Weak Convergence of Markov Random Evolutions in a Multidimensional Space

Abstract

We study Markov symmetrical and nonsymmetrical random evolutions in 𝐑^𝑛. 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 𝐃_𝐑^𝑛_×Θ[0,∞) of the families of Markov random evolutions is also shown.

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 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 𝐑² 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 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.

2. 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: 𝑠(𝜃)=csoisn𝜃𝜃₁₁,sin𝜃₁cos𝜃₂,sin𝜃₁sin𝜃₂cos𝜃₃,…,sin𝜃₁⋯sin𝜃_𝑛−2cos𝜃_𝑛−1,⋯sin𝜃_𝑛−2sin𝜃_𝑛−1,𝜃_𝑛−1∈[0,2𝜋),𝜃_𝑖∈[0,𝜋),𝑖=1,𝑛−2.(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^𝜀𝑡∈Θ be the switching Poisson process.

Definition 2.1. Markov symmetrical random evolution (MSRE) is the process 𝜉^𝜀𝑡∈𝐑^𝑛, given by 𝜉^𝜀𝑡∶=𝑥+𝑣^𝑡0𝑠𝜃^𝜀𝜏𝑑𝜏.(2.2)
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^𝜀𝑡,𝜃^𝜀𝑡) at the test functions 𝜑(𝑥₁,…,𝑥_𝑛;𝜃)∈𝐶^∞0(𝐑^𝑛×Θ) can be described by the generator (see, e.g., [2]) 𝐿^𝜀𝜑𝑥₁,…,𝑥_𝑛;𝜃∶==𝜀𝜆₋₂𝑄𝜑(⋅;𝜃)+𝑣𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅𝑄𝜑(⋅;𝜃)+𝜀⁻¹𝑐𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅,(2.3) where 𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅∶=−(𝑠(𝜃),∇)𝜑𝑥₁,…,𝑥_𝑛;⋅,(2.4) here ∇𝜑=(𝜕𝜑/𝜕𝑥_𝑖,1≤𝑖≤𝑛), 𝑄𝜑(⋅;𝜃)∶=(Π−𝐼)𝜑(⋅;𝜃)∶=1𝑁_𝑆𝑛𝜑(⋅;𝜃)𝜇(𝑑𝜃)−𝜑(⋅;𝜃),(2.5) 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 𝑆_𝑛, that is equal to 𝜇(𝑑𝜃)∶=sin^𝑛−2𝜃₁sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝑑𝜃₁⋯𝑑𝜃_𝑛−1.(2.6)
Using well-known formula ^𝜋0sin^2𝑚𝑑𝜃=2⋅1⋅2⋅⋯⋅(2𝑚−1)_𝜋02⋅4⋅⋯⋅2𝑚⋅𝜋2,sin^2𝑚+1𝑑𝜃=2⋅2⋅4⋅⋯⋅2𝑚1⋅3⋅⋯⋅(2𝑚+1)(2.7) we can see _𝑆𝑛𝜇(𝑑𝜃)=𝑁.(2.8)
Operator Π𝜑(⋅;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.
For Π we have 𝑄Π=Π𝑄=0.(2.9)
Potential operator 𝑅₀ can be defined as the following: 𝑅₀∶=Π−𝐼.(2.10)
This operator has the following property: 𝑅₀𝑄=𝑄𝑅₀=𝐼−Π,(2.11) thus it is inverse for 𝑄 at the range of 𝑄, but for the function 𝜙 from the null space of 𝑄 we have 𝑅₀𝜙=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 𝑄₁ consists in the following.
We should find two vectors 𝜑^𝜀=𝜑+𝜀𝜑₁+𝜀²𝜑₂ and 𝜓, that satisfy asymptotic representation 𝜀⁻²𝑄+𝜀⁻¹𝑄₁𝜑^𝜀=𝜓+𝜀𝜃^𝜀,(2.13) with the vector 𝜃^𝜀, that is uniformly bounded by the norm and such that ‖𝜃^𝜀‖≤𝐶,𝜀⟶0.(2.14)
The left part of the equation can be rewritten as 𝜀⁻²=𝑄𝜀+₋₂𝜀⁻¹𝑄₁𝜑+𝜀𝜑₁+𝜀²𝜑₂𝑄𝜑+𝜀⁻¹𝑄𝜑₁+𝑄₁𝜑+𝑄𝜑₂+𝑄₁𝜑₁+𝜀𝑄₁𝜑₂.(2.15)
And as soon as it is equal to the right side, we obtain 𝑄𝜑₁𝑄𝜑=0,𝑄𝜑₂+𝑄₁𝜑=0,𝑄+𝑄₁𝜑₁=𝜓,₁𝜑₂=𝜃^𝜀.(2.16)
From the last equation we may see that the function 𝜑₂ 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 Π𝑄₁=0.(2.17) 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 𝜑₁=−𝑅₀𝑄₁𝜑.(2.18)
Thus, the main problem is to solve the following equation: 𝑄𝜑₂=𝜓−𝑄₁𝜑₁=𝜓+𝑄₁𝑅₀𝑄₁𝜑.(2.19)
The solvability condition for 𝑄 has the view Π𝑄Π𝜑₂=0=Π𝜓+Π𝑄₁𝑅₀𝑄₁Π𝜑,(2.20) and we finally obtain Π𝜓=−Π𝑄₁𝑅₀𝑄₁Π𝜑.(2.21)
For the function 𝜑₂ we obviously have 𝜑₂=𝑅₀−Π𝑄₁𝑅₀𝑄₁Π+𝑄₁𝑅₀𝑄₁𝜑.(2.22)
Equations (2.18)–(2.22) give the solution of singular perturbation problem.
In case of MSRE, the balance condition has the following view: Π𝑆(𝜃)𝟏(𝑥)=0,(2.23) where 𝟏(𝑥)=(𝑥₁,…,𝑥_𝑛). Really, every term under the integral contains either ∫^𝜋0sin^𝑛𝜃cos𝜃𝑑𝜃=0 or ∫^02𝜋sin𝜃𝑑𝜃=0.

3. Main Result for MSRE

Theorem 3.1. MSRE 𝜉^𝜀𝑡, converges weakly to the Wiener process 𝑤(𝑡)∶=𝜉^0𝑡 when 𝜀→0 as 𝜉^𝜀𝑡⟹𝜉^0𝑡,(3.1) where 𝜉^0𝑡∈𝐑^𝑛 is defined by the generator 𝐿⁰𝜑𝑥₁,…,𝑥_𝑛=𝑐²𝑛Δ𝜑𝑥¹,…,𝑥_𝑛.(3.2) Here Δ𝜑∶=((𝜕²/(𝜕𝑥²¹))+⋯+(𝜕²/(𝜕𝑥^2𝑛)))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 𝜑^𝜀𝑥₁,…,𝑥_𝑛;𝜃=𝜑𝑥₁,…,𝑥_𝑛+𝜀𝜑₁𝑥₁,…,𝑥_𝑛;𝜃+𝜀²𝜑₂𝑥₁,…,𝑥_𝑛;𝜃,(3.3) having bounded derivatives of any degree and compact support, the operator 𝐿^𝜀 has the asymptotic representation 𝐿^𝜀||𝜑𝑅^𝜀𝑥₁,…,𝑥_𝑛;𝜃=𝐿⁰𝜑𝑥₁,…,𝑥_𝑛+𝑅^𝜀(𝜃)𝜑(𝑥),_𝜀(𝜃)𝜑(𝑥)||⟶0,𝜀⟶0,𝜑(𝑥)∈𝐶^∞0(𝐑^𝑛),(3.4) where 𝐿⁰ is defined in (3.2), 𝜑₁(𝑥₁,…,𝑥_𝑛;𝜃),𝜑₂(𝑥₁,…,𝑥_𝑛;𝜃) and 𝑅^𝜀(𝜃)𝜑(𝑥) are the following: 𝐿⁰Π=𝜑−𝑐²Π𝑆(𝜃)𝑅₀𝑆(𝜃)Π,₁𝜑=−𝑐𝑅₀𝑆(𝜃)𝜑,₂𝑅=𝑐²𝑅₀𝑆(𝜃)𝑅₀𝑆(𝜃)𝜑,_𝜀(𝜃)𝜑=𝜀𝑐³𝑆(𝜃)𝑅₀𝑆(𝜃)𝑅₀𝑆(𝜃)𝜑.(3.5)

Proof. Let us solve singular perturbation problem for the operator (2.3). To do this, we use the following correlation: 𝐿^𝜀𝜑^𝜀𝑥₁,…,𝑥_𝑛;𝜃==𝜀𝜀⁻²₋₂𝑄+𝜀⁻¹𝑐𝑆(𝜃)𝜑+𝜀𝜑₁+𝜀²𝜑₂𝑄𝜑+𝜀⁻¹𝑄𝜑₁+𝑐𝑆(𝜃)𝜑+𝑄𝜑₂+𝑐𝑆(𝜃)𝜑₁+𝜀𝑐𝑆(𝜃)𝜑₂.(3.6)
Thus, we obtain equations 𝑄𝜑₁𝑄𝜑=0,𝐿+𝑐𝑆(𝜃)𝜑=0,₀𝑅𝜑_𝜀=𝑄𝜑₂+𝑐𝑆(𝜃)𝜑₁,𝜑(𝜃)=𝜀𝑐𝑆(𝜃)𝜑₂.(3.7)
From the first equation we see that 𝜑(𝑥₁,…,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 𝜑₁=−𝑐𝑅₀𝑆(𝜃)𝜑.(3.8)
By substitution into the third equation and using the solvability condition, we can see 𝐿⁰Π𝜑+𝑐²Π𝑆(𝜃)𝑅₀𝑆(𝜃)Π𝜑=0.(3.9)
From the last equation of (2.23) we have 𝑅^𝜀(𝜃)𝜑(𝑥)=𝜀𝑐³𝑆(𝜃)𝑅₀𝑆(𝜃)𝑅₀𝑆(𝜃)𝜑(𝑥)⟶0when𝜀⟶0,𝜑(𝑥)∈𝐶^∞0(𝐑^𝑛).(3.10)
Let’s find the generator of the limit process 𝐿⁰ by the formula (3.5) as 𝐿⁰𝜑=𝑐²Π𝑆(𝜃)(𝐼−Π)𝑆(𝜃)Π𝜑=𝑐²Π𝑆²(𝜃)Π𝜑−𝑐²Π𝑆(𝜃)Π𝑆(𝜃)Π𝜑.(3.11)
The last term equals to 0 by the balance condition (2.23). Thus, finally 𝐿⁰=𝑐²Π𝑆²(𝜃),(3.12) or 𝐿⁰=𝑐²𝑁_𝑆𝑛𝑆²(𝜃)𝜇(𝑑𝜃).(3.13)
Let us calculate the following integral:𝑐²𝑁^𝜋0⋯^𝜋0^02𝜋cos²𝜃₁𝜕²𝜕𝑥²¹+sin²𝜃₁cos²𝜃₂𝜕²𝜕𝑥²²+⋯+sin²𝜃₁⋯sin²𝜃_𝑛−2cos²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛−1+sin²𝜃₁⋯sin²𝜃_𝑛−2sin²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛+sin𝜃₁cos𝜃₁cos𝜃₂𝜕²𝜕𝑥₁𝜕𝑥₂+⋯+sin²𝜃₁⋯sin²𝜃_𝑛−2sin𝜃_𝑛−1×cos𝜃_𝑛−1𝜕²×sin_𝑛−2𝜕𝑥_𝑛−1𝜕𝑥_𝑛𝜃₁sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝑑𝜃₁⋯𝑑𝜃_𝑛−1=𝑐²𝑁^𝜋0⋯^𝜋0^02𝜋cos²𝜃₁sin^𝑛−2𝜃₁sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝜕²𝜕𝑥²¹+sin^𝑛𝜃₁cos²𝜃₂sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝜕²𝜕𝑥²²+⋯+sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin³𝜃_𝑛−2cos²𝜃_𝑛−1×𝜕²𝜕𝑥^2𝑛−1+sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin²𝜃_𝑛−2sin²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛+sin^𝑛−1𝜃₁cos𝜃₁cos𝜃₂sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝜕²𝜕𝑥₁𝜕𝑥₂+⋯+sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin³𝜃_𝑛−2sin𝜃_𝑛−1cos𝜃_𝑛−1𝜕²×𝑑𝜃₁𝜕𝑥_𝑛−1𝜕𝑥_𝑛…𝑑𝜃_𝑛−1=𝑐²𝑁^𝜋0^𝜋0⋯^𝜋0sin^𝑛−2𝜃₁sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2−sin^𝑛𝜃₁sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝜕²𝜕𝑥²¹+sin^𝑛𝜃₁sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2×−si𝜕n^𝑛₂𝜃₁sin^𝑛−1𝜃₂sin^𝑛−4𝜃₃⋯sin𝜃_𝑛−2𝜕𝑥²²+⋯+sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin³𝜃_𝑛−2−sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin³𝜃_𝑛−2sin²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛−1+sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin³𝜃_𝑛−2sin²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛+sin^𝑛−1𝜃₁cos𝜃₁cos𝜃₂sin^𝑛−3𝜃₂⋯sin𝜃_𝑛−2𝜕²𝜕𝑥₁𝜕𝑥₂+⋯+sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin³𝜃_𝑛−2sin𝜃_𝑛−1cos𝜃_𝑛−1𝜕²×𝑑𝜃₁𝜕𝑥_𝑛−1𝜕𝑥_𝑛⋯𝑑𝜃_𝑛−1.(3.14)
Every term in braces has a multiplier of the type ∫^𝜋0sin^𝑛𝜃cos𝜃𝑑𝜃=0 or ∫^02𝜋sin𝜃cos𝜃𝑑𝜃=0, thus the corresponding integral equals to 0.
Integration of the correlations in the parentheses gives ^𝜋0⋯^𝜋0^02𝜋sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin^{𝑛−𝑘+2}𝜃_𝑘−1sin^{𝑛−𝑘−1}𝜃_𝑘sin^{𝑛−𝑘−2}𝜃_𝑘+1⋯sin𝜃_𝑛−2−sin^𝑛𝜃₁sin^𝑛−1𝜃₂⋯sin^{𝑛−𝑘+2}𝜃_𝑘−1sin^{𝑛−𝑘+1}𝜃_𝑘sin^{𝑛−𝑘−2}𝜃_𝑘+1⋯sin𝜃_𝑛−2𝑑𝜃₁⋯𝑑𝜃_𝑛−1=𝑁(2𝑚−1)(2𝑚−2)(2𝑚−3)⋅⋯⋅(2𝑚−𝑘+1)−(22𝑚𝑚(−2𝑚1)−(21𝑚)(−2𝑚2)−(22𝑚)⋅−⋯3)⋅⋅(2⋯𝑚⋅−(2𝑘𝑚+−2)𝑘)=𝑁2𝑚−2𝑚𝑘(+2𝑚1−1)(2𝑚−2)⋅⋯⋅(2𝑚−𝑘+1)or𝑁(2𝑚)(22𝑚𝑚−1)−(22𝑚𝑚2−𝑚−2𝑘)⋅⋯,⋅i(f2𝑛𝑚=−2𝑘𝑚+2)−(22𝑚𝑚+(21𝑚)2−𝑚1()2(𝑚2𝑚−−1)2⋅)⋯⋅⋯⋅(⋅2(𝑚2𝑚−−𝑘+𝑘+3)1)=𝑁2(𝑚2𝑚−+𝑘1+)22𝑚(2𝑚−1)⋅⋯⋅(2𝑚−𝑘+2)=𝑁2𝑚+1−2𝑚2𝑚−+𝑘+11,if𝑛=2𝑚+1𝑛.(3.15)
Finally, we have 𝐿⁰𝜑𝑥₁,…,𝑥_𝑛=𝑐²𝑛Δ𝜑𝑥¹,…,𝑥_𝑛.(3.16)
Lemma is proved.

Proof of Theorem 3.1. In Lemma 3.3, we proved that 𝐿^𝜀𝜑^𝜀⇒𝐿⁰𝜑 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^𝜀𝑡,𝜃^𝜀𝑡) 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 ||𝐿^𝜀𝜑(𝑢)||<𝐶_𝜑,(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 𝜑₀(𝑢)=√1+𝑢², 𝐿^𝜀𝜑₀(𝑢)≤𝐶_𝑙𝜑₀(𝑢),|𝑢|≤𝑙,(3.18) where the constant 𝐶_𝑙 depends on the function 𝜑₀, but does not depend on 𝜀>0.
Then, the family (𝜉^𝜀𝑡,𝜃^𝜀𝑡), 𝑡≥0, 𝜀>0is relatively compact in 𝐃_𝐑^𝑛_×Θ[0,∞).

Let us write the action of the generator (2.3) at the test function 𝜑^𝜀(𝑥₁,…,𝑥_𝑛;𝜃)=𝜑(𝑥₁,…,𝑥_𝑛)+𝜀𝜑₁(𝑥₁,…,𝑥_𝑛;𝜃), where 𝜑₁(𝑥₁,…,𝑥_𝑛;𝜃)=−𝑐𝑅₀𝑆(𝜃)𝜑(𝑥₁,…,𝑥_𝑛).

We have 𝐿^𝜀𝜑^𝜀𝑥₁,…,𝑥_𝑛;𝜃=+𝜀𝑐⁻²𝑆𝑄(𝜑𝜃)𝜑𝑥₁₁,…,𝑥_𝑛+𝜀⁻¹𝑄𝜑₁+𝑐𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛𝑥₁,…,𝑥_𝑛;𝜃.(3.19)

It follows from (3.7) that the first two terms equal to 0. Let us estimate the following last term: 𝑐𝑆(𝜃)𝜑₁𝑥₁,…,𝑥_𝑛;𝜃==𝑐𝑐²₂𝑆(𝜃)𝑅₀𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛=𝑐²𝑆²(𝜃)𝜑𝑥₁,…,𝑥_𝑛cos²𝜃₁𝜕²𝜕𝑥²¹+sin²𝜃₁cos²𝜃₂𝜕²𝜕𝑥²²+⋯+sin²𝜃₁⋯sin²𝜃_𝑛−2cos²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛−1+sin²𝜃₁⋯sin²𝜃_𝑛−2sin²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛sin𝜃₁+cos𝜃₁cos𝜃₂×𝜕²𝜕𝑥₁𝜕𝑥₂+⋯+sin²𝜃₁⋯sin²𝜃_𝑛−2×sin𝜃_𝑛−1×cos𝜃_𝑛−1𝜕²𝜕𝑥_𝑛−1𝜕𝑥_𝑛𝜑𝑥¹,…,𝑥_𝑛≤𝐶_1,𝜑,(3.20) as soon as all the constants, functions, and their derivatives are bounded.

We also have from (3.7) 𝐿^𝜀𝜑^𝜀=𝐿^𝜀𝜑+𝜀𝐿^𝜀𝜑₁=𝐿^𝜀𝜑+𝜀𝐿^𝜀𝑅₀𝑆(𝜃)𝜑.(3.21)

Thus, 𝐿^𝜀𝜑=𝐿^𝜀𝜑^𝜀−𝜀𝑐𝐿^𝜀𝑅₀𝑆(𝜃)𝜑≤𝐶_1,𝜑−𝜀𝐶_2,𝜑<𝐶_𝜑,(3.22) for small 𝜀.

To prove the second condition, it’s enough to use the properties of the function 𝜑₀(𝑢)=√1+𝑢², namely: ||𝜑^0(𝑢)||≤1≤𝜑₀(𝑢),||𝜑^0(𝑢)||≤𝜑₀(𝑢).(3.23)

So, the proof of the second condition is similar to the previous reasoning.

Thus, the family (𝜉^𝜀𝑡,𝜃^𝜀𝑡) 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.(C1)The family of processes (𝜉^𝜀(𝑡),𝑥^𝜀(𝑡)), 𝑡≥0, 𝜀>0 is relatively compact.(C2)There exists the family of test-functions 𝜑^𝜀(𝑢,𝑥)∈𝐶³⁰(𝐑^𝑛×𝐸), such that lim_𝜀→0𝜑^𝜀(𝑢,𝑥)=𝜑(𝑢)(3.24) uniformly by 𝑢, 𝑥.(C3) The following uniform convergence is true lim_𝜀→0𝐿^𝜀𝜑^𝜀(𝑢,𝑥)=𝐿𝜑(𝑢)(3.25) uniformly by 𝑢,𝑥. The family 𝐿^𝜀𝜑^𝜀, 𝜀>0 is uniformly bounded, moreover 𝐿^𝜀𝜑^𝜀 and 𝐿𝜑 belong to 𝐶(𝐑^𝑛×𝐸).(C4)Convergence by probability of initial values 𝜉s^𝜀u(p0_𝜀>0)⟶𝜉̂(0),𝜀⟶0,𝐄||𝜉^𝜀(0)||≤𝐶<+∞(3.26) is true.
Then we have the weak convergence 𝜉^𝜀(𝑡)⟹𝜉̂(𝑡),𝜀⟶0.(3.27)

According to Theorem 3.5, we may confirm the weak convergence in 𝐃_𝐑^𝑛[0,∞)𝜉^𝜀𝑡⟹𝜉^0𝑡.(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.

4. Description of MNRE

We study the same particle in 𝐑^𝑛, 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 process 𝜉̃^𝜀𝑡∈𝐑^𝑛, given by 𝜉̃^𝜀𝑡∶=𝑥+^𝑡0𝑣𝜃^𝜀𝜏𝑠𝜃^𝜀𝜏𝑑𝜏.(4.1)
Our aim is to prove the weak convergence of MNRE to a diffusion process with drift when 𝜀→0.
Two-component Markov process (𝜉̃^𝜀𝑡,𝜃^𝜀𝑡) at the test functions 𝜑(𝑥₁,…,𝑥_𝑛;𝜃)∈𝐶^∞0(𝐑^𝑛×Θ) is defined by a generator (see, e.g., [2]) 𝐿^𝜀𝜑𝑥₁,…,𝑥_𝑛;𝜃∶==𝜆𝜀𝑄₋₂𝜑(⋅;𝜃)+𝑣(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅+𝑐𝑄₁𝜑(⋅;𝜃)+𝜀⁻¹𝑐(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅,(4.2) where 𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;⋅∶=−(𝑠(𝜃),∇)𝜑𝑥₁,…,𝑥_𝑛;⋅.(4.3)
An important condition that let us confirm weak convergence is balance condition Π𝑐(𝜃)𝑆(𝜃)=0.(4.4)

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: Π𝑐₁(𝜃)𝑆(𝜃)=(𝑑,∇)≠0.(4.5)

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 ∫^𝜋0sin^𝑛𝜃cos𝜃𝑑𝜃=0 or ∫^02𝜋sin𝜃𝑑𝜃=0.
Another example is the function 𝑐(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 12𝜋^02𝜋sin𝜃cos𝜃𝜕𝜕𝑥₁+sin𝜃𝜕𝜕𝑥₂𝑑𝜃=12𝜕𝜕𝑥₂.(4.6)
Another example is the following function in 𝐑^𝑛: 𝑐₁(𝜃)=𝑐¹0,,𝜃𝜃_{𝑛−1𝑛−1}∈[𝜋,2𝜋),∈[0,𝜋).(4.7) Again, all terms under the integral, except the last one, contain ∫^𝜋0sin^𝑛𝜃cos𝜃𝑑𝜃=0, thus only one term is not trivial as 1𝑁𝑐¹^𝜋0⋯^𝜋0^𝜋2𝜋sin^𝑛−1𝜃₁sin^𝑛−2𝜃₂⋯sin²𝜃_𝑛−1sin𝜃_𝑛−1𝑑𝜃₁⋯𝑑𝜃_𝑛−1.(4.8)
By simple calculations, we see Π𝑐₁(𝜃)𝑆(𝜃)=⎧⎪⎨⎪⎩−𝑐2¹23⋅⋅54⋅⋅⋯⋯⋅⋅((𝑛𝑛−−12))𝜕𝜕𝑥_𝑛−𝑐,𝑛=2𝑚+1,₁𝜋[1⋅2⋅⋯(𝑛−2)]𝜕𝜕𝑥_𝑛,𝑛=2𝑚.(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. Main Result for MNRE

Theorem 5.1. MNRE 𝜉̃^𝜀𝑡, converges weakly to the process 𝜉̃^0𝑡 when 𝜀→0 as 𝜉̃^𝜀𝑡⟹𝜉̃^0𝑡.(5.1)
The limit process 𝜉̃^0𝑡∈𝐑^𝑛 is defined by a generator 𝐿⁰𝜑𝑥₁,…,𝑥_𝑛=(𝑑,∇)𝜑𝑥₁,…,𝑥_𝑛+𝜎²,Δ𝜑𝑥₁,…,𝑥_𝑛,(5.2) where Δ𝜑(𝑥₁,…,𝑥_𝑛)∶=((𝜕²/𝜕𝑥²¹)+⋯+(𝜕²/𝜕𝑥^2𝑛))𝜑(𝑥₁,…,𝑥_𝑛), (𝑑,∇)∶=−1𝑁_𝑆𝑛𝜎𝑐₁(𝜃)(𝑠(𝜃),∇)𝜇(𝑑𝜃),₂,Δ∶=1𝑁_𝑆𝑛𝑐²(𝜃)(𝑠(𝜃),∇)²𝜇(𝑑𝜃).(5.3)
We need the following Lemma to prove the Theorem.

Lemma 5.2. At the perturbed test functions 𝜑^𝜀𝑥₁,…,𝑥_𝑛;𝜃=𝜑𝑥₁,…,𝑥_𝑛+𝜀𝜑₁𝑥₁,…,𝑥_𝑛;𝜃+𝜀²𝜑₂𝑥₁,…,𝑥_𝑛;𝜃,(5.4) having bounded derivatives of any degree and compact support, the operator 𝐿^𝜀 has the following asymptotic representation: 𝐿^𝜀𝜑^𝜀||𝑅𝑥₁,…,𝑥_𝑛;𝜃=𝐿⁰𝜑𝑥₁,…,𝑥_𝑛+𝑅^𝜀(𝜃)𝜑(𝑥),_𝜀(𝜃)𝜑(𝑥)||⟶0,𝜀⟶0,𝜑(𝑥)∈𝐶^∞0(𝐑^𝑛),(5.5) where 𝐿⁰ is defined in (5.2), 𝜑₁(𝑥₁,…,𝑥_𝑛;𝜃),𝜑₂(𝑥₁,…,𝑥_𝑛;𝜃), and 𝑅^𝜀(𝜃)𝜑(𝑥) are defined by the following correlations: 𝐿⁰Π=−Π𝑐(𝜃)𝑆𝜑(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)Π+Π𝑐₁(𝜃)𝑆(𝜃)Π𝜑,₁𝜑=−𝑅₀𝑐(𝜃)𝑆(𝜃)𝜑,₂𝑅=𝑅₀𝑐(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝜑,_𝜀(𝜃)𝜑=𝜀+𝜀𝑐₂(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)+𝑐₁(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝑐₁(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝜑.(5.6)

Proof. We solve singular perturbation problem for the operator (4.2). Using the view of test-functions (5.4), we have 𝐿^𝜀𝜑^𝜀𝑥₁,…,𝑥_𝑛;𝜃==𝜀𝜀⁻²₋₂𝑄+𝜀⁻¹𝑐(𝜃)𝑆(𝜃)+𝑐₁(𝜃)𝑆(𝜃)𝜑+𝜀𝜑₁+𝜀²𝜑₂+𝜀𝑄𝜑𝑐(+𝜃)𝜀𝑆⁻¹(𝜃𝑄)𝜑𝜑₁₂+𝑐(𝜃)𝑆(𝜃)𝜑+𝑄𝜑₂+𝑐(𝜃)𝑆(𝜃)𝜑₁+𝑐₁(𝜃)𝑆(𝜃)𝜑+𝑐₁(𝜃)𝑆(𝜃)𝜑₁+𝜀²𝑐₁(𝜃)𝑆(𝜃)𝜑₂.(5.7)
Thus, we obtain the following equations: 𝑄𝜑₁𝑄𝜑=0,𝐿+𝑐(𝜃)𝑆(𝜃)𝜑=0,₀𝑅𝜑=𝑄𝜑₂+𝑐(𝜃)𝑆(𝜃)𝜑₁+𝑐₁(𝜃)𝑆(𝜃)𝜑,_𝜀𝜑(𝜃)=𝜀𝑐(𝜃)𝑆(𝜃)𝜑₂+𝑐₁(𝜃)𝑆(𝜃)𝜑₁+𝜀²𝑐₁(𝜃)𝑆(𝜃)𝜑₂.(5.8)
According to the first one, 𝜑(𝑥₁,…,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 𝜑₁=−𝑅₀𝑐(𝜃)𝑆(𝜃)𝜑.(5.9)
By substitution into the third equation and using the solvability condition, we obtain: 𝐿⁰Π𝜑+Π𝑐(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)Π𝜑−Π𝑐₁(𝜃)𝑆(𝜃)Π𝜑=0.(5.10)
From the last equation, we have 𝑅^𝜀(𝜃)𝜑(𝑥)=𝜀𝑐+(𝜀𝜃₂)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)+𝑐₁(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃)𝑆(𝜃)𝑐₁(𝜃)𝑆(𝜃)𝑅₀𝑐(𝜃w)h𝑆e(n𝜃)𝜀𝑅⟶₀𝑐(𝜃0),𝜑𝑆((𝑥𝜃))∈𝜑𝐶(_∞0𝑥)⟶0(𝐑^𝑛).(5.11)
We calculate the operator 𝐿⁰ by the formula (5.6) as 𝐿⁰𝜑==ΠΠ𝑐𝑐₂(𝜃)𝑆(𝜃)(𝐼−Π)𝑐(𝜃)𝑆(𝜃)Π𝜑+Π𝑐₁(𝜃)𝑆(𝜃)Π𝜑(𝜃)𝑆²(𝜃)Π𝜑−Π𝑐(𝜃)𝑆(𝜃)Π𝑐(𝜃)𝑆(𝜃)Π𝜑+Π𝑐₁(𝜃)𝑆(𝜃)Π𝜑.(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 𝐿⁰=Π𝑐²(𝜃)𝑆²(𝜃)+Π𝑐₁(𝜃)𝑆(𝜃).(5.13)
Using the view of 𝑆(𝜃), we may write Π𝑐₁(𝜃)𝑆(𝜃)=−1𝑁_𝑆𝑛Π𝑐₂𝑐₁(𝜃)(𝑠(𝜃),∇)𝜇(𝑑𝜃)=∶(𝑑,∇),(𝜃)𝑆²(𝜃)=1𝑁_𝑆𝑛𝑐²(𝜃)(𝑠(𝜃),∇)²𝜇(𝑑𝜃)=∶𝜎²,Δ.(5.14)
Lemma is proved.

Proof of Theorem 5.1. We proved in Lemma 5.2 that 𝐿^𝜀𝜑^𝜀⇒𝐿⁰𝜑 at the class of test functions 𝐶^∞0(𝐑^𝑛×Θ), when 𝜀→0. To prove the weak convergence, we should show relative compactness of the family of processes (𝜉̃^𝜀𝑡,𝜃^𝜀𝑡) in 𝐃_𝐑^𝑛_×Θ[0,∞). To do this, we use Lemma 3.4.
We have the following view of the operator (4.2) at the test function 𝜑^𝜀(𝑥₁,…,𝑥_𝑛;𝜃)=𝜑(𝑥₁,…,𝑥_𝑛)+𝜀𝜑₁(𝑥₁,…,𝑥_𝑛;𝜃), where 𝜑₁(𝑥₁,…,𝑥_𝑛;𝜃)=−𝑅₀𝑐(𝜃)𝑆(𝜃)𝜑(𝑥₁,…,𝑥_𝑛): 𝐿^𝜀𝜑^𝜀𝑥₁,…,𝑥_𝑛;𝜃=+𝜀⁻²𝑐𝑄(𝜃𝜑)𝑆𝑥(₁𝜃,…)𝜑,₁𝑥_𝑛+𝜀⁻¹𝑄𝜑₁+𝑐(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛+𝜀𝑐₁𝑥₁,…,𝑥_𝑛;𝜃+𝑐₁(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;𝜃(𝜃)𝑆(𝜃)𝜑₁𝑥₁,…,𝑥_𝑛;𝜃.(5.15)
It follows from (5.8) that the first two terms equal to 0. Let us estimate the following third term: 𝑐(𝜃=)𝑆𝑐((𝜃𝜃))𝑆𝜑₁(𝜃)𝑥𝑅₁,₀…,𝑥_𝑛;𝜃+𝑐₁(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;𝜃=𝑐₂𝑐(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛+𝑐₁(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;𝜃=𝑐₂(𝜃)𝑆²(𝜃)𝜑𝑥₁,…,𝑥_𝑛+𝑐₁(𝜃)𝑆(𝜃)𝜑𝑥₁,…,𝑥_𝑛;𝜃(𝜃)cos²𝜃₁𝜕²𝜕𝑥²¹+sin²𝜃₁cos²𝜃₂𝜕²𝜕𝑥²²+⋯+sin²𝜃₁⋯sin²𝜃_𝑛−2cos²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛−1+sin²𝜃₁⋯sin²𝜃_𝑛−2sin²𝜃_𝑛−1𝜕²𝜕𝑥^2𝑛+sin𝜃₁cos𝜃₁cos𝜃₂𝜕²𝜕𝑥₁𝜕𝑥₂+⋯+sin²𝜃₁⋯sin²𝜃_𝑛−2sin𝜃_𝑛−1×cos𝜃_𝑛−1𝜕²𝜕𝑥_𝑛−1𝜕𝑥_𝑛𝜑𝑥¹cos𝜃₁,…,𝑥_𝑛+𝑐₁(𝜃)𝜕𝜕𝑥₁+sin𝜃₁cos𝜃₂𝜕𝜕𝑥₂+⋯+sin𝜃₁⋯sin𝜃_𝑛−2cos𝜃_𝑛−1𝜕𝜕𝑥_𝑛−1+sin𝜃₁⋯sin𝜃_𝑛−2sin𝜃_𝑛−1𝜕𝜕𝑥_𝑛𝜑𝑥¹,…,𝑥_𝑛≤𝐶_1,𝜑,(5.16) 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 𝐿^𝜀𝜑^𝜀=𝐿^𝜀𝜑+𝜀𝐿^𝜀𝜑₁=𝐿^𝜀𝜑+𝜀𝑐𝐿^𝜀𝑅₀𝜃𝑆(𝜃)𝜑.(5.17)
Thus, 𝐿^𝜀𝜑=𝐿^𝜀𝜑^𝜀−𝜀𝑐𝐿^𝜀𝑅₀𝑆(𝜃)𝜑≤𝐶_1,𝜑−𝜀𝐶_2,𝜑<𝐶_𝜑,(5.18) for 𝜀 that is small enough.
To prove the second condition of Lemma 3.4, we use the following properties of the function 𝜑₀(𝑢)=√1+𝑢²: ||𝜑^0(𝑢)||≤1≤𝜑₀(𝑢),||𝜑^0(𝑢)||≤𝜑₀(𝑢).(5.19)
We can see again that the proof of the second condition is similar to the previous reasoning.
Thus, the family of the processes (𝜉̃^𝜀𝑡,𝜃^𝜀𝑡) is relatively compact in 𝐃_𝐑^𝑛_×Θ[0,∞).
Using Theorem 3.5, we confirm the following weak convergence in 𝐃_𝐑^𝑛[0,∞): 𝜉̃^𝜀𝑡⟹𝜉̃^0𝑡.(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.

Example 5.3. Let us consider one more variant of evolution in 𝐑².
Let 𝑐𝑐(𝜃)=11,,𝜃𝜃==0𝜋,,₁(𝜃)=1,𝜃=𝜋2.(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 𝑐₁(𝜃).
The limit generator (5.2) has the view 𝐿⁰𝜑𝑥₁,𝑥₂=12𝜋𝜕𝜕𝑥₂𝜑𝑥₁,𝑥₂+1𝜋𝜕𝜕𝑥²²¹𝜑𝑥₁,𝑥₂.(5.22)
Thus, the limit process has two parts: the drift with velocity 1/2𝜋 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].

List of Symbols

𝑐,𝑐(𝜃),𝑐1(𝜃):	the components of velocity of random evolution’s motion
𝐶∞0(𝐑𝑛×Θ):	the space of bounded continuous functions on 𝐑𝑛×Θ having continuous derivatives of all orders vanishing at infinity
𝐶𝜑:	constant depending on the norm of function 𝜑
𝑑:	parameter of drift
𝐃𝐑𝑛×Θ[0,∞):	the space of 𝐑𝑛×Θ-valued right continuous functions having left limits (cadlag), defined on 𝐑+
𝐿𝜀:	the generator of random evolution
𝐿0:	the generator of limit process
𝑁:	volume of 𝑆𝑛
𝑛:	dimension of space
𝑄:	the generator of switching Poisson process
𝑄1:	perturbing operator
𝑅0:	potential of generator 𝑄
𝑅𝜀(𝜃):	negligible term
𝑆𝑛:	unit 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=(𝜃𝑖,𝑖=1,𝑛−1),  𝜃𝑛−1∈[0,2𝜋),  𝜃𝑖∈[0,𝜋),  𝑖=1,𝑛−2:	angles that define directions in 𝐑𝑛
Θ:	the set of all angles 𝜃
𝜃𝜀𝑡:	Poisson process that switches directions of random evolution’s motion
𝜆:	intensity of switching Poisson process
𝜇(𝑑𝜃):	small element of volume of 𝑆𝑛
𝜉𝜀𝑡,𝜉̃𝜀𝑡:	Markov random evolutions
𝜉0𝑡,𝜉̃0𝑡:	limit processes
Π:	projector at the nullspace of generator 𝑄
𝜎2:	parameter of diffusion.