Weak Convergence of Markov Random Evolutions in a Multidimensional Space

We study Markov symmetrical and nonsymmetrical random evolutions in R. 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.


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 R n , 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 in R 2 and R 3 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 in R n 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 R n .
The difference in the methods can be easily seen by the analysis of the papers 16, 17 , where similar problems are studied.
In Sections 3 and 4, we use the method proposed to prove weak convergence of Markov non-symmetrical random evolution MNRE in R n .The distinction of this model is that the limit process is a diffusion process with drift.

Description of MSRE
We study a particle in the space R n , that starts at t 0 from the point x x i , i 1, n .Possible directions of motion are given by the following vectors:

2.1
These vectors have initial point in the center of the unit n-dimentional sphere S n 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 set Θ {θ : s θ ∈ S n } and suppose θ ε t ∈ Θ be the switching Poisson process.
Definition 2.1.Markov symmetrical random evolution MSRE is the process ξ ε t ∈ R n , given by It is easy to see that when ε → 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 ξ ε t , θ ε t at the test functions ϕ x 1 , . . ., x n ; θ ∈ C ∞ 0 R n × Θ can be described by the generator see, e.g., 2 where and 2.6 Using well-known formula we can see Operator Πϕ •; θ : 1/N S n ϕ •; 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.
For Π we have Potential operator R 0 can be defined as the following: R 0 : Π − I.

2.10
This operator has the following property: thus it is inverse for Q at the range of Q, but for the function φ from the null space of Q we have R 0 φ 0.

2.12
Solution of singular perturbation problem in the series scheme with the small series parameter ε → 0 ε > 0 see 15 for reducibly invertible operator Q and perturbing operator Q 1 consists in the following.
We should find two vectors ϕ ε ϕ εϕ 1 ε 2 ϕ 2 and ψ, that satisfy asymptotic representation with the vector θ ε , that is uniformly bounded by the norm and such that The left part of the equation can be rewritten as

2.15
And as soon as it is equal to the right side, we obtain Qϕ 0,

2.16
From the last equation we may see that the function ϕ 2 should be smooth enough to provide boundness of Q 1 ϕ 2 .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 This condition means that the function Q 1 ϕ 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 Thus, the main problem is to solve the following equation: The solvability condition for Q has the view and we finally obtain

2.21
For the function ϕ 2 we obviously have Equations 2.18 -2.22 give the solution of singular perturbation problem.
In case of MSRE, the balance condition has the following view: where 1 x x 1 , . . ., x n .Really, every term under the integral contains either π 0 sin n θ cos θdθ 0 or 2π 0 sin θdθ 0.

Main Result for MSRE
Theorem 3.1.MSRE ξ ε t , converges weakly to the Wiener process w t : ξ 0 t when ε → 0 as where ξ 0 t ∈ R n is defined by the generator Here Δϕ : Remark 3.2.The generator 3.2 generalizes the generators obtained in 8 for the spaces R 2 and R 3 .
We use the following Lemma to prove the Theorem.

Lemma 3.3. At the perturbed test functions
having bounded derivatives of any degree and compact support, the operator L ε has the asymptotic representation where L 0 is defined in 3.2 , ϕ 1 x 1 , . . ., x n ; θ , ϕ 2 x 1 , . . ., x n ; θ and R ε θ ϕ x are the following:

3.5
Proof.Let us solve singular perturbation problem for the operator 2.3 .To do this, we use the following correlation:

3.7
From the first equation we see that ϕ x 1 , . . ., x n 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 By substitution into the third equation and using the solvability condition, we can see From the last equation of 2.23 we have

3.10
Let's find the generator of the limit process L 0 by the formula 3.5 as

3.11
The last term equals to 0 by the balance condition 2.23 .Thus, finally Let us calculate the following integral:

3.14
Every term in braces has a multiplier of the type π 0 sin n θ cos θ dθ 0 or 2π 0 sin θ cos θdθ 0, thus the corresponding integral equals to 0. Integration of the correlations in the parentheses gives

3.15
Finally, we have

3.16
Lemma is proved.
Proof of Theorem 3.1.In Lemma 3.3, we proved that L ε ϕ ε ⇒ L 0 ϕ at the class of functions C ∞ 0 R n × Θ when ε → 0. To prove the weak convergence of the process, we should show relative compactness of the family ξ ε t , θ ε t in D R n ×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 generators L ε , ε > 0 satisfy the inequalities 3.17 for any real-valued nonnegative function ϕ ∈ C ∞ 0 R n × Θ , where the constant C ϕ depends on the norm of ϕ, and for where the constant C l depends on the function ϕ 0 , but does not depend on ε > 0.

3.19
It follows from 3.7 that the first two terms equal to 0. Let us estimate the following last term:

3.20
as soon as all the constants, functions, and their derivatives are bounded.We also have from 3.7
To prove the second condition, it's enough to use the properties of the function ϕ 0 u √ 1 u 2 , namely: So, the proof of the second condition is similar to the previous reasoning.Thus, the family ξ ε t , θ ε t is relatively compact in D R n ×Θ 0, ∞ .Now we may use the following theorem Theorem 6.6 from 15 .Theorem 3.5.Let random evolution with Markov switching ξ ε t , x ε t ∈ D R n ×E 0, ∞ satisfies the following conditions.

C2 There exists the family of test-functions
uniformly by u, x.

C3
The following uniform convergence is true uniformly by u, x.
The family

C4 Convergence by probability of initial values
Then we have the weak convergence

3.27
According to Theorem 3.5, we may confirm the weak convergence in D R n 0, ∞ ξ ε t ⇒ ξ 0 t .

3.28
Really, all the conditions are satisfied.Namely, the family of processes is relatively compact, the generators converge at the test functions belonging to the class C ∞ 0 R n × Θ , initial conditions for the limit, and prelimit processes are equal.
Theorem is proved.

Description of MNRE
We study the same particle in R n , but its velocity is equal to v θ c θ ε −1 c 1 θ , where ε → 0 ε > 0 is the small parameter, the functions c θ , c 1 θ are bounded.Definition 4.1.Markov nonsymmetrical random evolution MNRE is the process ξ ε t ∈ R n , given by Our aim is to prove the weak convergence of MNRE to a diffusion process with drift when ε → 0.

ISRN Probability and Statistics
Two-component Markov process ξ ε t , θ ε t at the test functions ϕ x 1 , . . ., x n ; θ ∈ C ∞ 0 R n × Θ is defined by a generator see, e.g., 2 where An important condition that let us confirm weak convergence is balance condition 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 c 1 θ ≡ 0. In case of MNRE the absence of symmetry of the process is caused by the following condition: Another example is the function c θ sin θ 1 .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 R 2 this function does not have symmetry.
Example 4.4.Condition 4.5 can be also satisfied for different functions c 1 θ .For example, in R 2 for c 1 θ sin θ we obtain Another example is the following function in R n : Again, all terms under the integral, except the last one, contain π 0 sin n θ cos θdθ 0, thus only one term is not trivial as

4.8
By simple calculations, we see

4.9
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.

5.1
The limit process ξ 0 t ∈ R n is defined by a generator where Δϕ x 1 , . . ., x n :

5.3
We need the following Lemma to prove the Theorem.
Lemma 5.2.At the perturbed test functions having bounded derivatives of any degree and compact support, the operator L ε has the following asymptotic representation:

5.6
Proof.We solve singular perturbation problem for the operator 4.2 .Using the view of testfunctions 5.4 , we have

5.7
Thus, we obtain the following equations:

5.8
According to the first one, ϕ x 1 , . . ., x n 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 By substitution into the third equation and using the solvability condition, we obtain: From the last equation, we have

5.11
We calculate the operator L 0 by the formula 5.6 as

5.12
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 L 0 Πc 2 θ S 2 θ Πc 1 θ S θ .

5.13
Using the view of S θ , we may write

5.14
Lemma is proved.

5.15
Using Theorem 3.5, we confirm the following weak convergence in D R n 0, ∞ :

5.20
Really, all the conditions are satisfied.Namely, the family of processes is relatively compact, the generators at the test functions belonging to the class C ∞ 0 R n × Θ converge, and initial conditions for the limit and prelimit processes are equal.
Theorem is proved.

5.21
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 c 1 θ .
The limit generator 5.2 has the view

5.22
Thus, the limit process has two parts: the drift with velocity 1/2π in direction of x 2 coordinate and diffusion part in one-dimensional subspace, corresponding to x 1 coordinate that is similar to the limit process described in Kac model 1 .parameter of drift D R n ×Θ 0, ∞ : the space of R n × Θ-valued right continuous functions having left limits cadlag , defined on R L ε : the generator of random evolution L 0 : the generator of limit process N:

List of Symbols
volume of S n n: dimension of space Q: the generator of switching Poisson process perturbing operator R 0 : potential of generator Q R ε θ : negligible term S n : u n i t n-dimensional sphere S θ : operator that describes particle's motion s θ : vectors that define possible directions of random evolution's motion v, v θ : velocity of random evolution's motion ε: small parameter θ θ i , i 1, n − 1 , θ n−1 ∈ 0, 2π , θ i ∈ 0, π , i 1, n − 2: angles that define directions in R n Θ: the set of all angles θ θ ε t : Poisson process that switches directions of random evolution's motion λ: intensity of switching Poisson process μ dθ : small element of volume of S n ξ ε t , ξ ε t : Markov random evolutions ξ 0 t , ξ 0 t : limit processes Π: projector at the nullspace of generator Q σ 2 : parameter of diffusion.

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 π 0 sin n θ cos θ dθ 0 or 2π 0 sin θ dθ 0.

Example 5 . 3 . 2
Let us consider one more variant of evolution in R c, c θ , c 1 θ :the components of velocity of random evolution's motionC ∞ 0 R n × Θ :the space of bounded continuous functions on R n × Θ having continuous derivatives of all orders vanishing at infinity C ϕ : constant depending on the norm of function ϕ d: