MP ISRN Mathematical Physics 2090-4681 International Scholarly Research Network 327298 10.5402/2012/327298 327298 Research Article Explicit Asymptotic Velocity of the Boundary between Particles and Antiparticles Malyshev V. A. Manita A. D. Zamyatin A. A. Putkaradze V. Roy P. Znojil M. Faculty of Mechanics and Mathematics Lomonosov Moscow State University GSP-1, Moscow 119991 Russia msu.ru/ 2012 19 9 2012 2012 14 04 2012 21 06 2012 2012 Copyright © 2012 V. A. Malyshev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

On the real line initially there are infinite number of particles on the positive half line, each having one of K-negative velocities v1(+),,vK(+). Similarly, there are infinite number of antiparticles on the negative half line, each having one of L-positive velocities v1(-),,vL(-). Each particle moves with constant speed, initially prescribed to it. When particle and antiparticle collide, they both disappear. It is the only interaction in the system. We find explicitly the large time asymptotics of β(t)—the coordinate of the last collision before t between particle and antiparticle.

1. Introduction

We consider one-dimensional dynamical model of the boundary between two phases (particles and antiparticles, bears and bulls) where the boundary moves due to reaction (annihilation, transaction) of pairs of particles of different phases.

Assume that at time t=0 infinite number of (+)-particles and (-)-particles are situated correspondingly on R+ and R- and have one-point correlation functions: (1.1)f+(x,v)=i=1Kρi(+)(x)δ(v-vi(+)),f-(x,v)=j=1Lρj(-)(x)δ(v-vj(-)). Moreover, for any i,j, (1.2)vi(+)<0,vj(-)>0, that is, two phases move towards each other. Particles of the same phase do not see each other and move freely with the velocities prescribed initially. The only interaction in the system is the following, when two particles of different phases find themselves at the same point they immediately disappear (annihilate). It follows that the phases stay separated, and one might call any point in-between them the phase boundary (e.g., it could be the point of the last collision). Thus, the boundary trajectory β(t) is a random piece-wise constant function of time.

The main result of the paper is the explicit formula for the asymptotic velocity of the boundary as the function of 2(K+L) parameters—densities and initial velocities. It appears to be continuous but at some hypersurface some first derivatives in the parameters do not exist. This kind of phase transition has very clear interpretation: the particles with smaller activities (velocities) cease to participate in the boundary movement—they are always behind the boundary, that is, do not influence the market price β(t). In this paper, we consider only the case of constant densities ρi(+),ρi(-), that is, the period of very small volatility in the market. This simplification allows us to get explicit formulae. In , the case K=L=1 was considered, however, with nonconstant densities and random dynamics.

Main technical tool of the proof may seem surprising (and may be of its own interest), we reduce this infinite particle problem to the study of a special random walk of one particle in the orthant R+N with N=KL. The asymptotic behavior of this random walk is studied using the correspondence between random walks in R+N and dynamical systems introduced in .

The organization of the paper is the following. In Section 2, we give exact formulation of the model and of the main result. In Section 3, we introduce the correspondence between infinite particle process, random walks, and dynamical systems. In Sections 4, and 5 we give the proofs.

2. Model and the Main Result 2.1. Initial Conditions

At time t=0 on the real axis, there is a random configuration of particles, consisting of (+)-particles and (-)-particles. (+)-particles and (-)-particles differ also by the type: denote I+={1,2,,K} the set of types of (+)-particles, and I-={1,2,,L} the set of types of (-)-particles. Let (2.1)0<x1,k=x1,k(0)<···<xj,k=xj,k(0)<··· be the initial configuration of particles of type kI+, and let (2.2)···<yj,i=yj,i(0)<···<y1,i=y1,i(0)<0, be the initial configuration of particles of type iI-, where the second index is the type of the particle in the configuration. Thus, all (+)-particles are situated on R+ and all (-)-particles on R-. Distances between neighbor particles of the same type are denoted by (2.3)xj,k-xj-1,k=uj,k(+),kI+,j=1,2,,yj-1,i-yj,i=uj,i(-),iI-,j=1,2,, where we put x0,k=y0,i=0. The random configurations corresponding to the particles of different types are assumed to be independent. The random distances between neighbor particles of the same type are also assumed to be independent, and, moreover, identically distributed, that is, random variables uj,i(-),uj,k(+) are independent and their distribution depends only on the upper and second lower indices. Our technical assumption is that all these distributions are absolutely continuous and have finite means. Denote μi(-)=Euj,i(-),ρi(-)=(μi(-))-1,iI-, μk(+)=Euj,k(+),ρk(+)=(μk(+))-1,kI+.

2.2. Dynamics

We assume that all (+)-particles of the type kI+ move in the left direction with the same constant speed vk(+), where v1(+)<v2(+)<···<vK(+)<0. The (-)-particles of type iI- move in the right direction with the same constant speed vi(-), where v1(-)>v2(-)>···>vL(-)>0. If at some time t a (+)-particle and a (-)-particle are at the same point (we call this a collision or annihilation event), then both disappear. Collisions between particles of different phases is the only interaction, otherwise, they do not see each other. Thus, for example, at time t, the jth particle of type kI+ could be at the point: (2.4)xj,k(t)=xj,k(0)+vk(+)t, if it will not collide with some (-)-particle before time t. Absolute continuity of the distributions of random variables uj,i(-),uj,k(+) guaranties that the events, when more than two particles collide, have zero probability.

We denote this infinite particle process D(t).

We define the boundary β(t) between plus and minus phases to be the coordinate of the last collision which occurred at some time t<t. For t=0, we put β(0)=0. Thus, the trajectories of the random process β(t) are piecewise constant functions, we will assume them continuous from the left.

2.3. Main Result

For any pair (J-,J+) of subsets J-I-,J+I+, define the numbers: (2.5)V(J-,J+)=iJ-vi(-)ρi(-)+kJ+vk(+)ρk(+)iJ-ρi(-)+kJ+ρk(+),V=V(I-,I+). The following condition is assumed: (2.6){V(J-,J+):J-,J+  }  {v1(-),,vL(-),v1(+),,vK(+)}=. If the limit W=limt(β(t)/t) exists a.e., we call it the asymptotic speed of the boundary. Our main result is the explicit formula for W.

Theorem 2.1.

The asymptotic velocity of the boundary exists and is equal to (2.7)W=V({1,,L1},{1,,K1}), where (2.8)L1=max{l{1,,L}:vl(-)>V({1,,l},I+)},(2.9)K1=max{k{1,,K}:vk(+)<V(I-,{1,,k})}.

Note that the definition of L1 and K1 is not ambiguous because v1(-)>V({1},I+) and v1(+)<V(I-,{1}).

Now, we will explain this result in more detail. As vK(+)<0<vL(-), there can be 3 possible orderings of the numbers vL(-),vK(+),V:

vK(+)<V<vL(-): in this case (2.10)K1=K,L1=L,W=V;

if vK(+)>V, then V<0 and K1<K,L1=L, moreover, (2.11)W=V({1,,L},{1,,K1})=minkI+V({1,,L},{1,,k})  <  V  <  0;

if vL(-)<V, then V>0 and K1=K,L1<L, moreover, (2.12)W=V({1,,L1},I+)=maxlI-V({1,,l},I+)  >V  >0.

Item(1) is evident. Items (2) and (3) will be explained in Appendix B.

2.4. Another Scaling

Normally, the minimal difference between consecutive prices (a tick) is very small. Moreover, one customer can have many units of the commodity. That is why it is natural to consider the scaled densities: (2.13)ρj(+),ϵ=ϵ-1ρj(+),ρj(-),ϵ=ϵ-1ρj(-), for some fixed constants ρj(+),ρj(-). Then, the phase boundary trajectory β(ϵ)(t) will depend on ϵ. The results will look even more natural. Namely, it follows from the main theorem, that is, for any t>0, there exists the following limit in probability: (2.14)β(t)=limϵ0β(ϵ)(t), that is, the limiting boundary trajectory.

This scaling suggests a curious interpretation of the model—the simplest model of one instrument (e.g., a stock) market. Particle initially at x(0)R+ is the seller who wants to sell his stock for the price x(0), which is higher than the existing price β(0). There are K groups of sellers characterized by their activity to move towards more realistic price. Similarly, the (-)-particles are buyers who would like to buy a stock for the price lower than β(t). When seller and buyer meet each other, the transaction occurs and both leave the market. The main feature is that the traders do not change their behavior (speeds are constant), that is, in some sense the case of zero volatility.

There are models of the market having similar type (but very different from ours, see ). In physical literature, there are also other one-dimensional models of the boundary movement, see [6, 7].

2.5. Example of Phase Transition

The case K=L=1, that is, when the activities of (+)-particles are the same (and similarly for (-)-particles), is very simple. There is no phase transition in this case. The boundary velocity (2.15)W=v1(+)ρ1(+)+v1(-)ρ1(-)ρ1(+)+ρ1(-) depends analytically on the activities and densities. This is very easy to prove because the nth collision time is given by the simple formula: (2.16)tn=xn(+)(0)-xn(-)(0)-v1(+)+v1(-) and nth collision point is given by (2.17)xn(+)(0)+tnv1(+)=xn(-)(0)+tnv1(-).

More complicated situation was considered in . There, the movement of (+)-particles have random jumps in both directions with constant drift v1(+)0 (and similarly for (-)-particles). In , the order of particles of the same type can be changed with time. There are no such simple formulae as (2.16) and (2.17) in this case. The result is, however, the same as in (2.15).

The phase transition appears already in case when K=2,L=1, and, moreover, the (-)-particles stand still, that is, v1(-)=0. Denote ρ1(-)=ρ0,vi(+)=vi,ρi(+)=ρi,i=1,2. Consider the function: (2.18)V1(v1,ρ1)=ρ1v1ρ0+ρ1. It is the asymptotic speed of the boundary in the system where there is no (+)-particles of type 2 at all.

Then, the asymptotic velocity is the function: (2.19)W=V(v1,v2,ρ1,ρ2)=ρ1v1+ρ2v2ρ0+ρ1+ρ2 if v2<V1 and (2.20)W=V1(v1,ρ1)=ρ1v1ρ0+ρ1 if v2>V1. We see that at the point v2=V1 the function W is not differentiable in v2.

2.6. Balance Equations—Physical Evidence

Assume that the speed w of the boundary is constant. Then, the (-)-particle will meet the boundary if and only if vi(-)>w. Then, the mean number of (-)-particles of type i, meeting the boundary on the time interval (0,t), is (vi(-)-w)tρi(-). The total number of (-)-particles meeting the boundary during time t is (2.21)i:  vi(-)>w(vi(-)-w)tρi(-). Similarly, the number of (+)-particles meeting the boundary is (2.22)j:  vj(+)<w(w-vj(+))tρj(+).

These numbers should be equal (balance equations), and after dividing by t, this gives the equation with respect to w: (2.23)i:  vi(-)>w(vi(-)-w)ρi(-)=j:  vj(+)<  w(w-vj(+))ρj(+). Note that both parts are continuous in w. Moreover, the left (right) side is decreasing (increasing). This defines w uniquely. One can obtain the main result from this equation.

One could think that on this way one can get rigorous proof. However, it is not so easy. We develop here different techniques, that gives much more information about the process than simple balance equations.

3. Random Walk and Dynamical System in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M150"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula> 3.1. Associated Random Walk

One can consider the phase boundary as a special kind of server where the customers (particles) arrive in pairs and are immediately served. However, the situation is more involved than in standard queuing theory, because the server moves, and correlation between its movement and arrivals is sufficiently complicated. That is why this analogy does not help much. However, we describe the crucial correspondence between random walks in R+N and the infinite particle problem defined above, that allows to get the solution.

Denote bi(-)(t) (bk(+)(t)) the coordinate of the extreme right (left), and still existing at time t, that is, not annihilated at some time t<t, (-)-particle of type iI- ((+)-particle of type kI+). Define the distances di,k(t)=bk(+)(t)-bi(-)(t)0,iI-,kI+. The trajectories of the random processes bi(-)(t),bk(+)(t),di,k(t) are assumed left continuous. Consider the random process D(t)=(di,k(t),(i,k)I)R+N, where N=KL.

Denote 𝒟R+N the state space of D(t). Note that the distances di,k(t), for any t, satisfy the following conservation laws: (3.1)di,k(t)+dn,m(t)=di,m(t)+dn,k(t), where in and km. That is why the state space 𝒟 can be given as the set of nonnegative solutions of the system of (L-1)(K-1) linear equations: (3.2)d1,1+dn,m=d1,m+dn,1, where n,m1. It follows that the dimension of 𝒟 equals K+L-1. However, it is convenient to speak about random walk in R+N, taking into account that only subset of dimension K+L-1 is visited by the random walk.

Now, we describe the trajectories D(t) in more detail. The coordinates di,k(t) decrease linearly with the speeds vi(-)-vk(+) correspondingly until one of the coordinates di,k(t) becomes zero. Let di,k(t0)=0 at some time t0. This means that (-)-particle of type i collided with (+)-particle of type k. Let them have numbers j and l correspondingly. Then, the components of D(t) become (3.3)di,k(t0+0)=uj+1,i(-)+ul+1,k(+),di,m(t0+0)-di,m(t0)=uj+1,i(-),mk,dn,k(t0+0)-dn,k(t0)=ul+1,k(+),ni, and other components will not change at all, that is, do not have jumps.

Note that the increments of the coordinates dn,m(t0+0)-dn,m(t0) at the jump time do not depend on the history of the process before time t0, as the random variables. uj,i(-)(uj,k(+)) are independent and equally distributed for fixed type. It follows that D(t) is a Markov process. However, this continuous time Markov process has singular transition probabilities (due to partly deterministic movement). This fact, however, does not prevent us from using the techniques from  where random walks in Z+N were considered.

3.2. Ergodic Case

We call the process D(t) ergodic, if there exists a neighborhood A of zero, such that the mean value Eτx of the first hitting time τx of A from the point x is finite for any x𝒟. In the ergodic case, the correspondence between boundary movement and random walks is completely described by the following theorem.

Theorem 3.1.

Two following two conditions are equivalent:

the process D(t) is ergodic;

vK(+)<V<vL(-).

All other cases of boundary movement correspond to nonergodic random walks. Even more, we will see that in all other cases the process D(t) is transient. Condition (2.6), which excludes the set of parameters of zero measure, excludes in fact null recurrent cases.

To understand the corresponding random walk dynamics introduce a new family of processes.

3.3. Faces

Let ΛI=I-×I+. The face of R+N associated with Λ is defined as (3.4)(Λ)={xR+N:xi,k>0,(i,k)Λ,xi,k=0,(i,k)Λ¯}R+N. If Λ=, then (Λ)={0}. For shortness, instead of (Λ), we will sometimes write Λ. However, one should note that the inclusion like ΛΛ1 is always understood for subsets of I, not for the faces themselves.

Define the following set of “appropriate” faces 𝒢={Λ:Λ-=J-×J+,J-I-,J+I+}.

Lemma 3.2.

It holds that (3.5)𝒟=Λ0𝒢(𝒟Λ0).

The proof will be given in Appendix A. This lemma explains why in the study of the process D(t) one can consider only “appropriate” faces.

3.4. Induced Process

One can define a family D(t;J-,J+) of infinite particle processes, where J-I-,J+I+. The process D(t;J-,J+) is the process D(t) with ρj(+)=0,jJ+, and ρj(-)=0,jJ-. All other parameters (i.e., the densities and velocities) are the same as for D(t). Note that these processes are in general defined on different probability spaces. Obviously, D(t;I_,I+)=D(t).

Similarly to D(t), the processes D(t;J-,J+) have associated random walks D(t;J-,J+) in R+N1 with N1=|J-||J+|. Usefulness of these processes is that they describe all possible types of asymptotic behavior of the main process D(t).

Consider a face Λ𝒢, that is, such face that its complement Λ¯=J-×J+ where J-I- and J+I+. The process DΛ(t)=D(t;J-,J+)=(di,kΛ(t),(i,k)Λ-) will be called an induced process, associated with Λ. The coordinates di,kΛ(t) are defined in the same way as di,k(t)=di,kΛ(t), where Λ-={}. The state space of this process is 𝒟Λ-=𝒟(R|Λ|-), where |Λ-|=|J-||J+|. Face Λ is called ergodic if the induced process DΛ(t) is ergodic.

3.5. Induced Vectors

Introduce the plane: (3.6)(Λ)={xRN:  xi,k=0,(i,k)Λ-}RN.

Lemma 3.3.

Let Λ be ergodic with Λ¯=J-×J+, and let Dy(t) be the process D(t) with the initial point y(Λ). Then, there exists vector vΛ(Λ) such that for any y(Λ)t0, such that y+vΛt(Λ), one has, as M, (3.7)DyM(tM)My+vΛt.

This vector vΛ will be called the induced vector for the ergodic face Λ. We will see other properties of the induced vector below.

3.6. Nonergodic Faces

Let Λ be the face which is not ergodic (nonergodic face). Ergodic face  Λ1:  Λ1Λ will be called outgoing for Λ, if vi,kΛ1>0 for (i,k)Λ1Λ. Let (Λ) be the set of outgoing faces for the nonergodic face Λ.

Lemma 3.4.

The set (Λ) contains the minimal element Λ1 in the sense that, for any Λ2(Λ), one has Λ2Λ1.

This lemma will be proved in Section 5.2.

3.7. Dynamical System

We define now the piece-wise constant vector field v(x) in 𝒟, consisting of induced vectors, as follows: v(x)=vΛ if x belongs to ergodic face Λ, and v(x)=vΛ1 if x belongs to nonergodic face Λ, where Λ1 is the minimal element of (Λ). Let Ut be the dynamical system corresponding to this vector field.

It follows that the trajectories Γx=Γx(t) of the dynamical system are piecewise linear. Moreover, if the trajectory hits a nonergodic face, it leaves it immediately. It goes with constant speed along an ergodic face until it reaches its boundary.

We call the ergodic face Λ= final, if either = or all coordinates of the induced vector v are positive. The central statement is that the dynamical system hits the final face, stays on it forever, and goes along it to infinity, if .

The following theorem, together with Theorem 3.1, is parallel to Theorem 2.1. That is, in all 3 cases of Theorems 2.1, 3.1, and 3.5 describe the properties of the corresponding random walks in the orthant.

Theorem 3.5.

( 1 ) If D(t) is ergodic then the origin is the fixed point of the dynamical system Ut. Moreover, all trajectories of the dynamical system Ut hit 0.

( 2 ) Assume vK(+)>V. Then, the process D(t) is transient and there exists a unique ergodic final face , such that vi,k>0 for (i,k). This face is (3.8)(L,K1)={(i,k):i=1,,L,k=K1+1,,K}, where K1 is defined by (2.9). Moreover, all trajectories of the dynamical system Ut hit (L,K1) and stay there forever.

( 3 ) Assume vL(-)<V. Then, the process D(t) is transient and there exists a unique ergodic final face , such that vi,k>0 for (i,k). This face is (3.9)(L1,K)={(i,k):i=L1+1,,L,k=1,,K}, where L1 is defined by (2.8). Moreover, all trajectories of the dynamical system Ut hit (L1,K) and stay there forever.

( 4 ) For any initial point x, the trajectory Γx(t) has finite number of transitions from one face to another, until it reaches {0} or one of the final faces.

This theorem will be proved in Section 5.3.

3.8. Simple Examples of Random Walks and Dynamical Systems

If K=L=1, the process D(t) is a random process on R+. It is deterministic on R+{0}—it moves with constant velocity v(+)-v(-) towards the origin. When it reaches 0 at time t, it jumps backwards: (3.10)D(t+0)=η, where η has the same distribution as u1(+)+u1(-). The dynamical system coincides with D(t) inside R+ and has the origin as its fixed point.

If L=1,K=2 and, moreover, v1(-)=0, then the state space of the process is R+2={(d11,d12)}. Inside the quarter plane, the process is deterministic and moves with velocity (v1(+),v2(+)). From any point x of the boundary d12=0, it jumps to the random point x+η1, and from any point of the boundary d11=0, it jumps to the pointx+η2, whereη1,η2 have the same distributions as (uj,1(-),uj,1(-)+uj,2(+)) and (uj,1(-)+uj,1(+),uj,1(-)) correspondingly. The classification results for random walks in Z+2 can be easily transferred to this case; the dynamical system is deterministic and has negative components of the velocity inside R+2. When it hits one of the axes, it moves along it. The velocity is always negative along the first axis, however, along second axis, it can be either negative or positive. This is the phase transition we described above. Correspondingly, the origin is the fixed point in the first case and has positive value of the vector field along the second axis, in the second case.

4. Collisions 4.1. Basic Process

Now, we come back to our infinite particle process D(t). The collision of particles of the types iI-,kI+ we will call the collision of type (i,k). Denote (4.1)νi,k(T)=#{t:di,k(t)=0,t[0,T]} the number of collisions of type (i,k) on the time interval [0,T].

Lemma 4.1.

If the process D(t) is ergodic, then the following positive limits exist a.s. (4.2)πi,k=limTνi,k(T)T>0,(i,k)I and satisfy the following system of linear equations: (4.3)vi(-)-vk(+)=(n,m)I-×I+(δ(n,i)μi(-)+δ(m,k)μk(+))πn,m,(i,k)I.

Proof.

Remind that the collisions can be presented as follows. If di,k(t0)=0, then for any n,m(4.4)dn,m(t0+0)-dn,m(t0)=δ(n,i)uj+1,i(-)+δ(m,k)ul+1,k(+), where δ(n,i)=1 for n=i and δ(n,i)=0 for ni. Note that the proof of (4.2) is similar to the proof of the corresponding assertion in . For large t, we have (4.5)di,k(t)=-(vi(-)-vk(+))t+(n,m)I-×I+(δ(n,i)μi(-)+δ(m,k)μk(+))νn,m(t)+o(t). Note that this is exact equality, if instead of μi(-) and μk(+), we take random distances between particles. By the law of large numbers and by (4.2), the system (4.3) follows.

We will need below the following new notation, (4.3) can be rewritten in the new variables πi(-),πk(+) as follows (4.6)vi(-)-vk(+)=πi(-)μi(-)+πk(+)μk(+), where (4.7)πi(-)=m=1Kπi,m,πk(+)=n=1Lπn,k. Obviously, the following balance equation holds: (4.8)i=1Lπi(-)=k=1Kπk(+)=i=1Lk=1Kπi,k. Rewrite the system (4.3) in a more convenient form, using the variables ri(-)=πi(-)μi(-), rk(+)=πk(+)μk(+). Then, (4.9)vi(-)-vk(+)=ri(-)+rk(+),(i,k)I,i=1Lri(-)ρi(-)=k=1Krk(+)ρk(+). It follows that, for all (i,k)I, (4.10)vi(-)-ri(-)=rk(+)+vk(+). Introduce the variable w=vi(-)-ri(-)=rk(+)+vk(+). We get the following system of equations with respect to the variables ri(-),rk(+),w: (4.11)vi(-)-ri(-)=w,iI-,rk(+)+vk(+)=w,kI+,i=1Lri(-)ρi(-)=k=1Krk(+)ρk(+). It is easy to see that this system has the unique solution: (4.12)ri(-)=vi(-)-w,rk(+)=-vk(+)+w,w=V, where V is defined by (2.5). If D(t) is ergodic, then by Lemma 4.1 we have ri(-),rk(+)>0 for any iI-,kI+.

Lemma 4.2.

Let the process D(t) be ergodic. Then,

vK(+)<V<vL(-),

the speed of the boundary W=V.

Proof .

( 1 ) If D(t) is ergodic, then by Lemma 4.1,πi(-)>0 and πk(+)>0 for all iI-,  kI+. So, by (4.12), we have (4.13)ri(-)=vi(-)-V>0,rk(+)=-vk(+)+V>0.

( 2 ) Let νi(-)(T) be the number of particles of type iI-, which had collisions during time T. Then, (4.14)j=1νi(-)(T)uj,i(-) is the initial coordinate of the particle of type iI, which was the last annihilated among the particle of this type. Let Ti be the annihilation time of this particle. Then, (4.15)β(Ti+0)+j=1νi(-)(T)uj,i(-)Ti=  vi(-). Rewrite this expression as follows: (4.16)β(Ti+0)-β(T)+β(T)+j=1νi(-)(T)uj,i(-)T=TiTvi(-). It follows that (4.17)β(T)T=TiTvi(-)-j=1νi(-)(T)uj,i(-)T+β(T)-β(Ti+0)T. By Lemma 4.1 and the strong law of large numbers, (4.18)j=1νi(-)(T)uj,i(-)T=νi(-)(T)Tj=1νi(-)(T)uj,i(-)νi(-)(T)πi(-)μi(-)=ri(-),a.e. as T. At the same time, ergodicity of the process D(t) gives that as T(4.19)T-TiT0,β(T)-β(Ti+0)T0,a.e. Thus, for any iI-, a.e. (4.20)limTβ(T)T=vi(-)-ri(-)=V. Similarly, one can prove that for all kI+, (4.21)limTβ(T)T=vk(+)+rk(+). It follows from (4.11) and (4.12) that the boundary velocity is defined by (2.5). Lemma is proved.

4.2. Induced Process

Consider the faces Λ such that Λ¯=J-×J+, where J-I- and J+I+. Let (4.22)νi,kΛ(T)=#{t:  di,kΛ(t)=0,t[0,T]} be the number of collisions of type (i,k) on the time interval [0,T] in the process D(t;J-,J+).

The following lemma is quite similar to Lemma 4.1.

Lemma 4.3.

If the process DΛ(t) is ergodic, then the following a.e. limits exist and are positive for all pairs (i,k)Λ¯, (4.23)πi,kΛ=limTνi,kΛ(T)T>0. They satisfy the following system of linear equations: (4.24)vi(-)-vk(+)=(n,m)Λ¯(δ(n,i)μi(-)+δ(m,k)μk(+))πn,mΛ,(i,k)Λ¯.

Introduce the following notation: (4.25)πi(Λ,-)=kJ+πi,kΛ,iJ-,πk(Λ,+)=iJ-πi,kΛ,kJ+,ri(Λ,-)=μi(-)πi(Λ,-),iJ-,rk(Λ,+)=μk(+)πk(Λ,+),kJ+. For Λ=, Λ¯=I-×I+, we have πi(Λ,-)=πi(-), πk(Λ,+)=πk(+) and ri(Λ,-)=ri(-), rk(Λ,+)=rk(+).

Due to (4.24), for (i,k)Λ¯, we have (4.26)vi(-)-vk(+)=(n,m)Λ¯(δ(n,i)μi(-)+δ(m,k)μk(+))πn,mΛ=μi(-)πi(Λ,-)+μk(+)πk(Λ,+)=ri(Λ,-)+rk(Λ,+). It follows that vi(-)-ri(Λ,-)=rk(Λ,+)+vk(+) for all (i,k)Λ¯. Put wΛ¯=vi(-)-ri(Λ,-)=rk(Λ,+)+vk(+). In this way, we have obtained the following system of linear equations (similar the system (4.11)) with respect to variables ri(Λ,-),rk(Λ,+),wΛ¯: (4.27)vi(-)-ri(Λ,-)=wΛ¯,iI-,rk(Λ,+)+vk(+)=wΛ¯,kI+,iJ-ρi(-)ri(Λ,-)=kJ+ρk(+)rk(Λ,+). As previously, this system has the unique solution: (4.28)ri(Λ,-)=vi(-)-wΛ¯,rk(Λ,+)=-vk(+)+wΛ¯,wΛ¯=VΛ¯=V(J-,J+).

For any process D(t;J-,J+) or for the corresponding induced process DΛ(t) (see Section 3), we also define the boundary βΛ(t) as the coordinate of the last collision (i,k)Λ¯ before t. Let us assume that βΛ(0)=0. The trajectories of the random process βΛ(t) are also piece-wise constant, we will assume them left continuous. The following lemma is completely analogous to Lemma 4.2.

Lemma 4.4.

Let Λ¯=J-×J+={il,,i1}×{k1,,km}, where il>···>i1 and k1<···<km, and letΛbe an ergodic face. Then,

vil(-)>VΛ¯=V(J-,J+) and vkm(+)<VΛ¯=V(J-,J+),

the boundary velocity for the process D(t;J-,J+) (or for the corresponding DΛ(t)) equals (with the a.e. limit) (4.29)limtβΛ(t)t=VΛ¯=V(J-,J+).

Note that VΛ¯=V for Λ=.

Lemma 4.5.

For any ergodic face Λ (Λ¯=J-×J+), the vector vΛ(Λ) with the coordinates equal to (4.30)vi,kΛ=-vi(-)+vk(+)+1(iJ-)μi(-)πi(Λ,-)+1(kJ+)μk(+)πk(Λ,+),(i,k)Λ, is the induced vector in the sense of Lemma 3.3.

This is quite similar to Lemma 2.2, page 143 of and Lemma 4.3.2, page 87 of .

It follows from (4.30) and (4.28), that the coordinates of the induced vector are given by (4.31)vi,kΛ=-vi(-)+VΛ¯,(i,k)Λ,iJ-,kJ+,(4.32)vi,kΛ=vk(+)-VΛ¯,(i,k)Λ,iJ-,kJ+,(4.33)vi,kΛ=-vi(-)+vk(+),(i,k)Λ,iJ-,kJ+,(4.34)vi,kΛ=0,(i,k)Λ¯.

Note that by condition (2.6) for all induced vectors vi,kΛ0 if (i,k)Λ.

Intuitive interpretation of this formula is the following. For example, the inequality vi,kΛ=-vi(-)+VΛ¯<0,(i,k)Λ,iJ-,kJ+ means that (-)-particles of type iI- overtake the boundary which moves with velocity VΛ¯. In the contrary case, vi,kΛ=-vi(-)+VΛ¯>0, that is, (-)-particles of type iI- fall behind the boundary.

5. Proofs 5.1. Proof of Theorem <xref ref-type="statement" rid="thm2">3.1</xref>

The implication 12 has been proved in Lemma 4.2. Now, we prove that (2) implies (1). We will use the method of Lyapounov functions to prove ergodicity. Define the Lyapounov function: (5.1)f(y)=(i,k)Ipi,kyi,k=(p,y), where vector p with coordinates pi,k>0 will be defined below. One has to verify the following condition: there exists δ>0 such that for any ergodic face Λ, Λ{0}, (5.2)f(y+vΛ)-f(y)=(p,vΛ)<-δ, where vΛ is the induced vector corresponding to the face Λ, see .

The system (4.3) can be written in the matrix form: (5.3)v=Aπ, where A is the N×N matrix (5.4)A={a(i,k),(n,m)=δ(n,i)μi(-)+δ(m,k)μk(+)}, with the elements indexed by (i,k)I, and the vector (5.5)v={v(i,k)=vi(-)-vk(+),(i,k)I}. It is easy to see that the coordinates of the vector Aπ are equal to (5.6)(Aπ)i,k=μi(-)πi(-)+μk(+)πk(+).

If the assumption (2) of the theorem holds, then the system of (4.11) has a positive solution, that is, ri(-),rk(+)>0. One can choose positive pi,k so that the following condition holds: (5.7)πi(-)=m=1Kpi,m,πk(+)=n=1Lpn,k, where πi(-)=ρi(-)ri(-) and πk(+)=ρk(+)rk(+). For example, one can put (5.8)pi,m=C-1πi(-)πk(+), where (5.9)C=i=1Lπi(-)=k=1Kπk(+). Let the vector p have coordinates pi,k. Then, p satisfies the system (5.3), that is, v=Ap.

For ergodic face Λ, define the vector πΛ with coordinates πi,kΛ, where πi,kΛ for (i,k)Λ¯ are defined in (4.23) and we put πi,kΛ=0 for (i,k)Λ. It follows from (4.26) and (4.30) that the induced vector can be written as (5.10)vΛ=-v+AπΛ, with the matrix A and the vector v defined in (5.4) and (5.5). By (5.10), we have (5.11)vΛ=-v+AπΛ=-A(p-πΛ). As the vector A(p-πΛ) belongs to the face Λ and PrΛπΛ=0, then (5.12)f(y+vΛ)-f(y)=(p,vΛ)=-(p,A(p-πΛ))=-(p-πΛ,A(p-πΛ)).

Note that the matrix A in (5.3) is a nonnegative operator. In fact, for any vector y=(yi,j)RN, (5.13)(Ay,y)=i,k(μi(-)yi(-)+μk(+)yk(+))yi,k=i=1Lμi(-)(yi(-))2+k=1Kμk(+)(yk(+))20, where (5.14)yi(-)=m=1Kyi,m,yk(+)=n=1Lyn,k. Let for definiteness Λ¯=J-×J+. By formula (5.13), (5.15)-(p-πΛ,A(p-πΛ))=-i=1Lμi(-)(πi(-)-πi(Λ,-))2-k=1Kμk(+)(πk(+)-πk(Λ,+))2<-iJ-μi(-)(πi(-))2-kJ+μk(+)(πk(+))2<0, as πi(-),πk(+)>0, πi(Λ,-)=0 for iJ-, πk(Λ,+)=0 if kJ+. As the number of faces is finite, one can always choose δ>0, so that (5.16)f(y+vΛ)-f(y)=-(p-πΛ,A(p-πΛ))<-δ.

The theorem is proved.

5.2. Proof of Lemma <xref ref-type="statement" rid="lem3">3.4</xref>

For any nonergodic face Λ with Λ¯=J-×J+={i1,,il}×{m1,,mk}, where i1<···<il and m1<···<mk, define (5.17)q=max{n{1,,l}:vin(-)>V({i1,,in},{m1,,mk})},(5.18)r=max{j{1,,k}:vmj(+)<V({i1,,il},{m1,,mj})}. This definition is correct because always (5.19)vi1(-)>V({i1},{m1,,mk}),  vm1(+)<V({i1,,il},{m1}).

Introduce the face Λ1 such that Λ¯1={i1,,iq}×{m1,,mr}. If r=k,q=l, then vmk(+)<V(J-,J+)<vil(-) and Λ1=Λ. By Theorem 3.1, the induced process DΛ(t) is ergodic and the face Λ is ergodic.

So, there can be two possible cases.

If r<k,q=l, then Λ¯1={i1,,il}×{m1,,mr}, vmk(+)>V(J-,J+) andV(J-,J+)<0.

If r=k,q<l, then Λ¯1={i1,,iq}×{m1,,mk}, vil(-)<V(J-,J+) andV(J-,J+)>0.

By construction, we have Λ1Λ.

We show that Λ1 is the minimal ergodic outgoing face for Λ. Consider the first case, namely, r<k,q=l. The second one is quite similar. Because of vmr(+)<V({i1,,il},{m1,,mr})<vil(-), we can apply Theorem 3.1 and so the induced process DΛ1(t) is ergodic. This gives ergodicity of the face Λ1.

By formula (4.32) for all (in,mj)Λ1Λ={i1,,il}×{mr+1,,mk}(5.20)vin,mjΛ1=vmj(+)-V({i1,,il},{m1,,mr}) and by formula (5.18), (5.21)vmj(+)>V({i1,,il},{m1,,mr,mr+1,,mj}). It follows from Lemma B.1 that (5.22)V({i1,,il},{m1,,mr})<V({i1,,il},{m1,,mr,mr+1,,mj}). Thus, we get vin,mjΛ1>0 for all (in,mj)Λ1Λ. It means that the face Λ1 is outgoing for Λ.

To finish the proof of Lemma 3.4, it is sufficient to show that the constructed face Λ1 is the minimal outgoing face for Λ. We give the proof by contradiction. Let there exist an ergodic outgoing ( for Λ) face Λ0Λ such that Λ0Λ1 and Λ1Λ0Λ1. Put (5.23)Λ¯0=J-0×J+0Λ¯={i1,,il}×{m1,,mk}. By (4.31)–(4.33), the coordinates vi,kΛ0 of the induced vector vΛ0 are given for (i,k)Λ0Λ as follows: (5.24)vi,kΛ0=-vi(-)+V(J-0,J+0),(i,k)(J-J-0)×J+0,vi,kΛ0=vk(+)-V(J-0,J+0),(i,k)J-0×(J+J+0),vi,kΛ0=-vi(-)+vk(+),(i,k)J-J-0×J+J+0. As the face Λ0 is outgoing, we must have vi,kΛ0>0 for all (i,k)Λ0Λ. Thus, the only two situations are possible: Λ¯0=J-0×{m1,,mk} or Λ¯0={i1,,il}×J+0. In the first case, we have (5.25)vi,jΛ0=-vi(-)+V(J-0,{m1,,mk})>0,(i,j)(J-J-0)×{m1,,mk}, and so V(J-0,{m1,,mk})>0. But then V(J-,J+)>0 and this contradicts the assumption V(J-,J+)<0.

So Λ¯0={i1,,il}×J+0. Show that J+0={m1,,mr}.

Let J+0{m1,,mr} and there is j{m1,,mr} such that jJ+0. Then, by Lemma B.1, (5.26)vi,jΛ0=vj(+)-V(J-0,J+0)<0, and, hence, the face Λ0 cannot be outgoing for Λ. If {m1,,mr}J+0, there exists some point (in,mj)Λ0Λ, where j{r+1,,k}, and by (5.18), (5.27)vmj(+)>V({i1,,il},{m1,,mr,mr+1,,mj}). It follows from Theorem 3.1 that the induced process DΛ0(t) is nonergodic and, hence, the face Λ0 is also nonergodic. This contradicts the assumption on ergodicity of the face Λ0. So J+0={m1,,mr}. The Lemma is proved.

5.3. Proof of Theorem <xref ref-type="statement" rid="thm3">3.5</xref>

The first goal of this subsection is to study trajectories Γ(t) of the dynamical system Ut. After that, using the obtained knowledge about behavior of Γ(t), we will prove Theorem 3.5. Let Γx(t) be the trajectory of the dynamical system, starting in the point Γx(0)=xR+N.

According to the definition of Ut, any trajectory Γx(t), t0, visits some sequence of faces. In general, this sequence depends on the initial point x and contains ergodic and nonergodic faces. It is very complicated to give a precise list of all faces visited by the concrete trajectory started from a given point x. Our idea is to find a common finite subsequence Λ1,Λ2,,Λn of ergodic faces in the order they are visited by any trajectory. We find this subsequence together with the time moments t1, t2,,tn, where tk is the first time the trajectory enters the closure of Λk. Moreover, it will follow from our proof that the intervals tk-tk-1 are finite, the dimensions of the ergodic faces in this sequence decrease and any trajectory, after hitting the closure of some face in this sequence, will never leave this closure.

Proposition 5.1.

There exists a monotone sequence of faces: (5.28)Λ1Λ2ΛrΛn,dimFi>dimFi+1, and a sequence of time moments: (5.29)t1t2trtn<+, depending on x, and having the following property: (5.30)Γx(t)Fr,ttr, where Fr=cl(Λr) denotes the closure of Λr in R+N. Moreover, the sequence Λ1,Λ2,,Λn depends only on the parameters of the model (i.e., on the velocities and densities), but the sequence of time moments t1, t2,,tn depends also on the initial point x of the trajectory Γx(t). Thus, any trajectory will hit the final set Ffin=Fn in finite time.

The proof of Proposition 5.1 will be given at the end of this subsection.

First, we will present here some algorithm for constructing the sequence Λ1,Λ2,,Λn. By Lemma 3.2, we can consider only faces Λ, such that Λ¯=J(-)×J(+). Algorithm consists of several number of steps and constructs a sequence Λ1¯, Λ2¯, , (5.31)Λ¯p=Jp(-)×Jp(+)={(l,k)lJp(-),kJp(+)}. In fact, it constructs a sequence {(Jp(-),Jp(+))}p=1n. We prefer here to use notation: (5.32)(Jp(-),Jp(+))=Tp=(Jp(-)    Jp(+)), and to call Tp a group consisting of particle types listed in Jp(-),Jp(+).

Notation VTi has the same meaning as earlier: (5.33)VTi=lJi(-)vl(-)ρl(-)+kJi(+)vk(+)ρk(+)lJi(-)ρl(-)+kJi(+)ρk(+).

Algorithm 5.2.

( 1 ) Put T1=(1    1) and find VT1.

( 2 a ) If VT1<0, compare -v2(+) and |VT1|.

(i) If -v2(+)>|VT1|, then T2=(1    1,2).

(ii) If -v2(+)<|VT1|, then T2=(2,1    1).

( 2 b ) If VT1>0, compare v2(-) and VT1.

(i) If v2(-)>VT1, then T2=(2,1    1).

(ii) If v2(-)<VT1, then T2=(1    1,2).

We have already constructed group: (5.34)Tr-1=(b,b-1,,11,,a-1,a).

Find VTr-1. If a<K and b<L hold, then apply the following steps (r-a) and (r-b).

If VTr-1<0 and a<K, compare -va+1(+) and|VTr-1|.

If -va+1(+)>|VTr-1|, then Tr=(b,,11,,a,a+1).

If -va+1(+)<|VTr-1|, then Tr=(b+1,b,,11,,a).

If VTr-1>0 and b<L, we compare vb+1(-) and VTr-1.

If vb+1(-)>VTr-1, then Tr=(b+1,b,,11,,K).

If vb+1(-)<VTr-1, then the algorithm is finished and the group Tr-1=(b,,1  |  1,,K) is declared to be the final group Tfin of the algorithm.

If a=K, and b<L, we compare vb+1(-) and VTr-1.

If vb+1(-)>VTr-1, then Tr=(b+1,b,,11,,K).

If vb+1(-)<VTr-1, then the algorithm is finished and the group Tr-1=(b,,11,,K) is declared to be the final group Tfin of the algorithm.

If a<K, and b=L, we compare va+1(+) and VTr-1.

If va+1(+)<VTr-1, then Tr=(L,,11,,a,a+1).

If va+1(+)>VTr-1, then the algorithm is finished and the group Tr-1=(L,,11,,a) is declared to be the final group Tfin of the algorithm.

If the algorithm did not stop at the steps (r-c), (r-d), or (r-e), then the step r+1 should be fulfilled, and so forth. It is clear that the algorithm stops after finite number of steps, and as the result, we get a final group Tfin, which will have one of the following types: (5.35)(L,,11,,K),(L,,11,,K1),(L1,,11,,K),

where K1<K, L1<L.

We need not only the final group, corresponding to the face along which the trajectory escapes to infinity, but also the whole chain: (5.36)T1=(11)T2T3Tfin.   As it follows from the algorithm, this chain is uniquely defined by the parameters of the model.

Let us remark that in the algorithm we excluded cases where some of VTr-1 are zero. We will show below (see Remark 5.5) how to modify the algorithm to take into account these cases as well.

The next lemma is needed for the proof of the Theorem 3.5. It is convenient, however, to give this proof here, as it is essentially based on the details of the algorithm defined above.

Lemma 5.3.

( 1 ) If Tfin=(L,,11,,K), then simultaneously vL(-)>VTfin and vK(+)<VTfin hold.

( 2 ) If Tfin=(L,,11,,K1), whereK1<K, then VTfin<0 and vK(+)>VTfin.

( 3 ) If Tfin=(L1,,11,,K), where L1<L, then VTfin>0 and vL(-)<VTfin.

Proof of Lemma <xref ref-type="statement" rid="lem9">5.3</xref>.

In fact, if Tfin=(L,,11,,K1), where K1<K, then the algorithm stops on some step (r0-d), and thus, the condition vK1+1(+)>VTfin will hold. As 0>vK(+)vK1+1(+), then we get the proof of the part (2) of the lemma. Part (3) is quite similar.

To prove assertion (1) of the lemma consider the face, previous to the final one: (5.37)Tfin=(L,,11,,K). Two cases are possible: (5.38)Tf-1=(L,,11,,K-1)orTf-1=(L-1,,11,,K).

Consider the case Tf-1=(L,,11,,K-1) and the final fragment of the trajectory in the algorithm: (5.39)(L-1,,11,,q)(L,,11,,q)···Tf-1=(L,,11,,K-1)Tfin. Two cases of the first transition in this chain are possible:

V(L-1,,11,,q)<0 and vq+1(+)>V(L-1,,11,,q);

V(L-1,,11,,q)>0 and vL(-)>V(L-1,,11,,q).

In both cases, one can claim that (5.40)vL(-)>V(L,,11,,q). To prove this consider both cases separately.

Case 1. As vL(-)>0, then we have vL(-)>V(L-1,,1    1,,q). Thus, vL(-)>V(L,,11,,q), as V(L,,11,,q) is the convex linear combination (CLC(CLC of the numbers x1,xn is iαixi for some numbers αi>0, i=1,n¯ such that iαi=1.))vL(-) and V(L-1,,11,,q).

Case 2 . Here, we assume vL(-)>V(L-1,,1    1,,q). From this, as above, we get that vL(-)>V(L,,11,,q).

Thus, the inequality (5.40) is proved. As V(L,,11,,K) is CLC of V(L,,11,,q) and negative numbers vq+1(+), , vK(+), then (5.41)V(L,,11,,K)<V(L,,11,,q). Then, we have V(L,,11,,K)<vL(-).

The latter transition in the chain occurs because vK(+)<V(L,,1    1,,k-1). Then, vK(+)<V(L,,1    1,,K), as V(L,,1    1,,K) is CLC of V(L,,1    1,,K-1) and vK(+).

This gives the proof.

Let ar and br are such that (5.42)Tr=(br,,11,,ar); The numbers ar and br are non-decreasing functions of r. Moreover ar+br increases by 1 if r increases by 1. What can be the difference between Tr-1 and Tr? There can be two cases:

Case Π r . Consider ar=ar-1+1,br=br-1.

Case U r . Consider ar=ar-1,br=br-1+1.

Remind that the face B(Λ)R+N is defined by the set of pairs of indices ΛI-×I+. Namely, to each pair (j,k)Λ corresponds positive coordinates dj,k>0 in the definition (3.4) of the face B(Λ) and vice versa. For shortness, we say that the face B(Λ) consists of pairs (j,k)Λ.

Proposition 5.4.

Let the chain (5.36) be given and Case Πr occur. For any ergodic face Λ, not containing the pairs: (5.43)(l,k),l1,br-1¯,k1,ar-1¯, the following holds true: for any pairs as (5.44)(b,ar),b1,br-1¯, belonging to Λ, the corresponding component of the vector field is negative: (5.45)vb,arΛ<0. If the Case Ur occurs, then for any ergodic face Λ, not containing the pairs (5.43), the following components of the vector field are negative: (5.46)vbr,aΛ<0,a1,ar-1¯, under the condition, of course, that (br,a)Λ.

Proof of Proposition <xref ref-type="statement" rid="prop5">5.4</xref>.

Remind the notation Tr=(br,,11,,ar). As it was mentioned above, the connection between Tr-1 and Tr can be of two kinds—Πr or Ur, which we write schematically as (5.47)Πr:Tr=Tr-1(ar),Ur:Tr=Tr-1(br). Consider only the Case Πr, as the Case Ur is symmetric. It is necessary to prove that for any ergodic face Λ, which does not contain (5.48)(l,k),l1,br-1¯,k1,ar-1¯, for any pairs (b,ar)Λ, where b1,br-1¯, the inequality, (5.49)vb,arΛ<0, holds. Thus, we mean the faces with (5.50)Λ¯=(lm,,lr,br-1,,11,,ar-1,ar^,kr+1,,kn). For such faces vb,arΛ=var(+)-VΛ¯.

Consider now the case when the set kr+1,,kn is not empty. As Λ¯ corresponds to ergodic group of particles, then by Lemma 5.7vkr+1(+)<VΛ¯. As ar<kr+1, then (5.51)var(+)<vkr+1(+)<VΛ¯var(+)-VΛ¯<0. The case when the set kr+1,,kn is empty corresponds to (5.52)Λ¯=(lm,,lr,br-1,,11,,ar-1). Case Πr includes two possible subcases: (5.53)VTr-1<0,var(+)<VTr-1,(5.54)VTr-1>0,vbr-1+1(-)<VTr-1. Consider firstly (5.54). If the set lm,,lr is not empty, then the subcase (5.54) contradicts the ergodicity assumption for (5.52), thus it is impossible. If the set lm,,lr is empty, then Λ¯=Tr-1 and the assumption (5.54) means that VΛ¯=VTr-1>0. As var(+)<0, we easily conclude that in this case: (5.55)vb,arΛ=var(+)-VΛ¯<0. Consider now (5.53). If the set lm,,lr is not empty, then due to the ergodicity of the group (5.52), we have strict inequality VΛ¯>VTr-1. If the set lm,,lr is empty, then Λ¯=Tr-1 and consequently VΛ¯=VTr-1. Finally, we conclude that in the subsituation (5.53) always (5.56)VΛ¯VTr-1. From (5.53), we have (5.57)var(+)<VTr-1var(+)<VTr-1VΛ¯, and it follows that vb,arΛ=var(+)-VΛ¯<0.

This ends the proof.

Proof of Proposition <xref ref-type="statement" rid="prop4">5.1</xref>.

Assume the above algorithm produces the chain of groups (5.36). Let B(Λ1), B(Λ2), , B(Λfin) be the faces in R+N, corresponding to the chain T1,T2,  , Tfin via the rule (5.31). Denote F1, F2, , Ffin the closures of these faces in R+N. That is, in notation (5.42), (5.58)Fi=cl(B(Λi))={xR+N:xj,k0,(j,k){1,,br}×{1,,ar},xj,k=0,(j,k){1,,br}×{1,,ar}xR+N:xj,k0,}. It is clear that F1F2Ffin, and, moreover, dimFi>dimFi+1. More exactly, dimFr-dimFr+1=br or ar in the Case Πr or Ur correspondingly.

Let Γx(t)=(γj,k(t),(j,k)I-×I+) be the coordinate description of the trajectory Γx. To prove that Γx(t)Fr one should check that γj,k(t')=0 for all (j,k){1,,br}×{1,,ar}. The trajectory goes along ergodic faces.

( 1 ) Maximal ergodic face is Λ0=R+N. The vector field vΛ0 on this face is such that v1,1Λ0=v1(+)-v1(-)<0. Note that also for any other ergodic face Λ, containing the pair (1,1), the component v1,1Λ will also be negative, as by (4.31)–(4.33) it can take only one of three following negative values: (5.59)v1(+)-v1(-),v1(+)-VΛ-orVΛ--v1(-). Thus, for any initial point x, there is t10 such that γ1,1(t1)=0, and, moreover, γ1,1(t)=0 for all tt1.

( 2 ) Thus, Γx(t1)F1. If the Case Π2 occurs, then we have to show the existence of t2t1 such that γ1,2(t)=0 for all  tt2. If it appeared that Γx(t1)F2, then just put t2=t1. If, however, Γx(t1)F2, that is, γ1,2(t1)>0, then Γx(t1) belongs to some ergodic face Λ(1,2). By Proposition 5.4v1,2Λ<0, and thus there is t2>t1 such that γ1,2(t2)=0 (i.e., Γx(t2)F2). In future, the dynamical system will never quit F2. In fact, assume the contrary. Note that Γx(t2) can belong either to Λ2, or to its boundary (remind that Λ2¯={(1,1),(1,2)} and cl(Λ2)=F2). For the trajectory to quit F2, it is necessary that it used some outgoing ergodic face Λ'. There are two possibilities to do this. The first possibility is (1,1)Λ'. But in this case (see (5.59)) v1,1Λ'<0, and we get contradiction with the hypothesis that Λ' is an ergodic outgoing face. The second possibility is (1,1)Λ' and (1,2)Λ'. But according to the Proposition 5.4 for any such face v1,2Λ'<0, and thus the dynamical system cannot quit F2 along such face Λ', This gives the contradiction.

If the Case U2 occurred then, quite similarly, one show existence of t2t1 such that γ2,1(t)=0 for all tt2.

(r) We can use further the induction, using subsequently Proposition 5.4, to show on the step (r), that there exists trtr-1 such that for any ttr:

γb,ar(tr)=0 for all  b1,br-1¯, if the Case Πr holds,

γbr,a(tr)=0 for alla1,ar-1¯, if the Case Ur holds.

Let us show now that in any Case Γx(t)Fr for all ttr. For concreteness, consider only the Case Πr, that is, when    (5.60)Fr-1={xR+N:xi,j=0,(i,j){br-1,,1}×{1,,ar-1}},Fr={xR+N:xi,j=0,(i,j){br-1,,1}×{1,,ar}},ar=ar-1+1. Assume that the trajectory of the dynamical system Γx(t), being at time t=tr in Fr, will leave it at some future moment. The set Fr is a finite union of faces having various dimensions. One should understand then which outgoing ergodic faces Λ' can be used. Again, there are two possibilities.

Case 1 . Consider Λ'{br-1,,1}×{1,,ar-1}=, that is, Λ'Fr-1. Then, there exists b{br-1,,1} such that (b,ar)Λ' (otherwise, Λ'Fr, which gives the contradiction). By Proposition 5.4, we have vb,arΛ<0. This contradicts to the fact that the face Λ' is outgoing.

Case 2. Consider Λ{br-1,,1}×{1,,ar-1}. Consider (5.61)q=min{n:Λ{bn,,1}×{1,,an}}. Assume for definiteness, that on step q of the algorithm, we have (5.62)Tq=Tq-1(bq). Then there exists such a{1,,aq-1}, that is, (bq,a)Λ'. Applying Proposition 5.4, to Λ', we get vbq,aΛ<0 and come to the contradiction because Λ' is outgoing.

Thus, there exists a time moment tfin>0 such that for ttfin the trajectory hits the final ergodic face Ffin, which is the complement to the final group (5.35).

Important remark is that the sequence of times (5.63)t1t2trtfin depends on the initial point. In particular, for some initial points, some consequent moments tr-1 and tr can coincide.

Remark 5.5.

Consider the following modification of the algorithm: in Cases (2a) and (r-a) change the conditions VT1<0 and VTr-1<0 on VT10 and VTr-10 correspondingly. All the rest we leave untouched. It is easy to see that all results of this section hold after such modification as well. In particular, our study covers the situation when (VTfin coincides with the asymptotic boundary velocity of our system (see Section 5.4) . )VTfin=0.

From the above, it follows that any trajectory Γx(t) reaches the final face in finite time. To proceed with the proof of Theorem 3.5,we will prove the following lemma.

Lemma 5.6.

For any initial point x, the path Γx(t) has finite number of transitions from one face to another, until it reaches one of the final faces. In other words, the sequence of faces, passed by the path Γx(t), is finite and the last element of this sequence is the final face.

Proof of Lemma <xref ref-type="statement" rid="lem11">5.6</xref>.

Consider an arbitrary trajectory Γx(t). Let {Λix} be a sequence of all faces visited by this trajectory. Denote {Tix} the sequence of the corresponding groups, where Tix=Λix¯. We want to show that the sequence {Λix} is finite.

Two cases are possible for the transition ΛixΛi+1x, or equivalently, for the transition TixTi+1x. If the face Λix is ergodic, then the group Ti+1x is obtained by adding some new particle type to the group Tix. During this transition, the dimension of Λix decreases. If the face Λix is nonergodic, then Λi+1x is the minimal outgoing face, containing Λix (see Lemma 3.4). In the transition ΛixΛi+1x from Tix, some types are deleted, and the dimension of Λix increases. Thus, the transition TixTi+1x can occur with two operations: adding some new type and deleting some types. The same type can be added and deleted several times. If we could show that addition and deletion are possible only finite number of times, that will give finiteness of the sequence {Λix}.

Note the following fact. Take, for example, some (+)-type k. Then, it can be deleted from the group on some step if and only if on the previous step we added to the group some (+)-type with smaller number (i.e., with greater velocity). That is why the type 1, plus or minus, can be added only once and cannot be deleted. (+)-type 2 can be deleted only after adding (+)-type 1. Similarly for (-)-type 2. That is why type 2, plus or minus, can be added to the group not more than twice and can be deleted not more than once. One can prove by induction that any type can be deleted and added not more that finite number of times.

Proof of Theorem <xref ref-type="statement" rid="thm3">3.5</xref>.

Let the chain (5.36) be the result of the algorithm. Three cases are possible, defined by simple inequalities between vL(-),vK(+), and VTfin.

v K ( + ) < V T fin < v L ( - ) : this corresponds to part (2.1) of Lemma 5.3, that is, Λfin=Tfin¯={0}. Thus, (Proposition 5.1), all trajectories of the dynamical system Ut reach 0 for finite time and finite number of changes. Note that from this, using well-known methods (see [2, 9]), one can get alternative proof of ergodicity of D(t), in addition to the one of Theorem 3.1. The first assertion of Theorem 3.5 is proved.

V T fin < v K ( + ) < 0 : this case corresponds to part 2 of Lemma 5.3, and thus, (5.64)Tfin=(L,,11,,K1), where K1<K. From the rules of the algorithm, it follows immediately that vK1+1(+)>VTfin, but vK1(+)<VTfin. Thus, (see Theorem 3.1), the process DTfin¯(t) is ergodic, and the face Λfin=Tfin¯ is also ergodic. Find now the vector vΛfin. Note that (5.65)Λfin=(L,K1)={(i,k):i=1,,L,k=K1+1,,K}. To find components of vi,kΛfin, we use the formulas (4.31)–(4.33): (5.66)vi,kΛfin=vk(+)-VΛfin¯>vK1+1(+)-VTfin>0(i,j){1,,L}×{K1+1,,K},vi,kΛfin=0(i,j){1,,L}×{1,,K1}. By Proposition 5.1 any trajectory, in finite time and after finite number of changes, will reach (L,K1), and will move along it with constant speed vΛfin, having strictly positive components (5.66). By standard methods of [2, 9], we conclude that D(t) is transient. The second assertion of Theorem 3.5 is proved.

0 < v L ( - ) < V T fin : this case corresponds to part (3) of Lemma 5.3, and the proof is completely similar to the previous case. That proves assertion (3) of Theorem 3.5.

The fourth assertion of theorem 3 is a corollary of Proposition 5.1 and Lemma 5.6.

Theorem 3.5 is proved.

5.4. Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>

If associated random walk D(t) is ergodic, then by Lemma 4.2, the speed of the boundary equals V which is defined by (2.5).

Let the process D(t) be nonergodic. Then, there are two possible cases: vK(+)>V or vL(-)<V. From the previous Section 5.3, it follows that any trajectory Γx(t) reaches the final face in finite time and during this time only finite number of changing the face occurs.

The following assertion is an obvious analog of the proposition 1.4.3 of .

Lemma 5.7.

For any t0 and any initial point x, (5.67)DxM(tM)MΓx(t), a.e. as M.

Let vK(+)>V. We have proved that any trajectory of the dynamical system Ut reaches the final face (L,K1), where the coordinates of the induced vector are positive. By Lemma 5.7 the coordinates dq,r(t) of the process D(t), where q=1,,L, r=K1+1,,K, grow linearly (a.e.) as t. In other words (+)-types with numbers r=K1+1,,K fall behind the boundary and do not contribute to its velocity. It means that the boundary velocity is defined only by the particles of types q=1,,L, r=1,,K1 and are given by formula (2.5). The case of vL(-)<V is quite similar.

Appendices A. Proof of Lemma <xref ref-type="statement" rid="lem1">3.2</xref>

Let the face Λ be such that Λ¯ is not the direct product. Put (A.1)I-Λ¯={iI-:kI+,(i,k)Λ¯},I+Λ¯={kI+:iI-,(i,k)Λ¯}. Choose an “appropriate” face Λ0 so that Λ¯0=I-Λ¯×I+Λ¯. To prove the lemma, it is sufficient to show that (A.2)𝒟Λ=𝒟Λ0. As ΛΛ0, we always have 𝒟Λ𝒟Λ0. Let us prove that 𝒟Λ𝒟Λ0. Let (i,k)Λ¯0 and (i,k)Λ¯. Then, there exist mI+ and nI- such that (i,m)Λ¯, (n,k)Λ¯ and the following equation holds: (A.3)di,k(t)+dn,m(t)=di,m(t)+dn,k(t). Take arbitrary element d=(dj,l) of the set 𝒟Λ. As its coordinates di,m(t)=dn,k(t)=0, then di,k=0 for all (i,k)Λ¯0. Thus, d𝒟Λ0, and the lemma is proved.

B. Technical Lemma

For shortness, denote (B.1)f(k)=V(I-,{1,,k}),g(l)=V({1,,l},I+).

Lemma B.1.

one has

vk+1(+)<f(k+1)f(k+1)<f(k), k=1,,K-1,

vk+1(+)>f(k+1)f(k+1)>f(k), k=1,,K-1

vk(+)>f(k)vk+1(+)>f(k+1), k=2,,K-1.

Similarly,

vl+1(-)<g(l+1)g(l+1)<g(l), l=1,,L-1

vl+1(-)>g(l+1)g(l+1)>g(l), l=1,,L-1

vl(-)<g(l)vl+1(-)<g(l+1), l=2,,L-1.

Proof.

We prove the first three items. The others are quite similar. Using (2.5), one can check (B.2)f(k+1)=αf(k+1)+βf(k+1)=αf(k)+βvk+1(+) for some α,β>0 such that α+β=1. It follows that (B.3)α(f(k+1)-f(k))=β(vk+1(+)-f(k+1)). Thus, vk+1(+)<f(k+1)f(k+1)<f(k). If vk(+)>f(k), using vk(+)<vk+1(+), we get (B.4)f(k+1)<αvk(+)+βvk+1(+)<αvk+1(+)+βvk+1(+)=vk+1(+).

The Lemma is proved.

Let K1 and L1 be defined by (2.9) and (2.8). It follows from the lemma that (B.5)f(1)>···>f(K1)<f(K1+1)<···<f(K),g(1)<···<g(L1)>g(L1+1)>···>g(L). So the minimum of f(k) is reached at point K1 and maximum of g(l) is reached at point L1.

Funding

A. D. Manita was supported by the Russian Foundation of Basic Research (Grants 12-01-00897 and 11-01-90421).

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