# Weak Convergence of Markov Random Evolutions in a Multidimensional Space

Authors: Igor V. Samoilenko
Journal: ISRN Probability and Statistics
 (2012)

---

## Abstract

We study Markov symmetrical and nonsymmetrical random evolutions inRn. Weak convergence of Markov symmetrical random evolution to Wiener process and of Markov non-symmetrical random evolution to a diffusion process with drift is proved using problems of singular perturbation for the generators of evolutions. Relative compactness in DRn×Θ[0,∞) of the families of Markov random evolutions is also shown.


---

## Body

## 1. Introduction

Markov symmetrical random evolutions (MSRE) in spaces of different dimensions were studied in the works of Kac [1], Pinsky [2], Orsingher (e.g., [3, 4]), and Kolesnik and Turbin (e.g., [5, 6]) (see also [7] for other references). Symmetry in this sense should be regarded as uniform stationary distribution of switching at a symmetrical structure in Rn, for instance at n+1-hedron [7], or at a unit sphere [8].Weak convergence of distributions of MSRE is also studied in some of these works, namely, convergence inR2 and R3 was proved by Kolesnik in [8, 9].In this work we not only generalize the results of Kolesnik on the multidimensional space, but solve another problem. We prove weak convergence of the process of Markov random evolution, that means not only proof of weak convergence of respective distributions (or generators) but also proof of compactness of the process. In majority of works that deal with random evolutions, compactness of the process is not considered at all. It is also shown that the symmetry of the process is closely connected with balance condition (see (2.23), (4.4)).We should note that the problem of weak convergence of random walks (partially, similar to MSRE) was studied by many authors. Among the most interesting works, we can point out [10–13]. Large bibliography as for this problem could be found in [10]. The methods proposed in these works allow us to solve a wide range of problems connected with convergence of random walks but do not let us obtain limit process to be averaged by the stationary measure of switching process.Such averaging can be found in the works of Anisimov and his students (see [14] and references therein), but here the averaging by the stationary measure is one of the conditions proposed for the prelimit process in the corresponding theorem.In this work, MSRE inRn is studied using the methods proposed in [15]. We find a solution of singular perturbation problem for the generator of the evolution, and thus the averaging by a stationary measure of switching process is obtained as a corollary of this solution. At the second stage we prove relative compactness of the family of MSRE. This method let us show weak convergence of the process of MSRE to the Wiener process in Rn.The difference in the methods can be easily seen by the analysis of the papers [16, 17], where similar problems are studied.In Sections3 and 4, we use the method proposed to prove weak convergence of Markov non-symmetrical random evolution (MNRE) in Rn. The distinction of this model is that the limit process is a diffusion process with drift.


## 2. Description of MSRE

We study a particle in the spaceRn, that starts at t=0 from the point x=(xi,i=1,n-). Possible directions of motion are given by the following vectors:
(2.1)s(θ)=(cos⁡θ1,sin⁡θ1cos⁡θ2,sin⁡θ1sin⁡θ2cos⁡θ3,…,sin⁡θ1⋯sin⁡θn-2cos⁡θn-1,sin⁡θ1⋯sin⁡θn-2sin⁡θn-1),θn-1∈[0,2π),θi∈[0,π),i=1,n-2-.
These vectors have initial point in the center of the unit n-dimentional sphere Sn and the terminal point at its surface. Choosing of every next direction is random and its time is distributed by the Poisson law. Thus, the switching process process is the Poisson one with intensity λ=ε-2. The velocity of particle’s motion is fixed and equals v=cε-1, where ε is a small parameter, ε→0(ε>0).Let us define a setΘ={θ:s(θ)∈Sn} and suppose θtε∈Θ be the switching Poisson process.Definition 2.1.
Markov symmetrical random evolution (MSRE) is the processξtε∈Rn, given by
(2.2)ξtε:=x+v∫0ts(θτε)dτ.
It is easy to see that whenε→0(ε>0),the velocity of the particle and intensity of switching decrease. Our aim is to prove weak convergence of MSRE to the Wiener process when ε→0. The main method is solution of singular perturbation problem for the generator of MSRE. Let us describe this generator.
Two-component Markov process(ξtε,θtε) at the test functions φ(x1,…,xn;θ)∈C0∞(Rn×Θ) can be described by the generator (see, e.g., [2])
(2.3)Lεφ(x1,…,xn;θ)≔λQφ(⋅;θ)+vS(θ)φ(x1,…,xn;⋅)=ε-2Qφ(⋅;θ)+ε-1cS(θ)φ(x1,…,xn;⋅),
where
(2.4)S(θ)φ(x1,…,xn;⋅):=-(s(θ),∇)φ(x1,…,xn;⋅),
here ∇φ=(∂φ/∂xi,1≤i≤n),
(2.5)Qφ(⋅;θ):=(Π-I)φ(⋅;θ):=1N∫Snφ(⋅;θ)μ(dθ)-φ(⋅;θ),
and N=(2π)n/2(1/(2·4·⋯·(n-2))) for even n, N=(2π)(n-1)/2(2/(3·5·⋯·(n-2))) for odd n; μ(dθ) is the element of volume of the sphere Sn, that is equal to
(2.6)μ(dθ):=sin⁡n-2θ1sin⁡n-3θ2⋯sin⁡θn-2dθ1⋯dθn-1.
Using well-known formula(2.7)∫0π‍sin⁡2mdθ=2⋅1⋅2⋅⋯⋅(2m-1)2⋅4⋅⋯⋅2m⋅π2,∫0π‍sin⁡2m+1dθ=2⋅2⋅4⋅⋯⋅2m1⋅3⋅⋯⋅(2m+1)
we can see
(2.8)∫Sn‍μ(dθ)=N.
OperatorΠφ(·;θ):=(1/N)∫Sn‍φ(·;θ)μ(dθ) is the projector at the null space of reducibly invertible operator Q, because by definition it maps functions to constants but constants to itself.
ForΠ we have
(2.9)QΠ=ΠQ=0.
Potential operatorR0 can be defined as the following:
(2.10)R0:=Π-I.
This operator has the following property:(2.11)R0Q=QR0=I-Π,
thus it is inverse for Q at the range of Q, but for the function ϕ from the null space of Q we have
(2.12)R0ϕ=0.
Solution of singular perturbation problem in the series scheme with the small series parameterε→0(ε>0) (see [15]) for reducibly invertible operator Q and perturbing operator Q1 consists in the following.
We should find two vectorsφε=φ+εφ1+ε2φ2 and ψ, that satisfy asymptotic representation
(2.13)[ε-2Q+ε-1Q1]φε=ψ+εθε,
with the vector θε, that is uniformly bounded by the norm and such that
(2.14)‖θε‖≤C,ε⟶0.
The left part of the equation can be rewritten as(2.15)[ε-2Q+ε-1Q1](φ+εφ1+ε2φ2)=ε-2Qφ+ε-1[Qφ1+Q1φ]+[Qφ2+Q1φ1]+εQ1φ2.
And as soon as it is equal to the right side, we obtain(2.16)Qφ=0,Qφ1+Q1φ=0,Qφ2+Q1φ1=ψ,Q1φ2=θε.
From the last equation we may see that the functionφ2 should be smooth enough to provide boundness of Q1φ2. Moreover, from the first equation we see that any function from the null space of Q can be taken as φ, and it does not depend on the variable that corresponds to the switching process.
An important condition of solvability for this problem is the balance condition(2.17)ΠQ1=0.
This condition means that the function Q1φ belongs to the range of Q, thus we may solve the second equation using the potential operator, that is, inverse to Q at its range
(2.18)φ1=-R0Q1φ.
Thus, the main problem is to solve the following equation:(2.19)Qφ2=ψ-Q1φ1=ψ+Q1R0Q1φ.
The solvability condition forQ has the view
(2.20)ΠQΠφ2=0=Πψ+ΠQ1R0Q1Πφ,
and we finally obtain
(2.21)Πψ=-ΠQ1R0Q1Πφ.
For the functionφ2 we obviously have
(2.22)φ2=R0[-ΠQ1R0Q1Π+Q1R0Q1]φ.
Equations (2.18)–(2.22) give the solution of singular perturbation problem.
In case of MSRE, the balance condition has the following view:(2.23)ΠS(θ)1(x)=0,
where 1(x)=(x1,…,xn). Really, every term under the integral contains either ∫0π‍sinnθcosθdθ=0 or ∫02π‍sinθdθ=0.


## 3. Main Result for MSRE

Theorem 3.1.
MSREξtε, converges weakly to the Wiener process w(t):=ξt0 when ε→0 as
(3.1)ξtε⟹ξt0,
where ξt0∈Rn is defined by the generator
(3.2)L0φ(x1,…,xn)=c2nΔφ(x1,…,xn).
Here Δφ:=((∂2/(∂x12))+⋯+(∂2/(∂xn2)))φ.Remark 3.2.
The generator (3.2) generalizes the generators obtained in [8] for the spaces R2 and R3.
We use the following Lemma to prove the Theorem.Lemma 3.3.
At the perturbed test functions(3.3)φε(x1,…,xn;θ)=φ(x1,…,xn)+εφ1(x1,…,xn;θ)+ε2φ2(x1,…,xn;θ),
having bounded derivatives of any degree and compact support, the operator Lε has the asymptotic representation
(3.4)Lεφε(x1,…,xn;θ)=L0φ(x1,…,xn)+Rε(θ)φ(x),|Rε(θ)φ(x)|⟶0,ε⟶0,φ(x)∈C0∞(Rn),
where L0 is defined in (3.2), φ1(x1,…,xn;θ),φ2(x1,…,xn;θ) and Rε(θ)φ(x) are the following:
(3.5)L0Π=-c2ΠS(θ)R0S(θ)Π,φ1=-cR0S(θ)φ,φ2=c2R0S(θ)R0S(θ)φ,Rε(θ)φ=εc3S(θ)R0S(θ)R0S(θ)φ.Proof.
Let us solve singular perturbation problem for the operator (2.3). To do this, we use the following correlation:
(3.6)Lεφε(x1,…,xn;θ)=[ε-2Q+ε-1cS(θ)][φ+εφ1+ε2φ2]=ε-2Qφ+ε-1[Qφ1+cS(θ)φ]+[Qφ2+cS(θ)φ1]+εcS(θ)φ2.
Thus, we obtain equations(3.7)Qφ=0,Qφ1+cS(θ)φ=0,L0φ=Qφ2+cS(θ)φ1,Rεφ(θ)=εcS(θ)φ2.
From the first equation we see thatφ(x1,…,xn) belongs to the null space of Q. It is easy to see from the balance condition (2.23) that S(θ)φ belongs to the range of Q, thus from the second equation of the system (3.7) we have
(3.8)φ1=-cR0S(θ)φ.
By substitution into the third equation and using the solvability condition, we can see(3.9)L0Πφ+c2ΠS(θ)R0S(θ)Πφ=0.
From the last equation of (2.23) we have
(3.10)Rε(θ)φ(x)=εc3S(θ)R0S(θ)R0S(θ)φ(x)⟶0whenε⟶0,φ(x)∈C0∞(Rn).
Let’s find the generator of the limit processL0 by the formula (3.5) as
(3.11)L0φ=c2ΠS(θ)(I-Π)S(θ)Πφ=c2ΠS2(θ)Πφ-c2ΠS(θ)ΠS(θ)Πφ.
The last term equals to 0 by the balance condition (2.23). Thus, finally
(3.12)L0=c2ΠS2(θ),
or
(3.13)L0=c2N∫Sn‍S2(θ)μ(dθ).
Let us calculate the following integral:(3.14)c2N∫0π⋯∫0π∫02π[cos⁡2θ1∂2∂x12+sin⁡2θ1cos⁡2θ2∂2∂x22+⋯+sin⁡2θ1⋯sin⁡2θn-2cos⁡2θn-1∂2∂xn-12+sin⁡2θ1⋯sin⁡2θn-2sin⁡2θn-1∂2∂xn2+{sin⁡θ1cos⁡θ1cos⁡θ2∂2∂x1∂x2+⋯+sin⁡2θ1⋯sin⁡2θn-2sin⁡θn-1×cos⁡θn-1∂2∂xn-1∂xn}]×sin⁡n-2θ1sin⁡n-3θ2⋯sin⁡θn-2dθ1⋯dθn-1=c2N∫0π⋯∫0π∫02π[cos⁡2θ1sin⁡n-2θ1sin⁡n-3θ2⋯sin⁡θn-2∂2∂x12+sin⁡nθ1cos⁡2θ2sin⁡n-3θ2⋯sin⁡θn-2∂2∂x22+⋯+sin⁡nθ1sin⁡n-1θ2⋯sin⁡3θn-2cos⁡2θn-1×∂2∂xn-12+sin⁡nθ1sin⁡n-1θ2⋯sin⁡2θn-2sin⁡2θn-1∂2∂xn2+{∂2∂x1∂x2sin⁡n-1θ1cos⁡θ1cos⁡θ2sin⁡n-3θ2⋯sin⁡θn-2∂2∂x1∂x2+⋯+sin⁡nθ1sin⁡n-1θ2⋯sin⁡3θn-2sin⁡θn-1cos⁡θn-1∂2∂xn-1∂xn}]×dθ1…dθn-1=c2N∫0π∫0π⋯∫0π[∂2∂xn-1∂xn(sin⁡n-2θ1sin⁡n-3θ2⋯sin⁡θn-2-sin⁡nθ1sin⁡n-3θ2⋯sin⁡θn-2)∂2∂x12+(sin⁡nθ1sin⁡n-3θ2⋯sin⁡θn-2-sin⁡nθ1sin⁡n-1θ2sin⁡n-4θ3⋯sin⁡θn-2)×∂2∂x22+⋯+(sin⁡nθ1sin⁡n-1θ2⋯sin⁡3θn-2-sin⁡nθ1sin⁡n-1θ2⋯sin⁡3θn-2sin⁡2θn-1)∂2∂xn-12+sin⁡nθ1sin⁡n-1θ2⋯sin⁡3θn-2sin⁡2θn-1∂2∂xn2+{sin⁡n-1θ1cos⁡θ1cos⁡θ2sin⁡n-3θ2⋯sin⁡θn-2∂2∂x1∂x2+⋯+sin⁡nθ1sin⁡n-1θ2⋯sin⁡3θn-2sin⁡θn-1cos⁡θn-1∂2∂xn-1∂xn}]×dθ1⋯dθn-1.
Every term in braces has a multiplier of the type∫0π‍sinnθcosθdθ=0 or ∫02π‍sinθcosθdθ=0, thus the corresponding integral equals to 0.
Integration of the correlations in the parentheses gives(3.15)∫0π⋯∫0π∫02π(sin⁡nθ1sin⁡n-1θ2⋯sin⁡n-k+2θk-1sin⁡n-k-1θksin⁡n-k-2θk+1⋯sin⁡θn-2-sin⁡nθ1sin⁡n-1θ2⋯sin⁡n-k+2θk-1sin⁡n-k+1θksin⁡n-k-2θk+1⋯sin⁡θn-2)dθ1⋯dθn-1=N((2m-1)(2m-2)(2m-3)⋅⋯⋅(2m-k+1)2m(2m-1)(2m-2)⋅⋯⋅(2m-k+2)-(2m-1)(2m-2)(2m-3)⋅⋯⋅(2m-k)2m(2m-1)(2m-2)⋅⋯⋅(2m-k+1))=N(2m-k+12m-2m-k2m),ifn=2morN((2m)(2m-1)(2m-2)⋅⋯⋅(2m-k+2)(2m+1)2m(2m-1)⋅⋯⋅(2m-k+3)-2m(2m-1)(2m-2)⋅⋯⋅(2m-k+1)(2m+1)2m(2m-1)⋅⋯⋅(2m-k+2))=N(2m-k+22m+1-2m-k+12m+1),ifn=2m+1=Nn.
Finally, we have(3.16)L0φ(x1,…,xn)=c2nΔφ(x1,…,xn).
Lemma is proved.Proof of Theorem3.1.
In Lemma3.3, we proved that Lεφε⇒L0φ at the class of functions C0∞(Rn×Θ) when ε→0. To prove the weak convergence of the process, we should show relative compactness of the family (ξtε,θtε) in DRn×Θ[0,∞). To do this, we use the methods proposed in [15, 18, 19]. Let us formulate Corollary 6.1 from [15] (see also Theorem  6.4 in [15]) as Lemma 3.4.Lemma 3.4.
Let the generatorsLε,  ε>0 satisfy the inequalities
(3.17)|Lεφ(u)|<Cφ,
for any real-valued nonnegative function  φ∈C0∞(Rn×Θ), where the constant  Cφ depends on the norm of  φ, and for  φ0(u)=1+u2,
(3.18)Lεφ0(u)≤Clφ0(u),|u|≤l,
where the constant  Cl depends on the function  φ0, but does not depend on  ε>0.
Then, the family(ξtε,θtε), t≥0, ε>0is relatively compact in DRn×Θ[0,∞).Let us write the action of the generator (2.3) at the test function φε(x1,…,xn;θ)=φ(x1,…,xn)+εφ1(x1,…,xn;θ), where φ1(x1,…,xn;θ)=-cR0S(θ)φ(x1,…,xn).We have(3.19)Lεφε(x1,…,xn;θ)=ε-2Qφ(x1,…,xn)+ε-1[Qφ1+cS(θ)φ(x1,…,xn)]+cS(θ)φ1(x1,…,xn;θ).It follows from (3.7) that the first two terms equal to 0. Let us estimate the following last term:
(3.20)cS(θ)φ1(x1,…,xn;θ)=c2S(θ)R0S(θ)φ(x1,…,xn)=c2S2(θ)φ(x1,…,xn)=c2[cos⁡2θ1∂2∂x12+sin⁡2θ1cos⁡2θ2∂2∂x22+⋯+sin⁡2θ1⋯sin⁡2θn-2cos⁡2θn-1∂2∂xn-12+sin⁡2θ1⋯sin⁡2θn-2sin⁡2θn-1∂2∂xn2+{sin⁡θ1cos⁡θ1cos⁡θ2×∂2∂x1∂x2+⋯+sin⁡2θ1⋯sin⁡2θn-2×sin⁡θn-1×cos⁡θn-1∂2∂xn-1∂xn}]φ(x1,…,xn)≤C1,φ,
as soon as all the constants, functions, and their derivatives are bounded.We also have from (3.7)
(3.21)Lεφε=Lεφ+εLεφ1=Lεφ+εLεR0S(θ)φ.Thus,(3.22)Lεφ=Lεφε-εcLεR0S(θ)φ≤C1,φ-εC2,φ<Cφ,
for small ε.To prove the second condition, it’s enough to use the properties of the functionφ0(u)=1+u2, namely:
(3.23)|φ0′(u)|≤1≤φ0(u),|φ0′′(u)|≤φ0(u).So, the proof of the second condition is similar to the previous reasoning.Thus, the family(ξtε,θtε) is relatively compact in DRn×Θ[0,∞).Now we may use the following theorem (Theorem  6.6 from [15]).Theorem 3.5.
Let random evolution with Markov switching(ξε(t),xε(t))∈DRn×E[0,∞) satisfies the following conditions.(C1)
The family of processes(ξε(t),xε(t)), t≥0, ε>0 is relatively compact.(C2)
There exists the family of test-functionsφε(u,x)∈C03(Rn×E), such that
(3.24)lim⁡ε→0φε(u,x)=φ(u)uniformly byu,  x.(C3)
The following uniform convergence is true(3.25)lim⁡ε→0Lεφε(u,x)=Lφ(u)uniformly byu,x.The familyLεφε, ε>0 is uniformly bounded, moreover  Lεφε and  Lφ belong to  C(Rn×E).(C4)
Convergence by probability of initial values(3.26)ξε(0)⟶ξ^(0),ε⟶0,sup⁡ε>0E|ξε(0)|≤C<+∞is true.
Then we have the weak convergence(3.27)ξε(t)⟹ξ^(t),ε⟶0.According to Theorem3.5, we may confirm the weak convergence in DRn[0,∞)(3.28)ξtε⟹ξt0.Really, all the conditions are satisfied. Namely, the family of processes is relatively compact, the generators converge at the test functions belonging to the classC0∞(Rn×Θ), initial conditions for the limit, and prelimit processes are equal.Theorem is proved.


## 4. Description of MNRE

We study the same particle inRn, but its velocity is equal to v(θ)=c(θ)ε-1+c1(θ), where ε→0(ε>0) is the small parameter, the functions c(θ),c1(θ) are bounded.Definition 4.1.
Markov nonsymmetrical random evolution (MNRE) is the processξ~tε∈Rn, given by
(4.1)ξ~tε:=x+∫0t‍v(θτε)s(θτε)dτ.
Our aim is to prove the weak convergence of MNRE to a diffusion process with drift whenε→0.
Two-component Markov process(ξ~tε,θtε) at the test functions φ(x1,…,xn;θ)∈C0∞(Rn×Θ) is defined by a generator (see, e.g., [2])
(4.2)Lεφ(x1,…,xn;θ)≔λQφ(⋅;θ)+v(θ)S(θ)φ(x1,…,xn;⋅)=ε-2Qφ(⋅;θ)+ε-1c(θ)S(θ)φ(x1,…,xn;⋅)+c1(θ)S(θ)φ(x1,…,xn;⋅),
where
(4.3)S(θ)φ(x1,…,xn;⋅):=-(s(θ),∇)φ(x1,…,xn;⋅).
An important condition that let us confirm weak convergence is balance condition(4.4)Πc(θ)S(θ)=0.Remark 4.2.
This is the last condition that defines symmetry or nonsymmetry of the process. In case of MSRE, the balance condition (4.4) is true, and c1(θ)≡0. In case of MNRE the absence of symmetry of the process is caused by the following condition:
(4.5)Πc1(θ)S(θ)=(d,∇)≠0.Example 4.3.
Condition (4.4) can be satisfied for different functions c(θ). Namely, in case of MSRE c(θ)=c=const. Then every term under the integral contains either ∫0π‍sinnθcosθdθ=0 or ∫02π‍sinθdθ=0.
Another example is the functionc(θ)=sinθ1. Really, we obtain the terms under the integral that are analogical to the previous ones. We note that the dimension of the space in this case should be more than 2, because in R2 this function does not have symmetry.Example 4.4.
Condition (4.5) can be also satisfied for different functions c1(θ). For example, in R2 for c1(θ)=sinθ we obtain
(4.6)12π∫02π‍sin⁡θ[cos⁡θ∂∂x1+sin⁡θ∂∂x2]dθ=12∂∂x2.
Another example is the following function inRn:
(4.7)c1(θ)={c1,θn-1∈[π,2π),0,θn-1∈[0,π).
Again, all terms under the integral, except the last one, contain ∫0π‍sinnθcosθdθ=0, thus only one term is not trivial as
(4.8)1Nc1∫0π‍⋯∫0π‍∫π2π‍sin⁡n-1θ1sin⁡n-2θ2⋯sin⁡2θn-1sin⁡θn-1dθ1⋯dθn-1.
By simple calculations, we see(4.9)Πc1(θ)S(θ)={-c123⋅5⋅⋯⋅(n-2)2⋅4⋅⋯⋅(n-1)∂∂xn,n=2m+1,-c1π[1⋅2⋅⋯(n-2)]∂∂xn,n=2m.
We obviously have a wide range of functions that preserve or, on the contrary, do not preserve symmetry. So, we can define the velocity of random evolution in different ways, depending on possible applications.


## 5. Main Result for MNRE

Theorem 5.1.
MNREξ~tε, converges weakly to the process ξ~t0 when ε→0 as
(5.1)ξ~tε⟹ξ~t0.
The limit processξ~t0∈Rn is defined by a generator
(5.2)L0φ(x1,…,xn)=(d,∇)φ(x1,…,xn)+(σ2,Δ)φ(x1,…,xn),
where Δφ(x1,…,xn):=((∂2/∂x12)+⋯+(∂2/∂xn2))φ(x1,…,xn),
(5.3)(d,∇):=-1N∫Sn‍c1(θ)(s(θ),∇)μ(dθ),(σ2,Δ):=1N∫Sn‍c2(θ)(s(θ),∇)2μ(dθ).
We need the following Lemma to prove the Theorem.Lemma 5.2.
At the perturbed test functions(5.4)φε(x1,…,xn;θ)=φ(x1,…,xn)+εφ1(x1,…,xn;θ)+ε2φ2(x1,…,xn;θ),
having bounded derivatives of any degree and compact support, the operator Lε has the following asymptotic representation:
(5.5)Lεφε(x1,…,xn;θ)=L0φ(x1,…,xn)+Rε(θ)φ(x),|Rε(θ)φ(x)|⟶0,ε⟶0,φ(x)∈C0∞(Rn),
where L0 is defined in (5.2), φ1(x1,…,xn;θ),φ2(x1,…,xn;θ), and Rε(θ)φ(x) are defined by the following correlations:
(5.6)L0Π=-Πc(θ)S(θ)R0c(θ)S(θ)Π+Πc1(θ)S(θ)Πφ,φ1=-R0c(θ)S(θ)φ,φ2=R0c(θ)S(θ)R0c(θ)S(θ)φ,Rε(θ)φ={ε2c1ε[c(θ)S(θ)R0c(θ)S(θ)R0c(θ)S(θ)+c1(θ)S(θ)R0c(θ)S(θ)]+ε2c1(θ)S(θ)R0c(θ)S(θ)R0c(θ)S(θ)}φ.Proof.
We solve singular perturbation problem for the operator (4.2). Using the view of test-functions (5.4), we have
(5.7)Lεφε(x1,…,xn;θ)=[ε-2Q+ε-1c(θ)S(θ)+c1(θ)S(θ)][φ+εφ1+ε2φ2]=ε-2Qφ+ε-1[Qφ1+c(θ)S(θ)φ]+[Qφ2+c(θ)S(θ)φ1+c1(θ)S(θ)φ]+ε[c(θ)S(θ)φ2+c1(θ)S(θ)φ1]+ε2c1(θ)S(θ)φ2.
Thus, we obtain the following equations:(5.8)Qφ=0,Qφ1+c(θ)S(θ)φ=0,L0φ=Qφ2+c(θ)S(θ)φ1+c1(θ)S(θ)φ,Rεφ(θ)=ε[c(θ)S(θ)φ2+c1(θ)S(θ)φ1]+ε2c1(θ)S(θ)φ2.
According to the first one,φ(x1,…,xn) belongs to the null space of Q. From the balance condition (4.4) we see that c(θ)S(θ)φ belongs to the range of Q, thus from the second equation of (5.8) we have
(5.9)φ1=-R0c(θ)S(θ)φ.
By substitution into the third equation and using the solvability condition, we obtain:(5.10)L0Πφ+Πc(θ)S(θ)R0c(θ)S(θ)Πφ-Πc1(θ)S(θ)Πφ=0.
From the last equation, we have(5.11)Rε(θ)φ(x)={ε2c1ε[c(θ)S(θ)R0c(θ)S(θ)R0c(θ)S(θ)+c1(θ)S(θ)R0c(θ)S(θ)]+ε2c1(θ)S(θ)R0c(θ)S(θ)R0c(θ)S(θ)}φ(x)⟶0whenε⟶0,φ(x)∈C0∞(Rn).
We calculate the operatorL0 by the formula (5.6) as
(5.12)L0φ=Πc(θ)S(θ)(I-Π)c(θ)S(θ)Πφ+Πc1(θ)S(θ)Πφ=Πc2(θ)S2(θ)Πφ-Πc(θ)S(θ)Πc(θ)S(θ)Πφ+Πc1(θ)S(θ)Πφ.
The second term equals 0 by the balance condition (4.4), the last one is not equal to 0 by (4.5). Thus, we finally have
(5.13)L0=Πc2(θ)S2(θ)+Πc1(θ)S(θ).
Using the view ofS(θ), we may write
(5.14)Πc1(θ)S(θ)=-1N∫Sn‍c1(θ)(s(θ),∇)μ(dθ)=:(d,∇),Πc2(θ)S2(θ)=1N∫Sn‍c2(θ)(s(θ),∇)2μ(dθ)=:(σ2,Δ).
Lemma is proved.Proof of Theorem5.1.
We proved in Lemma5.2 that Lεφε⇒L0φ at the class of test functions C0∞(Rn×Θ), when ε→0. To prove the weak convergence, we should show relative compactness of the family of processes (ξ~tε,θtε) in DRn×Θ[0,∞). To do this, we use Lemma 3.4.
We have the following view of the operator (4.2) at the test function φε(x1,…,xn;θ)=φ(x1,…,xn)+εφ1(x1,…,xn;θ), where φ1(x1,…,xn;θ)=-R0c(θ)S(θ)φ(x1,…,xn):
(5.15)Lεφε(x1,…,xn;θ)=ε-2Qφ(x1,…,xn)+ε-1[Qφ1+c(θ)S(θ)φ(x1,…,xn)]+[c(θ)S(θ)φ1(x1,…,xn;θ)+c1(θ)S(θ)φ(x1,…,xn;θ)]+εc1(θ)S(θ)φ1(x1,…,xn;θ).
It follows from (5.8) that the first two terms equal to 0. Let us estimate the following third term:
(5.16)c(θ)S(θ)φ1(x1,…,xn;θ)+c1(θ)S(θ)φ(x1,…,xn;θ)=c(θ)S(θ)R0c(θ)S(θ)φ(x1,…,xn)+c1(θ)S(θ)φ(x1,…,xn;θ)=c2(θ)S2(θ)φ(x1,…,xn)+c1(θ)S(θ)φ(x1,…,xn;θ)=c2(θ)[cos⁡2θ1∂2∂x12+sin⁡2θ1cos⁡2θ2∂2∂x22+⋯+sin⁡2θ1⋯sin⁡2θn-2cos⁡2θn-1∂2∂xn-12+sin⁡2θ1⋯sin⁡2θn-2sin⁡2θn-1∂2∂xn2+{sin⁡θ1cos⁡θ1cos⁡θ2∂2∂x1∂x2+⋯+sin⁡2θ1⋯sin⁡2θn-2sin⁡θn-1×cos⁡θn-1∂2∂xn-1∂xn}]φ(x1,…,xn)+c1(θ)[cos⁡θ1∂∂x1+sin⁡θ1cos⁡θ2∂∂x2+⋯+sin⁡θ1⋯sin⁡θn-2cos⁡θn-1∂∂xn-1+sin⁡θ1⋯sin⁡θn-2sin⁡θn-1∂∂xn]φ(x1,…,xn)≤C1,φ,
as soon as all the constants, functions, and their derivatives are bounded.
The last term may be estimated analogically.
From (5.8) we also have
(5.17)Lεφε=Lεφ+εLεφ1=Lεφ+εcLεR0θS(θ)φ.
Thus,(5.18)Lεφ=Lεφε-εcLεR0S(θ)φ≤C1,φ-εC2,φ<Cφ,
for ε that is small enough.
To prove the second condition of Lemma3.4, we use the following properties of the function φ0(u)=1+u2:
(5.19)|φ0′(u)|≤1≤φ0(u),|φ0′′(u)|≤φ0(u).
We can see again that the proof of the second condition is similar to the previous reasoning.
Thus, the family of the processes(ξ~tε,θtε) is relatively compact in DRn×Θ[0,∞).
Using Theorem3.5, we confirm the following weak convergence in DRn[0,∞):
(5.20)ξ~tε⟹ξ~t0.
Really, all the conditions are satisfied. Namely, the family of processes is relatively compact, the generators at the test functions belonging to the class C0∞(Rn×Θ) converge, and initial conditions for the limit and prelimit processes are equal.
Theorem is proved.Example 5.3.
Let us consider one more variant of evolution inR2.
Let(5.21)c(θ)={1,θ=0,1,θ=π,c1(θ)=1,θ=π2.
In other cases both functions equal to 0.
The balance condition (4.4) is true for c(θ), on the contrary, condition (4.5) is true for c1(θ).
The limit generator (5.2) has the view
(5.22)L0φ(x1,x2)=12π∂∂x2φ(x1,x2)+1π∂2∂x12φ(x1,x2).
Thus, the limit process has two parts: the drift with velocity1/2π in direction of x2 coordinate and diffusion part in one-dimensional subspace, corresponding to x1 coordinate that is similar to the limit process described in Kac model [1].



---

*Source: 509789-2012-08-16.xml*