TSWJ The Scientific World Journal 1537-744X 2356-6140 Hindawi Publishing Corporation 10.1155/2014/832861 832861 Research Article On Poisson Nonlinear Transformations http://orcid.org/0000-0001-6041-7973 Ganikhodjaev Nasir http://orcid.org/0000-0001-5160-5534 Hamzah Nur Zatul Akmar Altun Ishak 1 Department of Computational and Theoretical Sciences Faculty of Science International Islamic University Malaysia, 25710 Kuantan Malaysia iium.edu.my 2014 1772014 2014 01 04 2014 04 07 2014 17 7 2014 2014 Copyright © 2014 Nasir Ganikhodjaev and Nur Zatul Akmar Hamzah. 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.

We construct the family of Poisson nonlinear transformations defined on the countable sample space of nonnegative integers and investigate their trajectory behavior. We have proved that these nonlinear transformations are regular.

1. Introduction

Let (X,F) be a measurable space, where X is a state space and F is σ-algebra on X, and S(X,F) the set of all probability measures on (X,F).

Let {P(x,y,A):x,yX,AF} be a family of functions on X×X×F such that, for any fixed x,yXP(x,y,·)S(X,F), P(x,y,A) regarded as a function of two variables x and y with fixed AF is a measurable function on (X×X,FF) and P(x,y,A)=P(y,x,A) for any x,yX and AF.

We consider a nonlinear transformation called quadratic stochastic operator (qso) V:S(X,F)S(X,F) which is defined by (1)(Vλ)(A)=XP(x,y,A)dλ(x)dλ(y), where AF is an arbitrary measurable set.

If a state space X={1,2,,m} is a finite set and the corresponding σ-algebra is the power set P(X), that is, the set of all subsets of X, then the set of all probability measures on (X,F) has the following form: (2)Sm-1={i=1mx=(x1,x2,,xm)Rm:xi0foranyi,andi=1mxi=1} that is called a (m-1)-dimensional simplex.

In this case, the probabilistic measure P(i,j,·) is a discrete measure with k=1mP(ij,{k})=1, where P(ij,{k})Pij,k for any i,jX. In addition, the corresponding qso V has the following form: (3)(Vx)k=i,j=1mPij,kxixj, for any xSm-1 and the coefficients Pij,k satisfy the following conditions: (4)(a)Pij,k0;(b)Pij,k=Pji,k;(c)k=1mPij,k=1i,j,k{1,2,,m}. Such operator can be reinterpreted in terms of evolutionary operator of free population  and in this form it has a fair history.

In this paper, we consider nonlinear transformations defined on countable state space and investigate their limit behavior of trajectories.

2. A Poisson qso

Let X={0,1,} be a countable sample space and corresponding σ-algebra F a power set P(X), that is, the set of all subsets of X. In order to define a probability measure μ on countable sample space X, it is enough to define the measure μ({k}) of each singleton {k},k=0,1,. Thus, we will write μ(k) instead of μ({k}).

Let {P(i,j,k):i,j,kX} be a family of functions defined on X×X×F, which satisfy the following conditions:

P(i,j,·) is a probability measure on (X,F) for any fixed i,jX;

P(i,j,k)=P(j,i,k)Pij,k, where kX for any fixed i,jX.

In this case, a qso (1) on measurable space (X,F) is defined as follows: (5)Vμ(k)=i=0j=0Pij,kμ(i)μ(j), where kX for arbitrary measure μS(X,F).

In this paper, we consider a Poisson qso which is a Poisson distribution Pλ with a positive real parameter λ defined on X by the equation (6)Pλ(k)=e-λλkk!, for any kX.

Let S(X,F) be a set of all probability measures on (X,F) and let P(i,j,·) be a probability measure on (X,F) for any i,jX.

Definition 1.

A quadratic stochastic operator V (5) is called a Poisson qso if, for any i,jX, the probability measure P(i,j,·) is the Poisson distribution Pλ(i,j) with positive real parameters λ(i,j), where λ(i,j)=λ(j,i).

Assume that {Vnλ:n=0,1,2,} is the trajectory of the initial point λS(X,F), where Vn+1λ=V(Vnλ) for all n=0,1,2,, with V0λ=λ.

In this paper, we will study limit behavior of trajectories of Poisson qso.

3. Ergodicity and Regularity of qso

Let us consider a qso V (5) defined on countable set X. Let {Vnλ:n=0,1,2,} be the trajectory of the initial point λS(X,F), where Vn+1λ=V(Vnλ) for all n=0,1,2,.

Definition 2.

A measure μS(X,F) is called a fixed point of a qso V if Vμ=μ.

Let Fix(V) be the set of all fixed points of qso V.

Definition 3.

A qso V is called regular if, for any initial point μS(X,F), the limit (7)limnVn(μ) exists.

In measure theory, there are various notions of the convergence of measures: weak convergence, strong convergence, and total variation convergence. Below we consider strong convergence.

Definition 4.

For (X,F) a measurable space, a sequence μn is said to converge strongly to a limit μ if (8)limnμn(A)=μ(A), for every set AF.

If X is a countable set, then a sequence μn converges strongly to a limit μ if and only if (9)limnμn(k)=μ(k), for every singleton kX.

In statistical mechanics, the ergodic hypothesis proposes a connection between dynamics and statistics. In the classical theory, the assumption was made that the average time spent in any region of phase space is proportional to the volume of the region in terms of the invariant measure. More generally, such time averages may be replaced by space averages.

For nonlinear dynamical systems, Ulam  suggested an analogous measure-theoretic ergodicity with following ergodic hypothesis.

Definition 5.

A nonlinear operator V defined on S(X,F) is called ergodic, if the limit (10)limn1nk=0n-1Vkλ exists for any λS(X,F).

On the ground of numerical calculations for quadratic stochastic operators defined on S(X,F) with finite X, Ulam  conjectured that the ergodic theorem holds for any such qso V.

In 1977, Zakharevich  proved that this conjecture is false in general. He considered the following operator on S2: (11)x1=x12+2x1x2,x2=x22+2x2x3,x3=x32+2x1x3, and he proved that such operator is nonergodic transformation. Later in , the sufficient condition to be nonergodic transformation was established for qso defined on S2.

In the next section, we will show that Ulam’s conjecture is true for some class of Poisson qso.

4. Ergodicity and Regularity of Poisson qso

Let V defined in (5) be a Poisson qso. We consider the following cases.

4.1. Homogenious Poisson qso

We call a Poisson quadratic stochastic operator V (5) homogenious, if λ(i,j)=λ, for any i,jX, that is, Pij,k=e-λ(λk/k!). Then for arbitrary measure μS(X,F)(12)Vμ(k)=i=0j=0Pij,kμ(i)μ(j)=e-λλkk!, where kX, that is, Vμ=Pλ.

Thus Vnμ=Pλ for any n=1,2,, that is, Fix(V)=Pλ, and we have the following statement.

Proposition 6.

A homogenious Poisson qso is a regular transformation.

4.2. Poisson qso with Two Different Parameters

We consider a Poisson qso such that (13)Pij,k={e-λ1λ1kk!if  i+jis  even,e-λ2λ2kk!if  i+j  is  odd.

For any initial measure μS(X,F) let (14)A(μ)=n=0μ(2n),B(μ)=n=0μ(2n+1), where A(μ)+B(μ)=1. It is easy to show that for Poisson distribution Pλ(15)A(Pλ)=1+e-2λ2,B(Pλ)=1-e-2λ2. Then for any measure μS(X,F), we have (16)Vμ(k)=i=0j=0Pij,kμ(i)μ(j)=m,n=0[P2m,2n,kμ(2m)μ(2n)+P2m+1,2n+1,kμ(2m+1)μ(2n+1)]+m,n=0[P2m+1,2n,kμ(2m+1)μ(2n)+P2m,2n+1,kμ(2m)μ(2n+1)]=e-λ1λ1kk![A2(μ)+B2(μ)]+e-λ2λ2kk![2A(μ)B(μ)],V2μ(k)=i=0j=0Pij,kVμ(i)Vμ(j)=m,n=0[P2m,2n,kVμ(2m)Vμ(2n)+P2m+1,2n+1,kVμ(2m+1)Vμ(2n+1)]+m,n=0[P2m+1,2n,kVμ(2m+1)Vμ(2n)+P2m,2n+1,kVμ(2m)Vμ(2n+1)]=e-λ1λ1kk![A2(Vμ)+B2(Vμ)]+e-λ2λ2kk![2A(Vμ)B(Vμ)].

By simple calculations, we have (17)A(Vμ)=1+e-2λ12[A2(μ)+B2(μ)]hhhhhhhh+1+e-2λ22[2A(μ)B(μ)],B(Vμ)=1-e-2λ12[A2(μ)+B2(μ)]hhhhhhhh+1-e-2λ22[2A(μ)B(μ)].

Thus, by using induction on the sequence Vn(μ), we produce the following recurrent equation: (18)Vn+1μ(k)=e-λ1λ1kk![A2(Vnμ)+B2(Vnμ)]+e-λ2λ2kk![2A(Vnμ)B(Vnμ)], where n=0,1,, Besides, for parameters A(Vnμ) and B(Vnμ), we have the following recurrent equations: (19)A(Vn+1μ)=1+e-2λ12[A2(Vnμ)+B2(Vnμ)]hhhhhhhhhh+1+e-2λ22[2A(Vnμ)B(Vnμ)],B(Vn+1μ)=1-e-2λ12[A2(Vnμ)+B2(Vnμ)]hhhhhhhhhh+1-e-2λ22[2A(Vnμ)B(Vnμ)].

It is obvious that the limit behavior of the recurrent equation (18) is fully determined by limit behavior of recurrent equations (19).

Since A(Vnμ)+B(Vnμ)=1, where A(Vnμ)0 and B(Vnμ)0, the recurrent equations (19) are rewritten as follows: (20)x=A(λ1)(x2+y2)+2A(λ2)xy,y=B(λ1)(x2+y2)+2B(λ2)xy with x0, y0, and x+y=1.

Solving the following quadratic equation (21)x=A(λ1)(x2+(1-x)2)+2A(λ2)x(1-x), we have single fixed point and denoted it as (x*,y*) (see Figure 1). Using simple calculus (see Figure 1), one can show that any trajectory of the qso (20) defined on one-dimensional simplex S1 converges to this fixed point; that is, qso (20) is regular transformation, so that it is ergodic.

Graph of the function (21) for some fixed values λ1 and λ2.

When λ1=0.8 and λ2=0.4

When λ1=0.4 and λ2=1.4

Thus, for any initial measure μ, we have (22)limnA(Vnμ)=x*,limnB(Vnμ)=y*. Then, passing to limit in (18), for any singleton k, we have (23)limnVn+1μ(k)=limn{e-λ1λ1kk![A2(Vnμ)+B2(Vnμ)]+e-λ2λ2kk![2A(Vnμ)B(Vnμ)]}=e-λ1λ1kk![x*2+y*2]+e-λ2λ2kk![2x*y*]=[x*2+y*2]Pλ1(k)+[2x*y*]Pλ2(k).

Thus, for any initial measure μ, the strong limit of the sequence Vnμ exists and is equal to the convex linear combination (24)limnVnμ(k)=(x*2+y*2)Pλ1(k)+2x*y*Pλ2(k), of two Poisson measures Pλ1 and Pλ2. It is evident that Fix(V)=(x*2+y*2)Pλ1(k)+2x*y*Pλ2(k).

As corollary we have following statement.

Proposition 7.

A Poisson qso with two different parameters is a regular and, respectively, ergodic transformation with respect to strong convergence.

4.3. A Poisson qso with Three Different Parameters

We consider a Poisson qso such that (25)Pij,k={e-λ1λ1kk!ifi+j=0(mod3),e-λ2λ2kk!ifi+j=1(mod3),e-λ3λ3kk!ifi+j=2(mod3). For any initial measure μS(X,F), let (26)A(μ)=n=0μ(3n),B(μ)=n=0μ(3n+1),C(μ)=n=0μ(3n+2), where A(μ)+B(μ)+C(μ)=1. It is easy to show that, for Poisson distribution Pλ with parameter λ, we have (27)A(λ)=1+2e-(3/2)λcos(3/2)λ3,B(λ)=1-2e-(3/2)λcos((3/2)λ+π/3)3,C(λ)=1-2e-(3/2)λcos((3/2)λ-π/3)3. Then, for any measure μS(X,F), we have (28)Vμ(k)=i=0j=0Pij,kμ(i)μ(j)=m,n=0[P3m,3n,kμ(3m)μ(3n)+P3m+1,3n+2,kμ(3m+1)μ(3n+2)+P3m+2,3n+1,kμ(3m+2)μ(3n+1)]+m,n=0[P3m+1,3n,kμ(3m+1)μ(3n)+P3m,3n+1,kμ(3m)μ(3n+1)+P3m+2,3n+1,kμ(3m+2)μ(3n+1)]+m,n=0[P3m+2,3n,kμ(3m+2)μ(3n)+P3m,3n+2,kμ(3m)μ(3n+2)+P3m+1,3n+1,kμ(3m+1)μ(3n+1)]=e-λ1λ1kk![A2(μ)+2B(μ)C(μ)]+e-λ2λ2kk![2A(μ)B(μ)+C2(μ)]+e-λ3λ3kk![2A(μ)C(μ)+B2(μ)],V2μ(k)=i=0j=0Pij,kVμ(i)Vμ(j)=m,n=0[P3m,3n,kVμ(3m)Vμ(3n)+P3m+1,3n+2,kVμ(3m+1)Vμ(3n+2)+P3m+2,3n+1,kVμ(3m+2)Vμ(3n+1)]+m,n=0[P3m+1,3n,kVμ(3m+1)Vμ(3n)+P3m,3n+1,kVμ(3m)Vμ(3n+1)+P3m+2,3n+2,kVμ(3m+2)Vμ(3n+2)]+m,n=0[P3m+2,3n,kVμ(3m+2)Vμ(3n)+P3m,3n+2,kVμ(3m)Vμ(3n+2)+P3m+1,3n+1,kVμ(3m+1)Vμ(3n+1)]=e-λ1λ1kk![A2(Vμ)+B(Vμ)C(Vμ)]+e-λ2λ2kk![2A(Vμ)B(Vμ)+C2(Vμ)]+e-λ3λ3kk![2A(Vμ)C(Vμ)+B2(Vμ)]. By simple calculations, we have (29)A(Vμ)=A(λ1)[A2(μ)+2B(μ)C(μ)]+A(λ2)[2A(μ)B(μ)+C2(μ)]+A(λ3)[2A(μ)C(μ)+B2(μ)],B(Vμ)=B(λ1)[A2(μ)+2B(μ)C(μ)]+B(λ2)[2A(μ)B(μ)+C2(μ)]+B(λ3)[2A(μ)C(μ)+B2(μ)],C(Vμ)=C(λ1)[A2(μ)+2B(μ)C(μ)]+C(λ2)[2A(μ)B(μ)+C2(μ)]+C(λ3)[2A(μ)C(μ)+B2(μ)].

Thus, by using induction on sequence Vn(μ), we produce the following recurrent equation: (30)Vn+1μ(k)=e-λ1λ1kk![A2(Vnμ)+2B(Vnμ)C(Vnμ)]+e-λ2λ2kk![2A(Vnμ)B(Vnμ)+C2(Vnμ)]+e-λ3λ3kk![2A(Vnμ)C(Vnμ)+B2(Vnμ)], where n=0,1,. Besides, for parameters A(Vnμ), B(Vnμ) and C(Vnμ), we have the following recurrent equations: (31)A(Vn+1μ)=A(λ1)[A2(Vnμ)+2B(Vnμ)C(Vnμ)]hhhhhhhhhh+A(λ2)[2A(Vnμ)B(Vnμ)+C2(Vnμ)]hhhhhhhhhh+A(λ3)[2A(Vnμ)C(Vnμ)+B2(Vnμ)],B(Vn+1μ)=B(λ1)[A2(Vnμ)+2B(Vnμ)C(Vnμ)]hhhhhhhhhh+B(λ2)[2A(Vnμ)B(Vnμ)+C2(Vnμ)]hhhhhhhhhh+B(λ3)[2A(Vnμ)C(Vnμ)+B2(Vnμ)],C(Vn+1μ)=C(λ1)[A2(Vnμ)+2B(Vnμ)C(Vnμ)]hhhhhhhhhh+C(λ2)[2A(Vnμ)B(Vnμ)+C2(Vnμ)]hhhhhhhhhh+C(λ3)[2A(Vnμ)C(Vnμ)+B2(Vnμ)].

It is obvious that the limit behavior of the recurrent equation (30) is fully determined by limit behavior of recurrent equations (31).

Since A(Vnμ)+B(Vnμ)+C(Vnμ)=1, where A(Vnμ)0, B(Vnμ)0, and C(Vnμ)0, the recurrent equations (31) are rewritten as follows: (32)x=A(λ1)x2+A(λ3)y2+A(λ2)z2+2A(λ2)xy+A(λ3)xz+A(λ1)yz,y=B(λ1)x2+B(λ3)y2+B(λ2)z2+2B(λ2)xy+B(λ3)xz+B(λ1)yz,z=C(λ1)x2+C(λ3)y2+C(λ2)z2+2C(λ2)xy+C(λ3)xz+C(λ1)yz, where x+y+z=1.

Starting from arbitrary initial data, we iterate the recurrence equations (31) and observe their behavior after a large number of iterations. The resultant diagram in the space (λ1,λ2) with 0<λ1, λ22, and some fixed λ3 are shown in Figure 2. In this diagram, blue color corresponds to the converges of the trajectory.

Limit behavior of the dynamical system (31) 0<λ1, λ22 and some fixed values λ3.

Diagram when λ3=0.01

Diagram when λ3=100

One can prove that for any values of parameters λ1, λ2, and λ3 the nonlinear transformation (31) has a single fixed point (x*,y*,z*) and, respectively, it is regular transformation.

If these parameters are very small, for instance, λ1=3·10-15, λ2=2·10-15, and λ3=1·10-15, then any trajectory converges to (1,0,0). But, if they are rather large, for instance, λ1=25, λ2=50, and λ3=75, then any trajectory converges to (1/3,1/3,1/3).

As above, from (31) it follows that for any singleton kX the limit of the sequence Vnμ(k) exists and equals (33)limnVn+1μ(k)=e-λ1λ1kk![x*2+2y*z*]+e-λ2λ2kk![2x*y*+z*2]+e-λ3λ3kk![2x*z*+y*2]. Thus, the strong limit of the sequence Vnμ exists and equals convex linear combination (34)limnVn+1μ=(x*2+2y*z*)Pλ1+(2x*y*+z*2)Pλ2+(2x*z*+y*2)Pλ3, of three Poisson measures Pλ1, Pλ2, and Pλ3. It is evident that Fix(V)=(x*2+2y*z*)Pλ1+(2x*y*+z*2)Pλ2+(2x*z*+y*2)Pλ3.

As corollary we have following statement.

Proposition 8.

A Poisson qso with three different parameters is a regular and, respectively, ergodic transformation with respect to strong convergence.

5. Conclusion

In this paper, we present a construction of Poisson quadratic stochastic operators and prove their regularity when the number of different parameters λi is less than or equal to three. The Poisson quadratic stochastic operators with any finitely many different parameters λi and countably many different parameters λi will be considered in another paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by Ministry of Higher Education Malaysia (MOHE) under Grant FRGS14-116-0357.

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