and go trafficfor all x and t So, if the initial profile does not have waves of large amplitude,then the solution will not develop such waves at a later time

This of course does not mean that the model is worthless, it just means that

it is not useful for predicting the aforementioned phenomenon

Second Degree Models

Above we discussed problems related to the assumption that q = Q(ρ) In thissection we discuss two types of models that have been proposed to be able toproduce stop and go traffic, or so called “phantom” traffic jams (see Figure 1.2)

In the first kind of models, one assumes that drivers reduce their speed toaccount for an increasing density This assumption leads to the following rela-tionship q = Q(ρ, ρx) We consider only the special case

Q(ρ, ρx) = Qeq(ρ) − ερx (1.3.4)

where ε is some positive constant Substituting this into (1.3.2) gives the seconddegree equation

Later on, we will propose a new stochastic model, (2.1.11), based on this equation.The presence of the Laplacian ρxx will be essential for our analysis Solutions tothis equation also satisfy a maximum principle and therefore it does not solve theproblem discussed above.

In the second kind of models, one includes effects of momentum and inertia.This is often done by assuming that for a given traffic situation, drivers will adjusttheir speed towards some desired velocity It relates to the flux (1.3.4) by

vt+ vvx = dv

dt(x(t), t) =

1τ

Using q = ρv to couple this equation with (1.3.2) we arrive at the system

The model (1.3.8) can be derived heuristically from a microscopic car followingmodel, we refer the interested reader to [30]

Remark 1.3.4 In 1995 Daganzo [5] pointed out serious inconsistencies with thesecond degree models discussed so far, i.e (1.3.8) and (1.3.5), and all other models

of the same kind known at that time

• Some information travels faster than the cars, i.e cars respond from stimulifrom behind, which is not realistic in traffic, since experience tells us thatdrivers are mostly affected by what is happening in front of them In fluiddynamics terms we would phrase this by saying that the flow of cars is highlyanisotropic

• The second order term ερxx in (1.3.5) has a smoothing effect that causes themodel to predict cars moving backwards The system (1.3.8) has the the same

flaw; it predicts cars moving backwards This can be seen by consideringRiemann initial data (see Figure 1.3) Then the last cars at the traffic jam,

ρρ

x

x0

t = 0 t > 0

Figure 1.3: Riemann initial data

ending at x0, move backwards giving approximately the profile on the right

in Figure 1.3 after some small time t > 0

ρ

xyou

vyou

vhump > vyou

Figure 1.4: What would you do?

reasonable driver do? This is the situation at the time a driver enters a crowdedhighway; having waited for a decrease in the density before entering, and findingthat when on the highway the cars ahead are going faster The PW-model predictsslowing down since the density in front of the driver is increasing, i.e ρx > 0,

while most reasonable drivers would accelerate, i.e catch up with the “hump” infront.

This difference stems from the fact that cars are not travelling in the x direction

in the (t, x) plane, but rather in the direction (v, 1), as seen in Figure 1.5 Indeed

x

t(t, x) ∆t

∆xslope v

Figure 1.5: Direction of travel

we have

D(x,t)ρ(v, 1) = ∇ρ · (v, 1) = ρt+ vρx = −vxρ < 0where the last equality used is the conservation law (1.3.2) and the inequalitycomes from the fact that the cars ahead are going faster To incorporate this inthe model (1.3.8) we first write the PW model as

ρt+ (ρv)x = 0

vt+ vvx = 1

τ(V (ρ) − v) − p(ρ)x

(1.3.9)

were p is some increasing function of ρ

Remark 1.3.5 p(ρ)x = p′(ρ)ρx so the constants ε and τ have been incorporated

in the function p

Now the observation above about the travelling direction tells us that P (ρ)xshould be replaced by the convective or material derivative,

D := ∂t+ v∂x

of p(ρ), giving the system

ρt+ (ρv)x = 0D(v + p(ρ)) = 1

(1.3.11)

For more details see [22], [3]

This model can be derived from a microscopic model,

˙vi = Cvi−1− vi

xi−1− xi

+ A1τ

V

L

Continuous Kinetic Models

By analogy with the derivation of macroscopic models in particle physics, such

as the Boltzmann equation of fluids, one obtains a macroscopic model through acontinuum limit where the number of cars tends to infinity and the “total effect”

of their —sometimes probabilistic— behaviour is obtained via some kind of an

averaging process Indeed in gas dynamics one averages out the random collisions

of single atoms to get equations for the large-scale behaviour of the gas Thisprocedure typically leads to a system of deterministic PDEs

This approach was first used in the context of vehicular traffic by Prigogineand Herman in 1971, [21], and Paveri-Fontana in 1975, [19] Recently some morework has been done with this approach by Dirk Helbing, see [12]

Let us look a little closer to get a feeling for this method Let A be the set

of vehicles under study For example, in [12] the acceleration of vehicle α ∈ A, iswritten in the following very general form,

dvα

dt = fα(vα) +

Xβ6=α

fαβ(xα, vα, xβ, vβ) + ξα(t) (1.3.13)

where x, v are as usual, position and velocity, fα represents the driver’s will totravel at a certain speed (see below), fαβ represents the effect that car β has oncar α and ξα is a stochastic term that models variations in the driver’s behaviour

• One “natural” choice of fα would be

fα(v) = Vα− v

τ

where Vα is a desired velocity (possibly dependent on other factors such

as traffic density, vehicle type, etc.) and τ is as usual a relaxation time,measuring how fast the driver reacts

• On one lane, A ⊆ Z and vehicle n is just the n-th in line If on the otherhand there are multiple lanes there is no natural way to index the vehicles

• The model above is very general so one must make some simplifying

as-sumptions to be able to derive macroscopic equations.

From here, one proceeds by trying to mimic the effect of the fαβ with a mann-like interaction function A good example can be found in [12]

Boltz-In the next section we present a very simple heuristic method to obtain amacroscopic model from a particular case of (1.3.13)

We consider a large subclass of the car-following models, (1.3.13) We denote thevelocity and position of a leading car by w and y respectively, and by xc and vcfor the vehicle under consideration We assume

∆x = y − xc, ∆v = w − vc

Example 1.3.6 Examples of follow-the-leader models:

• The model (1.3.12) is given by F of the form

F (∆x, ∆v, vc) = α

V

L

∆x



− vc

+ β∆v

where V is a given function and α, β are constant

• The Intelligent Driver Model, IDM, (cf for example [10]) is given by

∆x

2!

where S is a given function, and α, δ are constant

Given a microscopic model, we now introduce a simple heuristic argument plaining how one can obtain macroscopic equations for the macroscopic variables,

ex-ρ and v, representing, as usual, the density and flow velocity respectively

First we list some properties of traffic flow that make it different from a normalfluid flow and that help justify our claims that follow

1 As we have already noted, the ratio of the size of the particles to the size ofthe region of interest is a lot larger than in traditional fluid mechanics

2 The particles think Thus in our model it is hard to justify that the particlesjust bounce around according to the laws of physics

3 Unidirectionality In traffic, information travels almost exclusively wards (at least on a one lane road), that is to say, we as drivers are affected,

back-to a very little extent, by the cars behind us while our driving behaviour is,

to a great extent, determined by what happens in front of us

We assume that macroscopic quantities ρ, v exist and search for an equationfor them The solution xc(t) to the equation

˙

xc(t) = v(xc(t), t), xc(0) = x0 (1.3.17)

should be the trajectory of a vehicle starting at the point x0 at time t = 0 Notehow this is different from the usual interpretation since most often v(x, t) is some

sort of limit average velocity around the position x, not the velocity of one particle.Indeed, for example in gas dynamics the particle trajectories are very wild, while

v is a lot smoother! But now if (1.3.17) describes the trajectory of a vehicle weshould get the same trajectory if we solved the microscopic equation (1.3.14), i.e

where P is some function to be determined later This is reasonable since thedensity on a road, in the eyes of a driver, is locally mostly determined by thedistance to the next car Indeed, as drivers, we do not care if the road ahead ispractically empty if we are just inches from the bumper of the car ahead of us.Any natural choice of P should insure that ρ ∈ [0, 1] and should be decreasingsince a bigger gap between cars represents lower density With this identificationcomes the following identification

w − vc = d

dt∆x ∼ dtdP (ρ) = P′(ρ)(ρt+ vρx) (1.3.20)

so equation (1.3.18) becomes

vt+ vvx = F (P (ρ), P′(ρ)(ρt+ vρx), v) (1.3.21)

Now there are two problems left First is to identify the “correct” P and thesecond to close this equation with another one since we have only one equationbut two unknowns Remember that in almost every model of highway traffic wehave the conservation equation ρt+ (ρv)x = 0 We, however, assumed nothingabout equations satisfied by the variables ρ and v so a priori there is no reason tobelieve that our representation is consistent with that equation Let’s make onemore natural identification, i.e

model of the form (1.3.14) a macroscopic one of the form

dtρ = ρt+ vρx, not ρx, but we do not have an interpretation of v in our setting.

At this time, we did not succeed in making a stochastic version of systems like(1.3.8)

We introduce a multi-lane model that includes the dynamics of lane shifting

It is a system of equations, one for each lane The dynamics are determined by

an equation of the form (2.1.1) with the addition of a lane shifting term, a sum

of two terms, one deterministic and one stochastic More precisely, if there are M

lanes then there are M + 1 stochastic processes, Wi, i = 1, , M + 1 where Wirepresents the randomness of exchange of mass between lanes i − 1 and i Lanes 0and M + 1 are outside the highway, on the left and right hand sides respectively.For example we can think of them as on- or off-ramps, see Figure 2.1 Even more

Lane 1

Lane 2

.

Lane M Lane 0

Lane M + 1

Entrance Exit

Figure 2.1: Lane numbering

precisely, the lane shifting from lane i − 1 to lane i will be written as

fi−1,i(ρi−1, ρi, x) + σi−1,i(ρi−1, ρi, x)Wi(dx, dt) (2.1.2)

where ρi−1, ρi are the densities on lanes i − 1 and i respectively (to be able to usethe same notation for all lanes we simply assume that ρ0 and ρM +1 are constant)

In order to write our model compactly, we define first the M × (M + 1)-matrix

Remark 2.1.1 The function fi := fi−1,i − fi,i+1, and the σi−1,i depend only on

ρi−1, ρi, ρi+1 and x Typical examples are

and

σi−1,i = Kρi(1 − ρi)ρi−1(1 − ρi−1) (2.1.8)

where k, K are constants.

Another typical example is the mass conserving case where

Our main result is the following theorem

Theorem 2.1.2 Assume we are given initial data ρ(0, x) = ρ0(x) where ρ0 =(ρ1

≤ C |U − V |

σi−1,i(U, x) − σi−1,i(V, x)

γ-continuous in t almost surely andits range lies between 0 and 1, i.e ρi(t, x) ∈ [0, 1] for all t, x

Remark 2.1.3 The Lipschitz continuity of the σi−1,i is only needed for theuniqueness part of the theorem For the existence part, it suffices that σi−1,i and

σi,i+1 are H¨older-1/2 continuous in the ith coordinate

Remark 2.1.4 Conditions (C1) and (C2) directly imply that there exists a K > 0such that

Qi(ρ, x) ≤ Kρ, for all ρ ≥ 0, x ∈ R

The outline of the rest of this chapter is as follows: We begin in Section 2.2 byexplaining exactly what we mean by equations (2.1.11) and their solutions Wealso give intuition for our hypothesis and explain the dynamics Then in Section2.3 we introduce a discretization of equations (2.1.11) We discretize only in space,leaving time continuous, resulting in an infinite system of SDEs In Section 2.4 weshow that as our space grid shrinks the approximations converge, and we identifythe limit as a solution of (2.1.11) We take care of the uniqueness part of Theorem(2.1.2) in Section 2.5.

The equations (2.1.11) above have to be interpreted in the weak sense since the

Wi(dx, dt) are not proper function but martingale measures

Here we also interpret (2.1.11) in the weak probability sense, i.e a solutionconsists of a probability space (Ω, F, P) on which we can define a vector of in-dependent white noises, W (dx, dt), and a stochastic process ρ = (ρi)M

i=1 suchthat

Z

R

ρi(t, x)ϕi(x)dx −

Z t 0

ZR

Qi(ρi(s, x), x)ϕ′i(x)dsdx

=ZR

ρi(0, x)ϕi(x)dx + ε

2

Z t 0

ZR

ρi(s, x)ϕ′′i(x)dxds+

Z t 0

ZR

fi(ρ(s, x), x)ϕi(x)dxds+

Z t 0

ZR(σ(ρ(s, x), x)ϕi(x)W (dx, ds))i,

In (2.2.1), and from here on, we interpret Wi(dx, dt) as a martingale measure

in the sense of [29]

We only prove weak convergence (tightness) of our scheme, however, we stronglybelieve that it converges also in the strong sense (it is almost trivial) That matterwill be settled in later publications

Remark 2.2.1 We do, however, prove strong uniqueness of solutions to (2.2.1)

so it does in fact have strong solutions in the probability sense

Despite the possibility of duplicating what we already explained in Chapter 1

we now give the interpretation of the hypothesis (C1)-(C4) and the various terms

of the equation (2.1.11) in the context of highway traffic:

• We interpret the ρi as car densities As such, our convention will be that

ρi = 0 means an empty road and ρi = 1 means bumper to bumper traffic,i.e we set ρi := (#cars per mile)/ρmax where

ρmax = #of cars per mile in bumper to bumper traffic

As we noted in the previous chapter, this is a fairly arbitrary definition sincethe maximum number of cars in bumper to bumper traffic is not a universallydefined number One can for example imagine, that we do not want themaximum number of side-by-side motorcycles per mile or the maximumnumber of eighteen wheel trucks per mile

• The functions Qi represent, as in (1.3.3), the equilibrium flux given a fixeddensity An example of a Qi can be seen in Figure 2.2, showing the genericshape we have in mind However, in what follows we will only use the

properties (C1), (C2) and not the specific form of Qi In the traffic literature,the graph of Q = Q(ρ) is called the fundamental diagram.

• Condition (C1) then represents the obvious criteria that there is no flow ofcars when the road is empty or in bumper to bumper traffic, and that theflow of cars is always in the same direction

• As we mentioned above, the fi and σi−1,i terms, model lane shifting Inour case, lane shifting is simply a transportation of mass from one lane toanother The fi give the average behavior while the stochastic terms withthe σi,i±1 represent deviations therefrom These deviations can for examplestem from drivers’ need to exit the highway, their longing to stay on someparticular lane or simply bad judgment

• Given what we said above, condition (C4) is completely natural, claimingthat no mass can enter an already full lane and no mass can leave an emptylane

• A less obvious condition is the technical condition (C3), which we use toshow that the ρi do not escape the interval [0, 1] We see that it is sufficientthat σi,i±1 decay faster than the square root when ρi approaches 0 or 1

In particular the condition implies that σi,i±1(ρ, x) vanishes for all ρ with

Q max

Figure 2.2: An example of a non-convex fundamental diagram

Similar equations have been studied under alternative hypothesis In [8] and[7], for example, existence and uniqueness is proven for a single equation, M = 1,similar to ours on a finite interval and for the stochastic Burgers equation, Q = ρ2,

on the whole line, respectively An example of a result about a system of diffusion equations can be found in [4]

be a discretization of the real line and Bi(t, x) be a family of independent Brownian

motions, i = 0, , M, x ∈ XN on some probability space (Ω, F, P) It will be

convenient to use the notation

x+= x + 1/N, x− = x − 1/N

The first and second discrete spatial derivatives of a vector Z = {Z(x)}x∈X N at a

point x ∈ Xn are defined as

∇xNZ := ∇x+Z = N(Z(x+) − Z(x))

∆xNZ := N2 Z(x+) − 2Z(x) + Z(x−)

(2.3.1)

Remark 2.3.1 Our choice of the right hand first derivative (instead of, say, the

left hand derivative) is arbitrary, it only matters when we show that ρi ∈ [0, 1] but

only to the extent that it will be easier to show boundedness from above, ρi ≤ 1,

than from below ρi ≥ 0

We will use the weighted l2 space

γx|ρx|2 < ∞

)

(2.3.2)

for some arbitrary summable weights γx

The goal of this section is to prove the following theorem

Theorem 2.3.2 Let M > 0 be an integer and Qi, σi−1,i, fi, ρ0, i = 1, , M,

satisfy the conditions of Theorem (2.1.2) Let (Ω, F, P) be a probability space on

which we have a countable collection of independent Brownian motions, Bj(t, x), x ∈

XN, j = 1, , M+1 Then if we let σ be as in (2.1.3) and B(t, x) := (B1(t, x), , BM +1(t, x))T,

then, the infinite system of SDEs

Furthermore, if N is large enough, the solution stays between 0 and 1, a.s P

Proof As in [26] we start with a finite system indexed by Xk

will be the solution to the M × 2kN dimensional system of SDEs (2.3.3) where

we now have x in the finite set Xk

N instead of XN,

ρi,kN(t) = (ρi,kN(t, x) : x ∈ XNk)

of the equation (2.1.11) in the context of highway traffic:

• We interpret the ρi as car densities As such,