The system of processes that describes evolution of the particles is constructed by martingale methods

MATHEMATICAL MODEL OF HEAVY DIFFUSION PARTICLES SYSTEM WITH DRIFT

VITALII KONAROVSKYI*

Abstract. In the article we consider the model of coalescing diffusion parti- cles which have some masses. At the moment of coalescing the masses of the particles are summed together and influence their motions. The system of processes that describes evolution of the particles is constructed by martingale methods. The Markov property of this system is stated and the asymptotic restriction on the mass growing of an individual particle is obtained.

1. Introduction

This paper is devoted to the construction a mathematical model of coalescing diffusion particles onR. We assume that every particle has a mass, which influences its diffusion and drift. The particles start from a finite or countable set of points, move independently up to the moment of meeting, after which they coalesce and their masses are summed.

Systems of coalescing diffusion particles were studied by Arratia R. A. [1, 2], Le Jan Y. [15], Norris J. [18], Evans S. S. [8], Dawson D. A. [3, 4], Dorogovtsev A. A. [5, 6], Konarovskyi V. V. [13, 14, 12] and others. Particular attention is paid to a fairly wide class of coalescing particles systems, in which every subsystem may be described as a separate system [15, 17, 8, 1, 9]. On the one hand, such systems are widely used in turbulence theory and statistical mechanics [18, 10], on the other hand they represent an important interest in terms of mathematics itself.

For example, the fact that the particles which start from an arbitrary compact set, instantly coalesce to the finite number [8], allows to integrate over a stochastic flow [5], and the latter, in turn, develops a new stochastic analysis. It should be noted that the ability to describe the motion of an arbitrary subsystem of the system, without taking into consideration all the particles of the system, allows to develop good methods for the study of appropriate mathematical models.

Often there is a need to assume that the particles transfer some mass. Models in which the particles have mass are actively studied. However, in some models a mass that is transferred does not influence their motion [3, 4, 21, 20], while in others, it influences but the particles don’t coalesce (smooth interaction) [5].

Received 2013-4-5; Communicated by A. Dorogovtsev.

2010Mathematics Subject Classification. Primary 60K35; Secondary 60G44.

Key words and phrases. Coalescing particles, change mass, stochastic differential equation, Wiener processes.

* This work was partially supported by the State fund for fundamental researches of Ukraine and the Russian foundation for basic research under project F40.1/023.

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Vol. 7, No. 4 (2013) 591-605

In the study of systems in which the particles transfer some mass, from the physical point of view it is natural to assume that in coalescing the mass is pre- served (the mass of the new particle is equal to the sum of the masses of particles, from which it was formed) and influences their motion. A system of Brownian particles which have masses that are summed together at the moment of coalesc- ing was constructively constructed in the papers [13, 14]. Moreover, the diffusion of particles changes only when particles are changing their masses. In this case, the random environment in which particles diffuse, is homogeneous. The desire to make the model that was under consideration earlier, more close to reality (to include heterogeneity and drift) leads to the fact that we have to consider the diffusion coefficients, which depend not only on the mass, but also on the position of the particles. So, it is assumed that the trajectory x(t), t ≥ 0, of a particle satisfies the following stochastic differential equation

dx(t) =a(x(t))

m(t) dt+σ(x(t)) pm(t)dw(t),

wherem(t) is a mass of the particles at the momentt,w(t), t≥0, some Wiener process, a, σ are bounded Lipschitz continuous functions and inf

x∈Rσ(x) >0. We call such system of the particles the heavy diffusion particles system with drift.

It should be noted that in this case the system can not be described by spec- ifying its finite subsystems, as it was done in the work [15, 17, 8, 9]. So, first a mathematical model of a finite number of particles is constructed, after we do the passing to the limit as the number of particles tend to infinity. Ability of passing to the limit ensures that the particles which are far from an isolated subsystem of finite system of particles, have little effect on it (Lemma 4.6).

This work consists of two parts. In the first part it is constructed the mathe- matical model of a finite number of particles (Section 2) and its Markov property is stated (Section 3). In the second part the passing to the limit as the number of particles tend to infinity is done and some properties of the infinite system of particles are shown (Sections 5, 6).

2. Finite Particles System

In this section it is studied the case of finite number of particles. The set of processes that describe such motion, is constructed by coalescing and rescaling of the solutions of stochastic differential equations

dx_i(t) =a(x_i(t))dt+σ(x_i(t))dw_i(t),

wherewi(t), t≥0, i= 1, . . . , N are independent Wiener processes.

LetN ∈Nis fixed. Denote [N] ={1,2, . . . , N}.

Definition 2.1. A setπ={π1, . . . , πp}of non-intersection subsets of [N] is called order partitioningof [N] if

1) Sp i=1

πi= [N];

2) if l, k∈πi then{l∧k, . . . , l∨k} ⊆πi, for alli= 1, . . . , p.

The set of all order partitioning of [N] is denoted by Π^N.

Every element π = {π1, . . . , πp} ∈ Π^N generates equivalence between [N] el- ements. We assume that i ∼π j if there exists a numberk such that i, j ∈πk. Denote an equivalence class that contain the elementi∈[N] bybiπ, i.e.biπ={j ∈ [N] : j ∼πi}.

Letγ: [N]→[N] be some bijection. Define i^γ_π=γ⁻¹

min

j∈bi_π

γ(j)

.

Remark 2.2. The mapγ will define range of particle. We will suppose thati-th particle has rangeγ(i).

If (R, r) is some metric space then we denote by C_R the space of continuous functions from [0,∞) toRwith metric

d_R(ξ, η) = X∞ k=1

1 2^k

max

t∈[0,k]

r(ξ(t), η(t))∧1

, ξ, η∈C_R.

Consider the subspace E^N ={x∈R^N : xi ≤xi+1, i = 1, . . . , N −1} of the spaceR^N and the setB^N ={b∈R^N : bi >0, i= 1, . . . , N}. Elements of space E^N andB^N will be used as start points and masses of particles, respectively.

Takeb∈B^N and construct a map Λ^b_γ from{ξ∈C_RN : ξ(0)∈E^N}to C_RN. It will be used to define a system of processes that describes the joint motion of the N particles system. Letξ∈C_RN andξ(0)∈E^N. Construct an elementζ= Λ^b_γξ by induction.

Takeπ⁰∈Π^N such that

i∼π⁰ j⇔ξi(0) =ξj(0).

Putτ0= 0 and

ζ_i⁰=ξ_i^γ

π0

P t

j∈bi_π0bj

!

, t≥0, i= 1, . . . , N.

Letπ^k,τk,ζ_i^k,i= 1, . . . , N are defined. Denote

τk+1= inf{t > τk: ζ_i^k(t) =ζ_j^k(t), i6∼π^k j, i, j= 1, . . . , N}. Ifτ_k+1=∞then putπ^k+1=π^k, else take an elementπ^k+1∈Π^N such that

i∼π^k+1j⇔ζ_i^k(τk+1) =ζ_j^k(τk+1).

Define

ζ_i^k+1(t) =





ζ_i^k(t), t < τk+1,

ζ_i^γ

πk+1

τk+1+

(t−τ_k+1)P

j∈biπk

b_j P

j∈bi πk+1b_j

, t≥τk+1. Put Λ^b_γξ=ζ^N−1.

Remark 2.3. Λγ is measurable map from the space (Ln,Ln) to (C_RN,B(C_RN)), whereLn =

f ∈C_RN :f(0)∈E^N ,Ln=B(C_RN)∩Ln. Let’s state the main result of this section.

Theorem 2.4. Let γ: [N]→[N] be some bijection, x∈E^N,b∈B^N andξi(t), t≥0,i= 1, . . . , N, be solutions of the stochastic differential equations

dξi(t) =a(ξi(t))dt+σ(ξi(t))dwi(t),

ξi(0) =xi, (2.1)

were wi(t), t≥0, i= 1, . . . , N, are independent Wiener processes,a, σ are some bounded Lipschitz continuous functions onRand inf

x∈Rσ(x)>0. Then the random process

ζ= Λ^b_γξ satisfies the following conditions

1^◦) Mi=ζi(·)−R^·

0 a(ζi(s))

m_i(s) dsis a continuous square integrable martingale with respect to the filtration

Ft^ζ =σ(ζ_i(s), s≤t, i= 1, . . . , N), wherem_i(t) = P

j∈Ai(t)

b_j,A_i(t) ={j: ∃s≤t ζ_j(s) =ζ_i(s)}; 2^◦) ζ_i(0) =x_i,i= 1, . . . , N;

3^◦) ζ_i(t)≤ζ_j(t),i < j,t≥0;

4^◦) hMiit= Rt 0

σ²(ζi(s))

m_i(s) ds,t≥0;

5^◦) hMi,MjitI_{t<τi,j}= 0,t≥0, where τi,j= inf{t:ζi(t) =ζj(t)}.

Remark 2.5. Further, unless otherwise stated we assume thata, σ are a bounded Lipschitz continuous functions onRand inf

x∈Rσ(x)>0.

The proof of the theorem follows from the construction of mapping Λ^b_γ and the next lemma.

Lemma 2.6. Let wi(t), t ≥ 0, i = 1, . . . , N, be a set of independent Wiener processes, τ be a stopping time with respect to the filtration Ft^w =σ(wi(s), s ≤ t, i= 1, . . . , N)and a random variableξbe strictly positive measurable with respect toFτ^w. Then

b w_i(t) =

wi(t), ift < τ,

wi(τ) +√¹

ξ[wi(τ+ (t−τ)ξ)−wi(τ)], else

are independent Wiener processes, moreover,τ is a stopping time with respect to Ft^wb = σ(wbi(s), s ≤ t, i = 1, . . . , N) and random variable ξ is measurable with respect to Fτ^wb.

Theorem 2.4 describes evolution of the finite heavy diffusion particles system with drift. Let’s prove that the conditions 1^◦)-5^◦) uniquely determine the distri- bution of such particles system.

Lemma 2.7. Suppose that a system of processesζi(t),t≥0,i= 1, . . . , N, satisfies the condition 1^◦)-5^◦) of Theorem 2.4 and γ: [N]→[N] is some bijection. Then there exists a system of independent Wiener processes wi(t), t≥0, i= 1, . . . , N, such that

ζ= Λ^b_γξ,

where ξi(t), t ≥0, i= 1, . . . , N, are solutions of the stochastic differential equa-

tions

dξi(t) =a(ξi(t))dt+σ(ξi(t))dwi(t), ξi(0) =xi.

Proof. Suppose that ζ_i(t), t ≥ 0, i = 1, . . . , N, satisfy the conditions 1^◦)-5^◦) of Theorem 2.4. We first show that the processes ζ_i andζ_j coalesce at the moment of the meeting, for alli, j= 1, . . . , N, i.e.

P{ζi(τi,j+t) =ζj(τi,j+t), t≥0|τi,j<∞}= 1.

Since √^σ(ζi(s))

m_i(s) >0,i= 1, . . . , N, than by the Doob theorem [16] there exists a system of Wiener processeswei(t),t≥0,i= 1, . . . , N, adapted to the filtrationFt^ζ

such that

ζ_i(t) =x_i+ Zt 0

a(ζ_i(s)) mi(s) ds+

Zt 0

σ(ζ_i(s)) pmi(s)dwe_i(s).

Takei < j,n∈Nand denote τ_i,jⁿ =τ_i,j∧n. From last equation we have

ζk(t+τ_i,jⁿ) =ζk(τ_i,jⁿ) + Zt

0

a(ζk(s+τ_i,jⁿ)) mk(s+τ_i,jⁿ) ds+

Zt 0

σ(ζk(s+τ_i,jⁿ )) q

m_k(s+τ_i,jⁿ )

dweⁿ_k(s).

wherewe_kⁿ(t) =wek(t+τ_i,jⁿ)−wek(τ_i,jⁿ ), t≥0, k=i, j. Using the Lipschitz continuity of the function aand equality mi(t+τ_i,jⁿ )I_{τ_i,jⁿ<n} =mj(t+τ_i,jⁿ)I_{τ_i,jⁿ<n}, t≥0, we obtain

(ζj(t+τ_i,jⁿ )−ζi(t+τ_i,jⁿ))I_{τ_i,jⁿ<n}≤L Zt 0

(ζj(s+τ_i,jⁿ )−ζi(s+τ_i,jⁿ))I_{τ_i,jⁿ<n}ds

+



 Zt 0

σ(ζj(s+τ_i,jⁿ)) q

m_j(s+τ_i,jⁿ)

dwe_jⁿ(s)− Zt 0

σ(ζi(s+τ_i,jⁿ)) q

m_i(s+τ_i,jⁿ ) dweⁿ_i(s)



I_{τ_i,jⁿ<n}.

Since the random variableI_{τ_i,jⁿ<n} is measurable with respect toFτ^ζ_i,jⁿ, E

h

(ζj(t+τ_i,jⁿ)−ζi(t+τ_i,jⁿ))I_{τ_i,jⁿ<n}

i

≤ Zt 0

E h

(ζj(s+τ_i,jⁿ)−ζi(s+τ_i,jⁿ ))I_{τ_i,jⁿ<n}

i ds.

From Gronwall’s inequality we have E

h

(ζ_j(t+τ_i,jⁿ )−ζ_i(t+τ_i,jⁿ))I_{τ_i,jⁿ<n}

i

= 0.

By Fatou’s lemma, E

(ζ_j(t+τ_i,j)−ζ_i(t+τ_i,j))I_{τ_i,j<∞}

= 0.

Hence, by virtue of the continuity of the processes ζi and ζj, we obtain needed equality.

Next calculate

hMi,MjitI_{t<τi,j}= Zt 0

σ(ζi(s))σ(ζj(s)) pm_i(s)p

m_j(s)dhwei,wejisI_{t<τi,j}= 0.

Hence

hwei,wejitI_{t<τi,j}= 0.

Let’s take a system of Wiener processes w⁰_i(t), t ≥ 0, i = 1, . . . , N, that are independent ofwei(t), t≥0, i= 1, . . . , N, and denote

δj= inf{t: ζ_γ−1(j)(t)∈ {ζ_γ−1(1)(t), . . . , ζ_γ−1(j−1)(t)}}, j= 2, . . . , N.

b w_i(t) =

we_i(t), ift < δ_γ(i),

e

wi(δ_γ(i)) +w⁰_i(t)−w⁰_i(δ_γ(i)), else,

where δ1 = +∞ andi= 1, . . . , N. By the Levi theorem (see Theorem 2.6.1 [11]) b

w_i(t), t ≥ 0, i = 1, . . . , N, are a system of independent Wiener processes. Let π⁰∈Π^N such thati∼π⁰ j⇔ζ_i(0) =ζ_j(0) andτ₀= 0. Set

τk = inf{t > τk−1: ζi(t) =ζj(t), i6∼π^k−1 j, i, j= 1, . . . , N} and ifτk =∞then putπ^k=π^k−1, else take an elementπ^k∈Π^N such that

i∼π^k j⇔ζ_i(τ_k) =ζ_j(τ_k).

Using the system of the processeswbi(t), t≥0, i= 1, . . . , N, the stopping timesτk

and the elementsπ^k,k= 0, . . . , N−1, one can construct a system of independent Wiener processesw_i(t), t≥0, i= 1, . . . , N, such that

ζ= Λ^b_γξ,

where ξi(t),t ≥0,i= 1, . . . , N, are solutions of the stochastic differential equa-

tions

dξi(t) =a(ξi(t))dt+σ(ξi(t))dwi(t), ξi(0) =xi.

The lemma is proved.

Corollary 2.8. The conditions 1^◦)-5^◦) of Theorem 2.4 uniquely determine the distribution of the process in the space(C_RN,B(C_RN)).

Definition 2.9. A system of processes is called the process of heavy diffusion particles with drift in the spaceE^N if it satisfies the conditions 1^◦)-5^◦) of Theo- rem 2.4.

3. Strictly Markov Property of the Process of Heavy Diffusion Particles with Drift in the SpaceE^N.

In this section the strictly Markov property of the heavy diffusion particles with drift is stated. Letγ: [N]→[N] be some bijection,x∈E^N, b∈B^N, ξ_i(t), t≥0,i= 1, . . . , N, be solutions of the stochastic differential equations (2.1) and ζ = Λ^b_γξ. Denote by P^ξx the distribution of the random process ξ in the space C_RN. As is well known (see for instance [7]),x→P^ξx(A) is Borel function, for all A∈ B(C_RN). Let

P^ζx=P^ξx◦(Λ^b_γ)⁻¹.

Then the mapx→P^ζx(A) is Borel function.

Theorem 3.1. The set of the distributions {P^ζx, x ∈ E^N} is strictly Markov system.

Proof. Let Ft(C_RN) = T

ε>0

T

x∈E^N

Bt+ε(C_RN)^P

ζ

x, where Bt(C_RN)^P

ζ

x denotes the σ- algebra of cylinder sets {y ∈ C_RN : y(s) ∈ B}, s ≤ t, B ∈ B R^N

, that is completed by all P^ζx-null sets. We will show that, for every bounded Ft(C_RN)- stopping timeτ,

P^ζx(A∩ {y: y(t+τ(y))∈Γ}) = Z

A

P^ζ_y0(τ(y⁰)){y: y(t)∈Γ}P^ζx(dy⁰),

where A ∈ Fτ(C_RN), Γ ∈ B R^N

, x ∈E^N. This will be sufficient to prove our theorem.

Fixi= 1, . . . , N. SinceNi =yi(·)−R^·

0 a(yi(s))

m_i(s) dsis a (P^ζx,Bt(C_RN))-martingale for each x∈ E^N and t → Ni(t) is right continuous, Ni is also a (P^ζx,Ft(C_RN))- martingale. By Doob’s optional sampling theoremNi(·+τ) is a (P^ζx,Ft+τ(C_RN))- martingale. In particular, fort > s A∈ Fs+τ(C_RN) andC∈ Fτ(C_RN), we have

Ex

(Ni(t+τ)−Ni(s+τ))I_{A∩C}

= 0.

This implies that

Ex[(Ni(t+τ)−Ni(s+τ))IA|Fτ(C_RN)] = 0 forP^ζx-a.a.y.

Therefore, ifPe(y, A) =P^ζx(θ⁻_τ¹(A)|Fτ(C_RN)), A∈ B(C_RN) is the regular condi- tional probability with respect toFτ(C_RN) (it exists by Theorem 1.3.1 [11]), where θτ : C_RN → C_RN is defined by (θτy)(t) = y(t+τ(y)), then P(y,e {y⁰ : y⁰(0) = y(τ(y))}) = 1 for P^ζx-a.a. y and Ni is a (Pe(y,·),Ft(C_RN))-martingale. Similarly, N²_i −R^t

0

σ²(y_i⁰(s))

m_i(s) dsand Ni(t∧τi,j)Nj(t∧τi,j) are (Pe(y,·),Ft(C_RN))-martingales.

Hence, by Lemma 2.7, Pe(y,·) = P^ζ_y(τ(y)). Thus, for every A ∈ Fτ(C_RN) and Γ∈ B R^N

, we have Z

A

P^ζ_y0(τ(y0)){y: y(t)∈Γ}P^ζx(dy⁰) = Z

A

Pe(y⁰,{y : y(t)∈Γ})P^ζx(dy⁰)

= Z

A

P^ζx({y: y(t+τ(y))∈Γ}|Fτ(C_RN))P^ζx(dy⁰) =P^ζx(A∩{y: y(t+τ(y))∈Γ}).

The theorem is proved.

4. Infinite Particle System

In this section the countable particles system is considered. The system of processes which describes the motion of the particles is constructed from a finite system of processes by passing to the limit. The following theorem holds.

Theorem 4.1. Leta, σ be bounded Lipschitz continuity functions and inf

x∈Rσ(x)>

0. Then for every non-decreasing sequence of real numbers {x_i, i ∈Z} and se- quence of strictly positive real numbers{b_i, i∈Z} such that

n→±∞lim {(xn+1−xn)∧bn+1∧bn}>0, (4.1) there exists a set of processes ζi(t), t≥0, i∈Z, satisfying

1^◦) M_i=ζ_i(·)−R^·

0 a(ζ_i(s))

mi(s) dsis a continuous square integrable martingale with respect to the filtration

Ft^ζ =σ(ζ_i(s), s≤t, i∈Z), wherem_i(t) = P

j∈Ai(t)

b_j,A_i(t) ={j: ∃s≤t ζ_j(s) =ζ_i(s)}; 2^◦) ζ_i(0) =x_i,i∈Z;

3^◦) ζ_i(t)≤ζ_j(t),i < j,t≥0;

4^◦) hMiit= Rt 0

σ²(ζi(s))

m_i(s) ds,t≥0;

5^◦) hMi,MjitI_{t<τi,j}= 0,t≥0, where τi,j= inf{t:ζi(t) =ζj(t)}. Remark 4.2. In case wheremi(t) =∞we assume that _m¹

i(t) = 0.

In order to prove the theorem, we will state several auxiliary lemmas.

Let{ni, i∈Z}be some strictly increasing sequence of real numbers. FixN ∈N and choose a bijectionγ^N : [2N+ 1]→[2N+ 1] as follows. Denote

D1={ni, ni+ 1, i∈Z} ∩[0, N] ={p¹₁, . . . , p¹_k1}, p¹₁< . . . < p¹_k1, D2={ni, ni+ 1, i∈Z} ∩[−N,0) ={p²₁, . . . , p²_k2}, p²₁< . . . < p²_k2. Let

Z∩[0, N]\D1={p³₁, . . . , p³_k3}, p³₁< . . . < p³_k3, Z∩[−N,0)\D2={p⁴₁, . . . , p⁴_k4}, p⁴₁< . . . < p⁴_k4. Put

γ^N(N+ 1 +p¹_i) =i, i= 1, . . . , k₁, γ^N(N+ 1−p²_i) =k1+i, i= 1, . . . , k2, γ^N(N+ 1 +p³_i) =k1+k2+i, i= 1, . . . , k3, γ^N(N+ 1−p⁴_i) =k1+k2+k3+i, i= 1, . . . , k4.

Lemma 4.3. Let {ni, i ∈Z} be a strictly increasing sequence of real numbers, γ^N : [2N + 1] → [2N + 1] be the bijection defined above, {x_i, i ∈ Z} be a non- decreasing a sequence of real numbers,{b_i, i∈Z} be a sequence of strictly positive numbers and{f_k, k∈Z} ⊂C(R),f_k(0) =x_k. Denote

(g^N_−N, . . . , g^N_N) = Λ^(b_γN^−N,...,bN)(f_−N, . . . , f_N), N ∈N.

(i) If for some m∈NandT >0 there existC >0andδ >0such that max

max

t∈[0,T]fk

t bk

, k∈ {ni, nj+ 1; i= 0, . . . , m, j= 0, . . . , m−1}

< C,

min

t∈[0,T]fn_m+1

t b_nm+1

> C+δ, then for allN > nm andk=−N, . . . , nm

max

t∈[0,T]g^N_k(t)≤ max

t∈[0,T]g_n^N_m(t)< C, min

t∈[0,T]g_n^N_m+1(t)> C+δ.

(ii) If for some −m∈Nthere existC <0andδ <0such that min

min

t∈[0,T]fk

t bk

; k∈ {ni, nj+ 1; i=m+ 1, . . . ,0 j=m, . . . ,0}

> C,

max

t∈[0,T]fn_m

t bn_m

< C+δ, then for allN >−n_mandk=n_m+ 1, . . . , N

min

t∈[0,T]g^N_k(t)≥ min

t∈[0,T]g_n^N_m+1(t)> C, max

t∈[0,T]g^N_nm(t)< C+δ.

The proof of the lemma immediately follows from the construction of the map Λ^(b_γN^−N,...,bN),N ∈N, and the choice of the bijectionγ^N,N ∈N.

Letf :D→Rbe some bounded function. Definekfk= sup

x∈D|f(x)|. Lemma 4.4. Leta, σ be a bounded Lipschitz continuous functions and inf

x∈Rσ(x)>

0. Then for eachδ >0andT >0 the solution of the equations

ξ(t) =x0+ Zt 0

a(ξ(s))ds+ Zt 0

σ(ξ(s))dw(s) (4.2)

satisfies following condition P

max

t∈[0,T]ξ(t)−x0< δ

≥P



w(t)< δ− kak inf

x∈Rσ(x)²t, t∈

0, T · kσk²



. Lemma 4.5. For every a∈R,δ >0and a Wiener processw(t), t≥0,

P{w(t) +at < δ, t∈[0, T]}>0.

The proof easily follows from the Girsanov theorem.

Lemma 4.6. Let yn, n∈N, be a non-decreasing sequence of real numbers such that inf

n≥1(yn+1−yn) =δ >0,ξn(t), t≥0, n∈N,be the solutions of the equations ξ_n(t) =y_n+

Zt 0

a(ξ_n(s))ds+ Zt 0

σ(ξ_n(s))dw_n(s),

wherewn(t), t≥0, n∈N,are a set of Wiener processes and ξ^max_n = max

t∈[0,T]ξn(t), ξ_n^min= min

t∈[0,T]ξn(t).

Then for everyδ₁∈ 0,^δ₂ , P

nlim→∞

max

k=1,...,nξ_k^max≤y_n+δ

2, ξ_n+1^min> y_n+1−δ₁

= 1.

The proof of the lemma is similar to the proof of Lemma 5 [13], considering Lemmas 4.5 and 4.4.

Proof of Theorem 4.1. Since the sequences {xi, i ∈ Z} and {bi, i ∈ Z} satisfy the inequality (4.1), there exists a strictly increasing sequence of real numbers {ni, i∈Z} such that

inf

i∈Z(xn_i+1−xn_i) =δ >0, sup

i∈Z

1 b_ni, 1

b_ni+1

<∞.

Let γ^N : [2N + 1] → [2N + 1] be a bijection constructed by the sequence {ni, i∈Z} andξn(t), t≥0, n∈Z,be the solutions of the equations

ξn(t) =xn+ Zt 0

a(ξn(s))ds+ Zt 0

σ(ξn(s))dwn(s),

wherewn(t), t≥0, n∈Z,are independent Wiener processes.

Put

(ζ₋^N_N, . . . , ζ_N^N) = Λ^(b_γN^−N,...,bN)(ξ₋N, . . . , ξN), for eachN∈N. FixT >0. By Lemmas 4.3 and 4.6

P

∃N ∀n≥N ζ_kⁿ(t) =ζ_k^N(t), t∈[0, T] = 1,

for anyk∈Z, i.e. the sequence{ζ_kⁿ(t), t∈[0, T]}n≥k is stabilized with probabil- ity 1, for all integerk. Denote the limit of{ζ_kⁿ, t∈[0, T]}n≥k byζk,T. From the stabilization of{ζ_kⁿ, t∈[0, T]}n≥k it follows that

P

∃N ∀n≥N mⁿ_k(t) =m^N_k (t), t∈[0, T] = 1, where mⁿ_k(t) = P

j∈Aⁿ_k(t)

bj, Aⁿ_k(t) ={j : ∃s≤t ζ_jⁿ(s) =ζ_kⁿ(s)}, t≥0. Let mk,T

denote the limit of the sequence {mⁿ_k, t ∈ [0, T]}n≥k. Denote ζk(t) = ζk,T(t) and m_k(t) = m_k,T(t), for someT ≥t. It is clear that such definition is correct, moreover,

mi(t) = X

j∈Ai(t)

bj, Ai(t) ={j: ∃s≤t ζj(s) =ζi(s)}.

From the stabilization it follows that one can construct a system of Wiener processeswe_n(t), t≥0, n∈Z, such thathwe_i,we_jitI_{t<τ_i,j}= 0 and

ζ_n(t) =x_n+ Zt 0

a(ζn(s)) m_n(s) ds+

Zt 0

σ(ζn(s))

pmn(s)dwe_n(s), n∈Z.

This implies that the system of the processes ζn(t), t≥0, n∈Z is found. The

theorem is proved.

Lemma 4.7. Let a sequence of random processes ζn, t ≥0, n ∈ Z, satisfy the conditions1^◦)-5^◦) of Theorem 4.1. Thenmn(t)<∞, for allt≥0 andn∈Z, i.e.

P{∃j₁, j₂ ζ_j1(s)< ζ_n(s)< ζ_j2(s), s≤t}= 1.

Proof. Let {ni, i∈ Z} be a strictly increasing sequence of integer number such that

inf

i∈Z(xn_i+1−xn_i) =δ >0, sup

i∈Z

1 b_ni, 1

b_ni+1

=C <∞. (4.3) Fixt >0 and takex >2tCkak. Let’s estimate following probability fori < j

P{Mn_j(s)−Mn_i(s)> x, s≤t}=

=P



ζnj(s)− Zs 0

a(ζ_nj(r))

mn_j(r) dr−ζni(s) + Zs 0

a(ζ_ni(r))

mn_i(r) dr > x, s≤t



≤

≤P

ζn_j(s)−ζn_i(s) + 2tCkak> x, s≤t ≤P

ζn_j(s)−ζn_i(s)>0, s≤t Next, let i ∈Z is fixed. For every m ∈N take y_m ∈R and a numbern_jm such that

P{Mni(s)< y_m, s≤t} ≥1− 1 2m² and

P{M_njm(s)> y_m+x, s≤t} ≥1− 1 2m². Write

P{Mn_jm(s)−Mn_i(s)> x, s≤t}

≥P{{Mn_jm(s)> ym+x, s≤t} ∩ {Mn_i(s)< ym, s≤t}} ≥1− 1 m². Hence

P

ζ_njm(s)−ζ_ni(s)>0, s≤t ≥1− 1 m². By Borel-Cantelli lemma,

P

inf

s≤t(ζ_njm(s)−ζ_ni(s)) = 0, for infinite numbersm

= 0.

So, we have

P

∃j∈Z inf

s≤t(ζn_j(s)−ζn_i(s))>0

= 1.

Similarly

P

∃j∈Z inf

s≤t(ζ_ni(s)−ζ_nj(s))>0

= 1.

This proves the lemma.

Theorem 4.8. The conditions 1^◦)-5^◦) of Theorem 4.1 uniquely determine the distribution of the process in the space(C_RZ,B(C_RZ)).

Proof. Let a system of random processesζn, t≥0, n∈Z, satisfy the conditions 1^◦)-5^◦) of Theorem 4.1. Then, by the Doob theorem [16], there exists a system of Wiener processeswe_n(t),t≥0,n∈Z, which areFt^ζ-adapted, such that

ζn(t) =xn+ Zt 0

a(ζn(s)) mn(s) ds+

Zt 0

σ(ζn(s))

pmn(s)dwen(s).

From condition 5^◦) we have

hwei,wejitI_{t<τi,j}= 0, t≥0, i, j∈Z.

Similarly to the proof of Lemma 2.7, it is easily seen thatζi(t) =ζj(t) fort≥τi,j

andi, j∈Z.

Next, let {ni, i ∈ Z} be a strictly increasing sequence of integer numbers satisfying (4.3). Construct a bijection γ_∞ : Z → N as follows. For every i ∈ Z define

γ_∞(i) = lim

N→∞γ^N(i+N+ 1),

where the bijectionγ^N,N ∈N, defined above. The existence of limit follows from the constructions ofγ^N,N ∈N.

Take a system of independent Wiener processes w_i⁰(t), t≥0, i∈Z, which are independent ofwe_i(t), t≥0, i∈Zand denote

δj = inf{t: ζ_γ−1

∞(j)(t)∈ {ζ_γ−1

∞(1)(t), . . . , ζ_γ−1

∞(j−1)(t)}}, j = 2,3, . . . . Put

b wi(t) =

wei(t), ift < δ_γ∞(i),

e

wi(δ_γ∞(i)) +w⁰_i(t)−w⁰_i(δ_γ∞(i)), else,

where δ1 = +∞ and i ∈ Z. By the Levi theorem (see Theorem 2.6.1 [11]), b

wi(t), t≥0, i∈Z, are independent Wiener processes.

Let N ∈ N and take π^0,N ∈ Π^2N+1 such that i ∼π^0,N j ⇔ ζ_i−N−1(0) = ζ_j−N−1(0) andτ_0,N = 0. Denote

τ_k,N = inf{t > τ_k−1,N : ζ_i−N−1(t) =ζ_j−N−1(t), i6∼πk−1,N j, i, j∈[2N+ 1]} and ifτk,N =∞then putπ^k,N =π^k−1,N, else takeπ^k,N ∈Π^2N+1 such that

i∼π^k,N j⇔ζi−N−1(τk,N) =ζj−N−1(τk,N).

Using the system of the processes wb_i(t), t ≥ 0, i =−N, . . . , N, stopping times τ_k,N and the elements π^k,N, k = 0, . . . ,2N, in reverse order (similar to how it was done in the proof of Theorem 2.4), one can construct a system of independent Wiener processesw_i^N(t), t≥0, i=−N, . . . , N.

From Lemma 4.7 and the construction ofw^N_i (t), t≥0, i=−N, . . . , N, N ∈N, it follows that for allk∈Zand T >0

Pn

∃N⁰ ∀N≥N⁰ w^N_k(t) =w^N_k⁰(t), t∈[0, T] o

= 1.

Definewk = lim

N→∞w^N_k ,k∈Z. It is clear thatwk(t), t≥0, k∈Z, are independent Wiener processes.

Let

(ζ₋^N_N, . . . , ζ_N^N) = Λ^(b_γN^−N,...,bN)(ξ_−N, . . . , ξ_N),