Table of Contents
ISRN Probability and Statistics
VolumeΒ 2012, Article IDΒ 509789, 19 pages
http://dx.doi.org/10.5402/2012/509789
Research Article

Weak Convergence of Markov Random Evolutions in a Multidimensional Space

Institute of Mathematics, NASU, Tereschenkivska Street, 3, Kiev 01601, Ukraine

Received 4 May 2012; Accepted 5 June 2012

Academic Editors: E.Β Orsingher and A.Β Pascucci

Copyright Β© 2012 Igor V. Samoilenko. 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.

Abstract

We study Markov symmetrical and nonsymmetrical random evolutions in 𝐑𝑛. 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 πƒπ‘π‘›Γ—Ξ˜[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 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 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 𝐑𝑛 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 Sections 3 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 space 𝐑𝑛, that starts at 𝑑=0 from the point π‘₯=(π‘₯𝑖,𝑖=1,𝑛). Possible directions of motion are given by the following vectors: 𝑠(πœƒ)=cosπœƒ1,sinπœƒ1cosπœƒ2,sinπœƒ1sinπœƒ2cosπœƒ3,…,sinπœƒ1β‹―sinπœƒπ‘›βˆ’2cosπœƒπ‘›βˆ’1,sinπœƒ1β‹―sinπœƒπ‘›βˆ’2sinπœƒπ‘›βˆ’1ξ€Έ,πœƒπ‘›βˆ’1∈[0,2πœ‹),πœƒπ‘–βˆˆ[0,πœ‹),𝑖=1,π‘›βˆ’2.(2.1) 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 πœ†=πœ€βˆ’2. The velocity of particle’s motion is fixed and equals 𝑣=π‘πœ€βˆ’1, where πœ€ is a small parameter, πœ€β†’0(πœ€>0).

Let us define a set Θ={πœƒβˆΆπ‘ (πœƒ)βˆˆπ‘†π‘›} and suppose πœƒπœ€π‘‘βˆˆΞ˜ be the switching Poisson process.

Definition 2.1. Markov symmetrical random evolution (MSRE) is the process πœ‰πœ€π‘‘βˆˆπ‘π‘›, given by πœ‰πœ€π‘‘ξ€œβˆΆ=π‘₯+𝑣𝑑0π‘ ξ€·πœƒπœ€πœξ€Έπ‘‘πœ.(2.2)
It is easy to see that when πœ€β†’0(πœ€>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 πœ€β†’0. The main method is solution of singular perturbation problem for the generator of MSRE. Let us describe this generator.
Two-component Markov process (πœ‰πœ€π‘‘,πœƒπœ€π‘‘) at the test functions πœ‘(π‘₯1,…,π‘₯𝑛;πœƒ)∈𝐢∞0(π‘π‘›Γ—Ξ˜) can be described by the generator (see, e.g., [2]) πΏπœ€πœ‘ξ€·π‘₯1,…,π‘₯𝑛π‘₯;πœƒβˆΆ=πœ†π‘„πœ‘(β‹…;πœƒ)+𝑣𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;β‹…=πœ€βˆ’2π‘„πœ‘(β‹…;πœƒ)+πœ€βˆ’1ξ€·π‘₯𝑐𝑆(πœƒ)πœ‘1,…,π‘₯𝑛,;β‹…(2.3) where 𝑆π‘₯(πœƒ)πœ‘1,…,π‘₯𝑛π‘₯;β‹…βˆΆ=βˆ’(𝑠(πœƒ),βˆ‡)πœ‘1,…,π‘₯𝑛;β‹…,(2.4) here βˆ‡πœ‘=(πœ•πœ‘/πœ•π‘₯𝑖,1≀𝑖≀𝑛), 1π‘„πœ‘(β‹…;πœƒ)∢=(Ξ βˆ’πΌ)πœ‘(β‹…;πœƒ)∢=π‘ξ€œπ‘†π‘›πœ‘(β‹…;πœƒ)πœ‡(π‘‘πœƒ)βˆ’πœ‘(β‹…;πœƒ),(2.5) and 𝑁=(2πœ‹)𝑛/2(1/(2β‹…4β‹…β‹―β‹…(π‘›βˆ’2))) for even 𝑛, 𝑁=(2πœ‹)(π‘›βˆ’1)/2(2/(3β‹…5β‹…β‹―β‹…(π‘›βˆ’2))) for odd 𝑛; πœ‡(π‘‘πœƒ) is the element of volume of the sphere 𝑆𝑛, that is equal to πœ‡(π‘‘πœƒ)∢=sinπ‘›βˆ’2πœƒ1sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2π‘‘πœƒ1β‹―π‘‘πœƒπ‘›βˆ’1.(2.6)
Using well-known formula ξ€œπœ‹0sin2π‘šπ‘‘πœƒ=2β‹…1β‹…2β‹…β‹―β‹…(2π‘šβˆ’1)β‹…πœ‹2β‹…4β‹…β‹―β‹…2π‘š2,ξ€œπœ‹0sin2π‘š+1π‘‘πœƒ=2β‹…2β‹…4β‹…β‹―β‹…2π‘š1β‹…3β‹…β‹―β‹…(2π‘š+1)(2.7) we can see ξ€œπ‘†π‘›πœ‡(π‘‘πœƒ)=𝑁.(2.8)
Operator βˆ«Ξ πœ‘(β‹…;πœƒ)∢=(1/𝑁)π‘†π‘›πœ‘(β‹…;πœƒ)πœ‡(π‘‘πœƒ) 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 Ξ  we have 𝑄Π=Π𝑄=0.(2.9)
Potential operator 𝑅0 can be defined as the following: 𝑅0∢=Ξ βˆ’πΌ.(2.10)
This operator has the following property: 𝑅0𝑄=𝑄𝑅0=πΌβˆ’Ξ ,(2.11) thus it is inverse for 𝑄 at the range of 𝑄, but for the function πœ™ from the null space of 𝑄 we have 𝑅0πœ™=0.(2.12)
Solution of singular perturbation problem in the series scheme with the small series parameter πœ€β†’0(πœ€>0) (see [15]) for reducibly invertible operator 𝑄 and perturbing operator 𝑄1 consists in the following.
We should find two vectors πœ‘πœ€=πœ‘+πœ€πœ‘1+πœ€2πœ‘2 and πœ“, that satisfy asymptotic representation ξ€Ίπœ€βˆ’2𝑄+πœ€βˆ’1𝑄1ξ€»πœ‘πœ€=πœ“+πœ€πœƒπœ€,(2.13) with the vector πœƒπœ€, that is uniformly bounded by the norm and such that β€–πœƒπœ€β€–β‰€πΆ,πœ€βŸΆ0.(2.14)
The left part of the equation can be rewritten as ξ€Ίπœ€βˆ’2𝑄+πœ€βˆ’1𝑄1ξ€»ξ€·πœ‘+πœ€πœ‘1+πœ€2πœ‘2ξ€Έ=πœ€βˆ’2π‘„πœ‘+πœ€βˆ’1ξ€Ίπ‘„πœ‘1+𝑄1πœ‘ξ€»+ξ€Ίπ‘„πœ‘2+𝑄1πœ‘1ξ€»+πœ€π‘„1πœ‘2.(2.15)
And as soon as it is equal to the right side, we obtain π‘„πœ‘=0,π‘„πœ‘1+𝑄1πœ‘=0,π‘„πœ‘2+𝑄1πœ‘1𝑄=πœ“,1πœ‘2=πœƒπœ€.(2.16)
From the last equation we may see that the function πœ‘2 should be smooth enough to provide boundness of 𝑄1πœ‘2. Moreover, from the first equation we see that any function from the null space of 𝑄 can be taken as πœ‘, 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 𝑄1πœ‘ 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 πœ‘1=βˆ’π‘…0𝑄1πœ‘.(2.18)
Thus, the main problem is to solve the following equation: π‘„πœ‘2=πœ“βˆ’π‘„1πœ‘1=πœ“+𝑄1𝑅0𝑄1πœ‘.(2.19)
The solvability condition for 𝑄 has the view Ξ π‘„Ξ πœ‘2=0=Ξ πœ“+Π𝑄1𝑅0𝑄1Ξ πœ‘,(2.20) and we finally obtain Ξ πœ“=βˆ’Ξ π‘„1𝑅0𝑄1Ξ πœ‘.(2.21)
For the function πœ‘2 we obviously have πœ‘2=𝑅0ξ€Ίβˆ’Ξ π‘„1𝑅0𝑄1Ξ +𝑄1𝑅0𝑄1ξ€»πœ‘.(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(πœƒ)𝟏(π‘₯)=0,(2.23) where 𝟏(π‘₯)=(π‘₯1,…,π‘₯𝑛). Really, every term under the integral contains either βˆ«πœ‹0sinπ‘›πœƒcosπœƒπ‘‘πœƒ=0 or ∫02πœ‹sinπœƒπ‘‘πœƒ=0.

3. Main Result for MSRE

Theorem 3.1. MSRE πœ‰πœ€π‘‘, converges weakly to the Wiener process 𝑀(𝑑)∢=πœ‰0𝑑 when πœ€β†’0 as πœ‰πœ€π‘‘βŸΉπœ‰0𝑑,(3.1) where πœ‰0π‘‘βˆˆπ‘π‘› is defined by the generator 𝐿0πœ‘ξ€·π‘₯1,…,π‘₯𝑛=𝑐2𝑛π‘₯Ξ”πœ‘1,…,π‘₯𝑛.(3.2) Here Ξ”πœ‘βˆΆ=((πœ•2/(πœ•π‘₯21))+β‹―+(πœ•2/(πœ•π‘₯2𝑛)))πœ‘.

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 πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛π‘₯;πœƒ=πœ‘1,…,π‘₯𝑛+πœ€πœ‘1ξ€·π‘₯1,…,π‘₯𝑛;πœƒ+πœ€2πœ‘2ξ€·π‘₯1,…,π‘₯𝑛,;πœƒ(3.3) having bounded derivatives of any degree and compact support, the operator πΏπœ€ has the asymptotic representation πΏπœ€πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛;πœƒ=𝐿0πœ‘ξ€·π‘₯1,…,π‘₯𝑛+π‘…πœ€||𝑅(πœƒ)πœ‘(π‘₯),πœ€||(πœƒ)πœ‘(π‘₯)⟢0,πœ€βŸΆ0,πœ‘(π‘₯)∈𝐢∞0(𝐑𝑛),(3.4) where 𝐿0 is defined in (3.2), πœ‘1(π‘₯1,…,π‘₯𝑛;πœƒ),πœ‘2(π‘₯1,…,π‘₯𝑛;πœƒ) and π‘…πœ€(πœƒ)πœ‘(π‘₯) are the following: 𝐿0Ξ =βˆ’π‘2Π𝑆(πœƒ)𝑅0πœ‘π‘†(πœƒ)Ξ ,1=βˆ’π‘π‘…0πœ‘π‘†(πœƒ)πœ‘,2=𝑐2𝑅0𝑆(πœƒ)𝑅0𝑆𝑅(πœƒ)πœ‘,πœ€(πœƒ)πœ‘=πœ€π‘3𝑆(πœƒ)𝑅0𝑆(πœƒ)𝑅0𝑆(πœƒ)πœ‘.(3.5)

Proof. Let us solve singular perturbation problem for the operator (2.3). To do this, we use the following correlation: πΏπœ€πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛=ξ€Ίπœ€;πœƒβˆ’2𝑄+πœ€βˆ’1𝑐𝑆(πœƒ)ξ€»ξ€Ίπœ‘+πœ€πœ‘1+πœ€2πœ‘2ξ€»=πœ€βˆ’2π‘„πœ‘+πœ€βˆ’1ξ€Ίπ‘„πœ‘1ξ€»+ξ€Ί+𝑐𝑆(πœƒ)πœ‘π‘„πœ‘2+𝑐𝑆(πœƒ)πœ‘1ξ€»+πœ€π‘π‘†(πœƒ)πœ‘2.(3.6)
Thus, we obtain equations π‘„πœ‘=0,π‘„πœ‘1𝐿+𝑐𝑆(πœƒ)πœ‘=0,0πœ‘=π‘„πœ‘2+𝑐𝑆(πœƒ)πœ‘1,π‘…πœ€πœ‘(πœƒ)=πœ€π‘π‘†(πœƒ)πœ‘2.(3.7)
From the first equation we see that πœ‘(π‘₯1,…,π‘₯𝑛) belongs to the null space of 𝑄. It is easy to see from the balance condition (2.23) that 𝑆(πœƒ)πœ‘ belongs to the range of 𝑄, thus from the second equation of the system (3.7) we have πœ‘1=βˆ’π‘π‘…0𝑆(πœƒ)πœ‘.(3.8)
By substitution into the third equation and using the solvability condition, we can see 𝐿0Ξ πœ‘+𝑐2Π𝑆(πœƒ)𝑅0𝑆(πœƒ)Ξ πœ‘=0.(3.9)
From the last equation of (2.23) we have π‘…πœ€(πœƒ)πœ‘(π‘₯)=πœ€π‘3𝑆(πœƒ)𝑅0𝑆(πœƒ)𝑅0𝑆(πœƒ)πœ‘(π‘₯)⟢0whenπœ€βŸΆ0,πœ‘(π‘₯)∈𝐢∞0(𝐑𝑛).(3.10)
Let’s find the generator of the limit process 𝐿0 by the formula (3.5) as 𝐿0πœ‘=𝑐2Π𝑆(πœƒ)(πΌβˆ’Ξ )𝑆(πœƒ)Ξ πœ‘=𝑐2Π𝑆2(πœƒ)Ξ πœ‘βˆ’π‘2Π𝑆(πœƒ)Π𝑆(πœƒ)Ξ πœ‘.(3.11)
The last term equals to 0 by the balance condition (2.23). Thus, finally 𝐿0=𝑐2Π𝑆2(πœƒ),(3.12) or 𝐿0=𝑐2π‘ξ€œπ‘†π‘›π‘†2(πœƒ)πœ‡(π‘‘πœƒ).(3.13)
Let us calculate the following integral:𝑐2π‘ξ€œπœ‹0β‹―ξ€œπœ‹0ξ€œ02πœ‹ξƒ¬cos2πœƒ1πœ•2πœ•π‘₯21+sin2πœƒ1cos2πœƒ2πœ•2πœ•π‘₯22+β‹―+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2cos2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2π‘›βˆ’1+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2sin2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2𝑛+ξ‚»sinπœƒ1cosπœƒ1cosπœƒ2πœ•2πœ•π‘₯1πœ•π‘₯2+β‹―+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2sinπœƒπ‘›βˆ’1Γ—cosπœƒπ‘›βˆ’1πœ•2πœ•π‘₯π‘›βˆ’1πœ•π‘₯𝑛×sinπ‘›βˆ’2πœƒ1sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2π‘‘πœƒ1β‹―π‘‘πœƒπ‘›βˆ’1=𝑐2π‘ξ€œπœ‹0β‹―ξ€œπœ‹0ξ€œ02πœ‹ξƒ¬cos2πœƒ1sinπ‘›βˆ’2πœƒ1sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2πœ•2πœ•π‘₯21+sinπ‘›πœƒ1cos2πœƒ2sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2πœ•2πœ•π‘₯22+β‹―+sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin3πœƒπ‘›βˆ’2cos2πœƒπ‘›βˆ’1Γ—πœ•2πœ•π‘₯2π‘›βˆ’1+sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin2πœƒπ‘›βˆ’2sin2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2𝑛+ξ‚»sinπ‘›βˆ’1πœƒ1cosπœƒ1cosπœƒ2sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2πœ•2πœ•π‘₯1πœ•π‘₯2+β‹―+sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin3πœƒπ‘›βˆ’2sinπœƒπ‘›βˆ’1cosπœƒπ‘›βˆ’1πœ•2πœ•π‘₯π‘›βˆ’1πœ•π‘₯π‘›ξ‚Όξ‚ΉΓ—π‘‘πœƒ1β€¦π‘‘πœƒπ‘›βˆ’1=𝑐2π‘ξ€œπœ‹0ξ€œπœ‹0β‹―ξ€œπœ‹0ξ‚Έξ€·sinπ‘›βˆ’2πœƒ1sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2βˆ’sinπ‘›πœƒ1sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2ξ€Έπœ•2πœ•π‘₯21+ξ€·sinπ‘›πœƒ1sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2βˆ’sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2sinπ‘›βˆ’4πœƒ3β‹―sinπœƒπ‘›βˆ’2ξ€ΈΓ—πœ•2πœ•π‘₯22ξ€·+β‹―+sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin3πœƒπ‘›βˆ’2βˆ’sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin3πœƒπ‘›βˆ’2sin2πœƒπ‘›βˆ’1ξ€Έπœ•2πœ•π‘₯2π‘›βˆ’1+sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin3πœƒπ‘›βˆ’2sin2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2𝑛+ξ‚»sinπ‘›βˆ’1πœƒ1cosπœƒ1cosπœƒ2sinπ‘›βˆ’3πœƒ2β‹―sinπœƒπ‘›βˆ’2πœ•2πœ•π‘₯1πœ•π‘₯2+β‹―+sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sin3πœƒπ‘›βˆ’2sinπœƒπ‘›βˆ’1cosπœƒπ‘›βˆ’1πœ•2πœ•π‘₯π‘›βˆ’1πœ•π‘₯π‘›ξ‚Όξ‚ΉΓ—π‘‘πœƒ1β‹―π‘‘πœƒπ‘›βˆ’1.(3.14)
Every term in braces has a multiplier of the type βˆ«πœ‹0sinπ‘›πœƒcosπœƒπ‘‘πœƒ=0 or ∫02πœ‹sinπœƒcosπœƒπ‘‘πœƒ=0, thus the corresponding integral equals to 0.
Integration of the correlations in the parentheses gives ξ€œπœ‹0β‹―ξ€œπœ‹0ξ€œ02πœ‹ξ€·sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sinπ‘›βˆ’π‘˜+2πœƒπ‘˜βˆ’1sinπ‘›βˆ’π‘˜βˆ’1πœƒπ‘˜sinπ‘›βˆ’π‘˜βˆ’2πœƒπ‘˜+1β‹―sinπœƒπ‘›βˆ’2βˆ’sinπ‘›πœƒ1sinπ‘›βˆ’1πœƒ2β‹―sinπ‘›βˆ’π‘˜+2πœƒπ‘˜βˆ’1sinπ‘›βˆ’π‘˜+1πœƒπ‘˜sinπ‘›βˆ’π‘˜βˆ’2πœƒπ‘˜+1β‹―sinπœƒπ‘›βˆ’2ξ€Έπ‘‘πœƒ1β‹―π‘‘πœƒπ‘›βˆ’1ξ‚΅=𝑁(2π‘šβˆ’1)(2π‘šβˆ’2)(2π‘šβˆ’3)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+1)βˆ’2π‘š(2π‘šβˆ’1)(2π‘šβˆ’2)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+2)(2π‘šβˆ’1)(2π‘šβˆ’2)(2π‘šβˆ’3)β‹…β‹―β‹…(2π‘šβˆ’π‘˜)ξ‚Άξ‚€2π‘š(2π‘šβˆ’1)(2π‘šβˆ’2)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+1)=𝑁2π‘šβˆ’π‘˜+1βˆ’2π‘š2π‘šβˆ’π‘˜ξ‚ξ‚΅2π‘š,if𝑛=2π‘šor𝑁(2π‘š)(2π‘šβˆ’1)(2π‘šβˆ’2)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+2)βˆ’(2π‘š+1)2π‘š(2π‘šβˆ’1)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+3)2π‘š(2π‘šβˆ’1)(2π‘šβˆ’2)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+1)ξ‚Άξ‚€(2π‘š+1)2π‘š(2π‘šβˆ’1)β‹…β‹―β‹…(2π‘šβˆ’π‘˜+2)=𝑁2π‘šβˆ’π‘˜+2βˆ’2π‘š+12π‘šβˆ’π‘˜+1=𝑁2π‘š+1,if𝑛=2π‘š+1𝑛.(3.15)
Finally, we have 𝐿0πœ‘ξ€·π‘₯1,…,π‘₯𝑛=𝑐2𝑛π‘₯Ξ”πœ‘1,…,π‘₯𝑛.(3.16)
Lemma is proved.

Proof of Theorem 3.1. In Lemma 3.3, we proved that πΏπœ€πœ‘πœ€β‡’πΏ0πœ‘ at the class of functions 𝐢∞0(π‘π‘›Γ—Ξ˜) when πœ€β†’0. To prove the weak convergence of the process, we should show relative compactness of the family (πœ‰πœ€π‘‘,πœƒπœ€π‘‘) in πƒπ‘π‘›Γ—Ξ˜[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 πΏπœ€, πœ€>0 satisfy the inequalities ||πΏπœ€||πœ‘(𝑒)<πΆπœ‘,(3.17) for any real-valued nonnegative function πœ‘βˆˆπΆβˆž0(π‘π‘›Γ—Ξ˜), where the constant πΆπœ‘ depends on the norm of πœ‘, and for πœ‘0√(𝑒)=1+𝑒2, πΏπœ€πœ‘0(𝑒)β‰€πΆπ‘™πœ‘0(𝑒),|𝑒|≀𝑙,(3.18) where the constant 𝐢𝑙 depends on the function πœ‘0, but does not depend on πœ€>0.
Then, the family (πœ‰πœ€π‘‘,πœƒπœ€π‘‘), 𝑑β‰₯0, πœ€>0is relatively compact in πƒπ‘π‘›Γ—Ξ˜[0,∞).

Let us write the action of the generator (2.3) at the test function πœ‘πœ€(π‘₯1,…,π‘₯𝑛;πœƒ)=πœ‘(π‘₯1,…,π‘₯𝑛)+πœ€πœ‘1(π‘₯1,…,π‘₯𝑛;πœƒ), where πœ‘1(π‘₯1,…,π‘₯𝑛;πœƒ)=βˆ’π‘π‘…0𝑆(πœƒ)πœ‘(π‘₯1,…,π‘₯𝑛).

We have πΏπœ€πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛;πœƒ=πœ€βˆ’2ξ€·π‘₯π‘„πœ‘1,…,π‘₯𝑛+πœ€βˆ’1ξ€Ίπ‘„πœ‘1ξ€·π‘₯+𝑐𝑆(πœƒ)πœ‘1,…,π‘₯𝑛+𝑐𝑆(πœƒ)πœ‘1ξ€·π‘₯1,…,π‘₯𝑛.;πœƒ(3.19)

It follows from (3.7) that the first two terms equal to 0. Let us estimate the following last term: 𝑐𝑆(πœƒ)πœ‘1ξ€·π‘₯1,…,π‘₯𝑛;πœƒ=𝑐2𝑆(πœƒ)𝑅0𝑆π‘₯(πœƒ)πœ‘1,…,π‘₯𝑛=𝑐2𝑆2ξ€·π‘₯(πœƒ)πœ‘1,…,π‘₯𝑛=𝑐2cos2πœƒ1πœ•2πœ•π‘₯21+sin2πœƒ1cos2πœƒ2πœ•2πœ•π‘₯22+β‹―+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2cos2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2π‘›βˆ’1+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2sin2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2𝑛+ξ‚»sinπœƒ1cosπœƒ1cosπœƒ2Γ—πœ•2πœ•π‘₯1πœ•π‘₯2+β‹―+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2Γ—sinπœƒπ‘›βˆ’1Γ—cosπœƒπ‘›βˆ’1πœ•2πœ•π‘₯π‘›βˆ’1πœ•π‘₯π‘›πœ‘ξ€·π‘₯ξ‚Όξ‚Ή1,…,π‘₯𝑛≀𝐢1,πœ‘,(3.20) as soon as all the constants, functions, and their derivatives are bounded.

We also have from (3.7) πΏπœ€πœ‘πœ€=πΏπœ€πœ‘+πœ€πΏπœ€πœ‘1=πΏπœ€πœ‘+πœ€πΏπœ€π‘…0𝑆(πœƒ)πœ‘.(3.21)

Thus, πΏπœ€πœ‘=πΏπœ€πœ‘πœ€βˆ’πœ€π‘πΏπœ€π‘…0𝑆(πœƒ)πœ‘β‰€πΆ1,πœ‘βˆ’πœ€πΆ2,πœ‘<πΆπœ‘,(3.22) for small πœ€.

To prove the second condition, it’s enough to use the properties of the function πœ‘0√(𝑒)=1+𝑒2, namely: ||πœ‘ξ…ž0||(𝑒)≀1β‰€πœ‘0||πœ‘(𝑒),0ξ…žξ…ž||(𝑒)β‰€πœ‘0(𝑒).(3.23)

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

Thus, the family (πœ‰πœ€π‘‘,πœƒπœ€π‘‘) is relatively compact in πƒπ‘π‘›Γ—Ξ˜[0,∞).

Now we may use the following theorem (Theorem  6.6 from [15]).

Theorem 3.5.   Let random evolution with Markov switching (πœ‰πœ€(𝑑),π‘₯πœ€(𝑑))βˆˆπƒπ‘π‘›Γ—πΈ[0,∞) satisfies the following conditions.(C1)The family of processes (πœ‰πœ€(𝑑),π‘₯πœ€(𝑑)), 𝑑β‰₯0, πœ€>0 is relatively compact.(C2)There exists the family of test-functions πœ‘πœ€(𝑒,π‘₯)∈𝐢30(𝐑𝑛×𝐸), such that limπœ€β†’0πœ‘πœ€(𝑒,π‘₯)=πœ‘(𝑒)(3.24) uniformly by 𝑒, π‘₯.(C3) The following uniform convergence is true limπœ€β†’0πΏπœ€πœ‘πœ€(𝑒,π‘₯)=πΏπœ‘(𝑒)(3.25) uniformly by 𝑒,π‘₯. The family πΏπœ€πœ‘πœ€, πœ€>0 is uniformly bounded, moreover πΏπœ€πœ‘πœ€ and πΏπœ‘ belong to 𝐢(𝐑𝑛×𝐸).(C4)Convergence by probability of initial values πœ‰πœ€Μ‚(0)βŸΆπœ‰(0),πœ€βŸΆ0,supπœ€>0𝐄||πœ‰πœ€||(0)≀𝐢<+∞(3.26) is true.
Then we have the weak convergence πœ‰πœ€Μ‚(𝑑)βŸΉπœ‰(𝑑),πœ€βŸΆ0.(3.27)

According to Theorem 3.5, we may confirm the weak convergence in 𝐃𝐑𝑛[0,∞)πœ‰πœ€π‘‘βŸΉπœ‰0𝑑.(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(π‘π‘›Γ—Ξ˜), initial conditions for the limit, and prelimit processes are equal.

Theorem is proved.

4. Description of MNRE

We study the same particle in 𝐑𝑛, but its velocity is equal to 𝑣(πœƒ)=𝑐(πœƒ)πœ€βˆ’1+𝑐1(πœƒ), where πœ€β†’0(πœ€>0) is the small parameter, the functions 𝑐(πœƒ),𝑐1(πœƒ) are bounded.

Definition 4.1. Markov nonsymmetrical random evolution (MNRE) is the process Μƒπœ‰πœ€π‘‘βˆˆπ‘π‘›, given by Μƒπœ‰πœ€π‘‘ξ€œβˆΆ=π‘₯+𝑑0π‘£ξ€·πœƒπœ€πœξ€Έπ‘ ξ€·πœƒπœ€πœξ€Έπ‘‘πœ.(4.1)
Our aim is to prove the weak convergence of MNRE to a diffusion process with drift when πœ€β†’0.
Two-component Markov process (Μƒπœ‰πœ€π‘‘,πœƒπœ€π‘‘) at the test functions πœ‘(π‘₯1,…,π‘₯𝑛;πœƒ)∈𝐢∞0(π‘π‘›Γ—Ξ˜) is defined by a generator (see, e.g., [2]) πΏπœ€πœ‘ξ€·π‘₯1,…,π‘₯𝑛π‘₯;πœƒβˆΆ=πœ†π‘„πœ‘(β‹…;πœƒ)+𝑣(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;β‹…=πœ€βˆ’2π‘„πœ‘(β‹…;πœƒ)+πœ€βˆ’1ξ€·π‘₯𝑐(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;β‹…+𝑐1ξ€·π‘₯(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛,;β‹…(4.2) where 𝑆π‘₯(πœƒ)πœ‘1,…,π‘₯𝑛π‘₯;β‹…βˆΆ=βˆ’(𝑠(πœƒ),βˆ‡)πœ‘1,…,π‘₯𝑛;β‹….(4.3)
An important condition that let us confirm weak convergence is balance condition Π𝑐(πœƒ)𝑆(πœƒ)=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(πœƒ)≑0. In case of MNRE the absence of symmetry of the process is caused by the following condition: Π𝑐1(πœƒ)𝑆(πœƒ)=(𝑑,βˆ‡)β‰ 0.(4.5)

Example 4.3. Condition (4.4) can be satisfied for different functions 𝑐(πœƒ). Namely, in case of MSRE 𝑐(πœƒ)=𝑐=const. Then every term under the integral contains either βˆ«πœ‹0sinπ‘›πœƒcosπœƒπ‘‘πœƒ=0 or ∫02πœ‹sinπœƒπ‘‘πœƒ=0.
Another example is the function 𝑐(πœƒ)=sinπœƒ1. 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(πœƒ). For example, in 𝐑2 for 𝑐1(πœƒ)=sinπœƒ we obtain 1ξ€œ2πœ‹02πœ‹ξ‚Έπœ•sinπœƒcosπœƒπœ•π‘₯1πœ•+sinπœƒπœ•π‘₯2ξ‚Ή1π‘‘πœƒ=2πœ•πœ•π‘₯2.(4.6)
Another example is the following function in 𝐑𝑛: 𝑐1𝑐(πœƒ)=1,πœƒπ‘›βˆ’1∈[πœ‹,2πœ‹),0,πœƒπ‘›βˆ’1∈[0,πœ‹).(4.7) Again, all terms under the integral, except the last one, contain βˆ«πœ‹0sinπ‘›πœƒcosπœƒπ‘‘πœƒ=0, thus only one term is not trivial as 1𝑁𝑐1ξ€œπœ‹0β‹―ξ€œπœ‹0ξ€œπœ‹2πœ‹sinπ‘›βˆ’1πœƒ1sinπ‘›βˆ’2πœƒ2β‹―sin2πœƒπ‘›βˆ’1sinπœƒπ‘›βˆ’1π‘‘πœƒ1β‹―π‘‘πœƒπ‘›βˆ’1.(4.8)
By simple calculations, we see Π𝑐1⎧βŽͺ⎨βŽͺβŽ©βˆ’π‘(πœƒ)𝑆(πœƒ)=123β‹…5β‹…β‹―β‹…(π‘›βˆ’2)πœ•2β‹…4β‹…β‹―β‹…(π‘›βˆ’1)πœ•π‘₯π‘›βˆ’π‘,𝑛=2π‘š+1,1πœ‹[]πœ•1β‹…2β‹…β‹―(π‘›βˆ’2)πœ•π‘₯𝑛,𝑛=2π‘š.(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 Μƒπœ‰πœ€π‘‘, converges weakly to the process Μƒπœ‰0𝑑 when πœ€β†’0 as Μƒπœ‰πœ€π‘‘βŸΉΜƒπœ‰0𝑑.(5.1)
The limit process Μƒπœ‰0π‘‘βˆˆπ‘π‘› is defined by a generator 𝐿0πœ‘ξ€·π‘₯1,…,π‘₯𝑛π‘₯=(𝑑,βˆ‡)πœ‘1,…,π‘₯𝑛+ξ€·πœŽ2ξ€Έπœ‘ξ€·π‘₯,Ξ”1,…,π‘₯𝑛,(5.2) where Ξ”πœ‘(π‘₯1,…,π‘₯𝑛)∢=((πœ•2/πœ•π‘₯21)+β‹―+(πœ•2/πœ•π‘₯2𝑛))πœ‘(π‘₯1,…,π‘₯𝑛), (1𝑑,βˆ‡)∢=βˆ’π‘ξ€œπ‘†π‘›π‘1(ξ€·πœŽπœƒ)(𝑠(πœƒ),βˆ‡)πœ‡(π‘‘πœƒ),2ξ€Έ1,Ξ”βˆΆ=π‘ξ€œπ‘†π‘›π‘2(πœƒ)(𝑠(πœƒ),βˆ‡)2πœ‡(π‘‘πœƒ).(5.3)
We need the following Lemma to prove the Theorem.

Lemma 5.2. At the perturbed test functions πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛π‘₯;πœƒ=πœ‘1,…,π‘₯𝑛+πœ€πœ‘1ξ€·π‘₯1,…,π‘₯𝑛;πœƒ+πœ€2πœ‘2ξ€·π‘₯1,…,π‘₯𝑛,;πœƒ(5.4) having bounded derivatives of any degree and compact support, the operator πΏπœ€ has the following asymptotic representation: πΏπœ€πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛;πœƒ=𝐿0πœ‘ξ€·π‘₯1,…,π‘₯𝑛+π‘…πœ€||𝑅(πœƒ)πœ‘(π‘₯),πœ€||(πœƒ)πœ‘(π‘₯)⟢0,πœ€βŸΆ0,πœ‘(π‘₯)∈𝐢∞0(𝐑𝑛),(5.5) where 𝐿0 is defined in (5.2), πœ‘1(π‘₯1,…,π‘₯𝑛;πœƒ),πœ‘2(π‘₯1,…,π‘₯𝑛;πœƒ), and π‘…πœ€(πœƒ)πœ‘(π‘₯) are defined by the following correlations: 𝐿0Ξ =βˆ’Ξ π‘(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)Ξ +Π𝑐1πœ‘(πœƒ)𝑆(πœƒ)Ξ πœ‘,1=βˆ’π‘…0πœ‘π‘(πœƒ)𝑆(πœƒ)πœ‘,2=𝑅0𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑅𝑐(πœƒ)𝑆(πœƒ)πœ‘,πœ€ξ€½πœ€ξ€Ί(πœƒ)πœ‘=𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)+𝑐1(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)+πœ€2𝑐1(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)πœ‘.(5.6)

Proof. We solve singular perturbation problem for the operator (4.2). Using the view of test-functions (5.4), we have πΏπœ€πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛=ξ€Ίπœ€;πœƒβˆ’2𝑄+πœ€βˆ’1𝑐(πœƒ)𝑆(πœƒ)+𝑐1(πœƒ)𝑆(πœƒ)ξ€»ξ€Ίπœ‘+πœ€πœ‘1+πœ€2πœ‘2ξ€»=πœ€βˆ’2π‘„πœ‘+πœ€βˆ’1ξ€Ίπ‘„πœ‘1ξ€»+ξ€Ί+𝑐(πœƒ)𝑆(πœƒ)πœ‘π‘„πœ‘2+𝑐(πœƒ)𝑆(πœƒ)πœ‘1+𝑐1ξ€»ξ€Ί(πœƒ)𝑆(πœƒ)πœ‘+πœ€π‘(πœƒ)𝑆(πœƒ)πœ‘2+𝑐1(πœƒ)𝑆(πœƒ)πœ‘1ξ€»+πœ€2𝑐1(πœƒ)𝑆(πœƒ)πœ‘2.(5.7)
Thus, we obtain the following equations: π‘„πœ‘=0,π‘„πœ‘1𝐿+𝑐(πœƒ)𝑆(πœƒ)πœ‘=0,0πœ‘=π‘„πœ‘2+𝑐(πœƒ)𝑆(πœƒ)πœ‘1+𝑐1(π‘…πœƒ)𝑆(πœƒ)πœ‘,πœ€πœ‘ξ€Ίπ‘(πœƒ)=πœ€(πœƒ)𝑆(πœƒ)πœ‘2+𝑐1(πœƒ)𝑆(πœƒ)πœ‘1ξ€»+πœ€2𝑐1(πœƒ)𝑆(πœƒ)πœ‘2.(5.8)
According to the first one, πœ‘(π‘₯1,…,π‘₯𝑛) belongs to the null space of 𝑄. From the balance condition (4.4) we see that 𝑐(πœƒ)𝑆(πœƒ)πœ‘ belongs to the range of 𝑄, thus from the second equation of (5.8) we have πœ‘1=βˆ’π‘…0𝑐(πœƒ)𝑆(πœƒ)πœ‘.(5.9)
By substitution into the third equation and using the solvability condition, we obtain: 𝐿0Ξ πœ‘+Π𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)Ξ πœ‘βˆ’Ξ π‘1(πœƒ)𝑆(πœƒ)Ξ πœ‘=0.(5.10)
From the last equation, we have π‘…πœ€ξ€½πœ€ξ€Ί(πœƒ)πœ‘(π‘₯)=𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)+𝑐1(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)+πœ€2𝑐1(πœƒ)𝑆(πœƒ)𝑅0𝑐(πœƒ)𝑆(πœƒ)𝑅0π‘ξ€Ύπœ‘(πœƒ)𝑆(πœƒ)(π‘₯)⟢0whenπœ€βŸΆ0,πœ‘(π‘₯)∈𝐢∞0(𝐑𝑛).(5.11)
We calculate the operator 𝐿0 by the formula (5.6) as 𝐿0πœ‘=Π𝑐(πœƒ)𝑆(πœƒ)(πΌβˆ’Ξ )𝑐(πœƒ)𝑆(πœƒ)Ξ πœ‘+Π𝑐1(πœƒ)𝑆(πœƒ)Ξ πœ‘=Π𝑐2(πœƒ)𝑆2(πœƒ)Ξ πœ‘βˆ’Ξ π‘(πœƒ)𝑆(πœƒ)Π𝑐(πœƒ)𝑆(πœƒ)Ξ πœ‘+Π𝑐1(πœƒ)𝑆(πœƒ)Ξ πœ‘.(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(πœƒ)𝑆2(πœƒ)+Π𝑐1(πœƒ)𝑆(πœƒ).(5.13)
Using the view of 𝑆(πœƒ), we may write Π𝑐1(1πœƒ)𝑆(πœƒ)=βˆ’π‘ξ€œπ‘†π‘›π‘1(πœƒ)(𝑠(πœƒ),βˆ‡)πœ‡(π‘‘πœƒ)=∢(𝑑,βˆ‡),Π𝑐2(πœƒ)𝑆21(πœƒ)=π‘ξ€œπ‘†π‘›π‘2(πœƒ)(𝑠(πœƒ),βˆ‡)2ξ€·πœŽπœ‡(π‘‘πœƒ)=∢2ξ€Έ.,Ξ”(5.14)
Lemma is proved.

Proof of Theorem 5.1. We proved in Lemma 5.2 that πΏπœ€πœ‘πœ€β‡’πΏ0πœ‘ at the class of test functions 𝐢∞0(π‘π‘›Γ—Ξ˜), when πœ€β†’0. To prove the weak convergence, we should show relative compactness of the family of processes (Μƒπœ‰πœ€π‘‘,πœƒπœ€π‘‘) in πƒπ‘π‘›Γ—Ξ˜[0,∞). To do this, we use Lemma 3.4.
We have the following view of the operator (4.2) at the test function πœ‘πœ€(π‘₯1,…,π‘₯𝑛;πœƒ)=πœ‘(π‘₯1,…,π‘₯𝑛)+πœ€πœ‘1(π‘₯1,…,π‘₯𝑛;πœƒ), where πœ‘1(π‘₯1,…,π‘₯𝑛;πœƒ)=βˆ’π‘…0𝑐(πœƒ)𝑆(πœƒ)πœ‘(π‘₯1,…,π‘₯𝑛): πΏπœ€πœ‘πœ€ξ€·π‘₯1,…,π‘₯𝑛;πœƒ=πœ€βˆ’2ξ€·π‘₯π‘„πœ‘1,…,π‘₯𝑛+πœ€βˆ’1ξ€Ίπ‘„πœ‘1ξ€·π‘₯+𝑐(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛+𝑐(πœƒ)𝑆(πœƒ)πœ‘1ξ€·π‘₯1,…,π‘₯𝑛;πœƒ+𝑐1ξ€·π‘₯(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;πœƒξ€Έξ€»+πœ€π‘1(πœƒ)𝑆(πœƒ)πœ‘1ξ€·π‘₯1,…,π‘₯𝑛.;πœƒ(5.15)
It follows from (5.8) that the first two terms equal to 0. Let us estimate the following third term: 𝑐(πœƒ)𝑆(πœƒ)πœ‘1ξ€·π‘₯1,…,π‘₯𝑛;πœƒ+𝑐1ξ€·π‘₯(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;πœƒ=𝑐(πœƒ)𝑆(πœƒ)𝑅0ξ€·π‘₯𝑐(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛+𝑐1(ξ€·π‘₯πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;πœƒ=𝑐2(πœƒ)𝑆2ξ€·π‘₯(πœƒ)πœ‘1,…,π‘₯𝑛+𝑐1ξ€·π‘₯(πœƒ)𝑆(πœƒ)πœ‘1,…,π‘₯𝑛;πœƒ=𝑐2(πœƒ)cos2πœƒ1πœ•2πœ•π‘₯21+sin2πœƒ1cos2πœƒ2πœ•2πœ•π‘₯22+β‹―+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2cos2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2π‘›βˆ’1+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2sin2πœƒπ‘›βˆ’1πœ•2πœ•π‘₯2𝑛+ξ‚»sinπœƒ1cosπœƒ1cosπœƒ2πœ•2πœ•π‘₯1πœ•π‘₯2+β‹―+sin2πœƒ1β‹―sin2πœƒπ‘›βˆ’2sinπœƒπ‘›βˆ’1Γ—cosπœƒπ‘›βˆ’1πœ•2πœ•π‘₯π‘›βˆ’1πœ•π‘₯π‘›πœ‘ξ€·π‘₯ξ‚Όξ‚Ή1,…,π‘₯𝑛+𝑐1ξ‚Έ(πœƒ)cosπœƒ1πœ•πœ•π‘₯1+sinπœƒ1cosπœƒ2πœ•πœ•π‘₯2+β‹―+sinπœƒ1β‹―sinπœƒπ‘›βˆ’2cosπœƒπ‘›βˆ’1πœ•πœ•π‘₯π‘›βˆ’1+sinπœƒ1β‹―sinπœƒπ‘›βˆ’2sinπœƒπ‘›βˆ’1πœ•πœ•π‘₯π‘›ξ‚Ήπœ‘ξ€·π‘₯1,…,π‘₯𝑛≀𝐢1,πœ‘,(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 πΏπœ€πœ‘πœ€=πΏπœ€πœ‘+πœ€πΏπœ€πœ‘1=πΏπœ€πœ‘+πœ€π‘πΏπœ€π‘…0πœƒπ‘†(πœƒ)πœ‘.(5.17)
Thus, πΏπœ€πœ‘=πΏπœ€πœ‘πœ€βˆ’πœ€π‘πΏπœ€π‘…0𝑆(πœƒ)πœ‘β‰€πΆ1,πœ‘βˆ’πœ€πΆ2,πœ‘<πΆπœ‘,(5.18) for πœ€ that is small enough.
To prove the second condition of Lemma 3.4, we use the following properties of the function πœ‘0√(𝑒)=1+𝑒2: ||πœ‘ξ…ž0||(𝑒)≀1β‰€πœ‘0||πœ‘(𝑒),0ξ…žξ…ž||(𝑒)β‰€πœ‘0(𝑒).(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 (Μƒπœ‰πœ€π‘‘,πœƒπœ€π‘‘) is relatively compact in πƒπ‘π‘›Γ—Ξ˜[0,∞).
Using Theorem 3.5, we confirm the following weak convergence in 𝐃𝐑𝑛[0,∞): Μƒπœ‰πœ€π‘‘βŸΉΜƒπœ‰0𝑑.(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 𝐢∞0(π‘π‘›Γ—Ξ˜) 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(πœƒ)=1,πœƒ=0,1,πœƒ=πœ‹,1πœ‹(πœƒ)=1,πœƒ=2.(5.21)
In other cases both functions equal to 0.
The balance condition (4.4) is true for 𝑐(πœƒ), on the contrary, condition (4.5) is true for 𝑐1(πœƒ).
The limit generator (5.2) has the view 𝐿0πœ‘ξ€·π‘₯1,π‘₯2ξ€Έ=1πœ•2πœ‹πœ•π‘₯2πœ‘ξ€·π‘₯1,π‘₯2ξ€Έ+1πœ‹πœ•2πœ•π‘₯21πœ‘ξ€·π‘₯1,π‘₯2ξ€Έ.(5.22)
Thus, the limit process has two parts: the drift with velocity 1/2πœ‹ in direction of π‘₯2 coordinate and diffusion part in one-dimensional subspace, corresponding to π‘₯1 coordinate that is similar to the limit process described in Kac model [1].

List of Symbols

𝑐,𝑐(πœƒ),𝑐1(πœƒ):the components of velocity of random evolution’s motion
𝐢∞0(π‘π‘›Γ—Ξ˜):the space of bounded continuous functions on π‘π‘›Γ—Ξ˜ having continuous derivatives of all orders vanishing at infinity
πΆπœ‘:constant depending on the norm of function πœ‘
𝑑:parameter of drift
πƒπ‘π‘›Γ—Ξ˜[0,∞):the space of π‘π‘›Γ—Ξ˜-valued right continuous functions having left limits (cadlag), defined on 𝐑+
πΏπœ€: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
π‘…πœ€(πœƒ):negligible term
𝑆𝑛:unit 𝑛-dimensional sphere
𝑆(πœƒ):operator that describes particle’s motion
𝑠(πœƒ):vectors that define possible directions of random evolution’s motion
𝑣,𝑣(πœƒ):velocity of random evolution’s motion
πœ€:small parameter
πœƒ=(πœƒπ‘–,𝑖=1,π‘›βˆ’1),β€‰β€‰πœƒπ‘›βˆ’1∈[0,2πœ‹),β€‰β€‰πœƒπ‘–βˆˆ[0,πœ‹),  𝑖=1,π‘›βˆ’2:angles that define directions in 𝐑𝑛
Θ:the set of all angles πœƒ
πœƒπœ€π‘‘:Poisson process that switches directions of random evolution’s motion
πœ†:intensity of switching Poisson process
πœ‡(π‘‘πœƒ):small element of volume of 𝑆𝑛
πœ‰πœ€π‘‘,Μƒπœ‰πœ€π‘‘:Markov random evolutions
πœ‰0𝑑,Μƒπœ‰0𝑑:limit processes
Ξ :projector at the nullspace of generator 𝑄
𝜎2:parameter of diffusion.

References

  1. M. Kac, β€œA stochastic model related to the telegrapher's equation,” The Rocky Mountain Journal of Mathematics, vol. 4, pp. 497–509, 1974. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. M. A. Pinsky, Lectures on Random Evolution, World Scientific Publishing, Singapore, 1991.
  3. E. Orsingher, β€œBessel functions of third order and the distribution of cyclic planar motions with three directions,” Stochastics and Stochastics Reports, vol. 74, no. 3-4, pp. 617–631, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. E. Orsingher and A. M. Sommella, β€œA cyclic random motion in R3 with four directions and finite velocity,” Stochastics and Stochastics Reports, vol. 76, no. 2, pp. 113–133, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. A. D. Kolesnik and A. F. Turbin, β€œInfinitesimal hyperbolic operator of Markovian random evolutions,” Doklady Akademii Nauk Ukrainskoj, vol. 1, pp. 11–14, 1991 (Russian). View at Google Scholar
  6. A. D. Kolesnik and A. F. Turbin, β€œSymmetric random evolutions in R2,” Doklady Akademii Nauk Ukrainskoj, vol. 2, pp. 10–11, 1990 (Russian). View at Google Scholar Β· View at Zentralblatt MATH
  7. I. V. Samoilenko, β€œMarkovian random evolution in Rn,” Random Operators and Stochastic Equations, vol. 9, no. 2, pp. 139–160, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  8. A. D. Kolesnik, β€œWeak convergence of the distributions of Markovian random evolutions in two and three dimensions,” Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3, pp. 41–52, 2003. View at Google Scholar
  9. A. D. Kolesnik, β€œWeak convergence of a planar random evolution to the Wiener process,” Journal of Theoretical Probability, vol. 14, no. 2, pp. 485–494, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks, vol. 118 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 2008. View at Publisher Β· View at Google Scholar
  11. K. Borovkov, β€œOn random walks with jumps scaled by cumulative sums of random variables,” Statistics & Probability Letters, vol. 35, no. 4, pp. 409–416, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. S. Foss, T. Konstantopoulos, and S. Zachary, β€œDiscrete and continuous time modulated random walks with heavy-tailed increments,” Journal of Theoretical Probability, vol. 20, no. 3, pp. 581–612, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. S. Zakhari and S. G. Foss, β€œOn the exact asymptotics of the maximum of a random walk with increments in a class of light-tailed distributions,” Siberian Mathematical Journal, vol. 47, no. 6, pp. 1034–1041, 2006. View at Publisher Β· View at Google Scholar
  14. V. V. Anisimov, Switching Processes in Queueing Models, Applied Stochastic Methods Series, ISTE, London, UK, John Wiley & Sons, Hoboken, NJ, USA, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. V. S. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific Publishing, Singapore, 2005. View at Publisher Β· View at Google Scholar
  16. V. V. Anīsīmov, β€œConvergence of accumulation processes with switchings,” Teorīya Imovīrnosteĭ ta Matematichna Statistika, no. 63, pp. 3–12, 2000, translation in Theory of Probability and Mathematical Statistics no. 63, 1–11, 2001-2002. View at Google Scholar
  17. I. V. Samoilenko, β€œConvergence of an impulsive storage process with jump switchings,” Ukraïni Matematicni Zhurnal, vol. 60, no. 9, pp. 1282–1286, 2008, translation in Ukrainian Mathematical Journal, vol. 60, no. 9, 1492–1497, 2008. View at Publisher Β· View at Google Scholar
  18. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1986. View at Publisher Β· View at Google Scholar
  19. D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, vol. 233 of Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1979.