On Poisson Nonlinear Transformations

Abstract

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,𝑦∈𝑋,𝐴∈ℱ} be a family of functions on 𝑋×𝑋×ℱ such that, for any fixed 𝑥,𝑦∈𝑋  𝑃(𝑥,𝑦,⋅)∈𝑆(𝑋,ℱ), 𝑃(𝑥,𝑦,𝐴) regarded as a function of two variables 𝑥 and 𝑦 with fixed 𝐴∈ℱ is a measurable function on (𝑋×𝑋,ℱ⊗ℱ) and 𝑃(𝑥,𝑦,𝐴)=𝑃(𝑦,𝑥,𝐴) for any 𝑥,𝑦∈𝑋 and 𝐴∈ℱ.

We consider a nonlinear transformation called quadratic stochastic operator (qso) 𝑉∶𝑆(𝑋,ℱ)→𝑆(𝑋,ℱ) which is defined by (𝑉𝜆)(𝐴)=_𝑋𝑃𝑥,𝑦,𝐴𝑑𝜆(𝑥)𝑑𝜆𝑦, (1) where 𝐴∈ℱ 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: 𝑆^𝑚−1=𝑥=𝑥₁,𝑥₂,…,𝑥_𝑚∈𝑅^𝑚∶𝑥_𝑖≥0forany𝑖,and^𝑚_𝑖=1𝑥_𝑖=1 (2) that is called a (𝑚−1)-dimensional simplex.

In this case, the probabilistic measure 𝑃(𝑖,𝑗,⋅) is a discrete measure with ∑^𝑚_𝑘=1𝑃(𝑖𝑗,{𝑘})=1, where 𝑃(𝑖𝑗,{𝑘})≡𝑃_{𝑖𝑗,𝑘} for any 𝑖,𝑗∈𝑋. In addition, the corresponding qso V has the following form: (𝑉𝑥)_𝑘=^𝑚_𝑖,𝑗=1𝑃_{𝑖𝑗,𝑘}𝑥_𝑖𝑥_𝑗, (3) for any 𝑥∈𝑆^𝑚−1 and the coefficients 𝑃_{𝑖𝑗,𝑘} satisfy the following conditions: (a)𝑃_{𝑖𝑗,𝑘}(b)𝑃≥0;_{𝑖𝑗,𝑘}=𝑃_{𝑗𝑖,𝑘};(c)^𝑚_𝑘=1𝑃_{𝑖𝑗,𝑘}=1∀𝑖,𝑗,𝑘∈{1,2,…,𝑚}. (4) Such operator can be reinterpreted in terms of evolutionary operator of free population [1–10] 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,𝑘∈𝑋} be a family of functions defined on 𝑋×𝑋×ℱ, which satisfy the following conditions:(i)𝑃(𝑖,𝑗,⋅) is a probability measure on (𝑋,ℱ) for any fixed 𝑖,𝑗∈𝑋;(ii)𝑃(𝑖,𝑗,𝑘)=𝑃(𝑗,𝑖,𝑘)≡𝑃_{𝑖𝑗,𝑘}, where 𝑘∈𝑋 for any fixed 𝑖,𝑗∈𝑋.

In this case, a qso (1) on measurable space (𝑋,ℱ) is defined as follows: 𝑉𝜇(𝑘)=^∞∞_𝑖=0𝑗=0𝑃_{𝑖𝑗,𝑘}𝜇(𝑖)𝜇𝑗, (5) where 𝑘∈𝑋 for arbitrary measure 𝜇∈𝑆(𝑋,ℱ).

In this paper, we consider a Poisson qso which is a Poisson distribution 𝑃_𝜆 with a positive real parameter 𝜆 defined on 𝑋 by the equation 𝑃_𝜆(𝑘)=𝑒^−𝜆𝜆^𝑘,𝑘! (6) for any 𝑘∈𝑋.

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,𝑗∈𝑋.

Definition 1. A quadratic stochastic operator 𝑉 (5) is called a Poisson qso if, for any 𝑖,𝑗∈𝑋, the probability measure 𝑃(𝑖,𝑗,⋅) is the Poisson distribution 𝑃_{𝜆(𝑖,𝑗)} with positive real parameters 𝜆(𝑖,𝑗), where 𝜆(𝑖,𝑗)=𝜆(𝑗,𝑖).

Assume that {𝑉^𝑛𝜆∶𝑛=0,1,2,…} is the trajectory of the initial point 𝜆∈𝑆(𝑋,𝐹), where 𝑉^𝑛+1𝜆=𝑉(𝑉^𝑛𝜆) 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 {𝑉^𝑛𝜆∶𝑛=0,1,2,…} be the trajectory of the initial point 𝜆∈𝑆(𝑋,ℱ), where 𝑉^𝑛+1𝜆=𝑉(𝑉^𝑛𝜆) for all 𝑛=0,1,2,….

Definition 2. A measure 𝜇∈𝑆(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 𝜇∈𝑆(𝑋,ℱ), the limit lim_𝑛→∞𝑉^𝑛𝜇 (7) 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 𝜇_𝑛 is said to converge strongly to a limit 𝜇 if lim_𝑛→∞𝜇_𝑛(𝐴)=𝜇(𝐴), (8) for every set 𝐴∈ℱ.

If 𝑋 is a countable set, then a sequence 𝜇_𝑛 converges strongly to a limit 𝜇 if and only if lim_𝑛→∞𝜇_𝑛(𝑘)=𝜇(𝑘), (9) for every singleton 𝑘∈𝑋.

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 [11] 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 lim_𝑛→∞1𝑛^𝑛−1_𝑘=0𝑉^𝑘𝜆 (10) exists for any 𝜆∈𝑆(𝑋,𝐹).

On the ground of numerical calculations for quadratic stochastic operators defined on 𝑆(𝑋,𝐹) with finite 𝑋, Ulam [11] conjectured that the ergodic theorem holds for any such qso 𝑉.

In 1977, Zakharevich [12] proved that this conjecture is false in general. He considered the following operator on 𝑆²: 𝑥^′₁=𝑥²₁+2𝑥₁𝑥₂,𝑥^′₂=𝑥²₂+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2𝑥₁𝑥₃, (11) and he proved that such operator is nonergodic transformation. Later in [13], 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,𝑗∈𝑋, that is, 𝑃_{𝑖𝑗,𝑘}=𝑒^−𝜆(𝜆^𝑘/𝑘!). Then for arbitrary measure 𝜇∈𝑆(𝑋,ℱ)𝑉𝜇(𝑘)=^∞∞_𝑖=0𝑗=0𝑃_{𝑖𝑗,𝑘}𝜇(𝑖)𝜇𝑗=𝑒^−𝜆𝜆^𝑘,𝑘! (12) where 𝑘∈𝑋, that is, 𝑉𝜇=𝑃_𝜆.

Thus 𝑉^𝑛𝜇=𝑃_𝜆 for any 𝑛=1,2,…, that is, Fix(𝑉)=𝑃_𝜆, 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 𝑃_{𝑖𝑗,𝑘}=𝑒^−𝜆1𝜆^𝑘₁𝑒𝑘!if𝑖+𝑗iseven,_−𝜆2𝜆^𝑘₂𝑘!if𝑖+𝑗isodd. (13)

For any initial measure 𝜇∈𝑆(𝑋,ℱ) let 𝐴𝜇=^∞_𝑛=0𝜇(2𝑛),𝐵𝜇=^∞_𝑛=0𝜇(2𝑛+1), (14) where 𝐴(𝜇)+𝐵(𝜇)=1. It is easy to show that for Poisson distribution 𝑃_𝜆𝐴𝑃_𝜆=1+𝑒^−2𝜆2,𝐵𝑃_𝜆=1−𝑒^−2𝜆2. (15) Then for any measure 𝜇∈𝑆(𝑋,ℱ), we have 𝑉𝜇(𝑘)=^∞∞_𝑖=0𝑗=0𝑃_{𝑖𝑗,𝑘}𝜇(𝑖)𝜇𝑗=^∞_𝑚,𝑛=0𝑃_{2𝑚,2𝑛,𝑘}𝜇(2𝑚)𝜇(2𝑛)+𝑃_{2𝑚+1,2𝑛+1,𝑘}𝜇()𝜇(+2𝑚+12𝑛+1)_{∞𝑚,𝑛=0}𝑃_{2𝑚+1,2𝑛,𝑘}𝜇(2𝑚+1)𝜇(2𝑛)+𝑃_{2𝑚,2𝑛+1,𝑘}𝜇(2𝑚)𝜇(2𝑛+1)=𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝜇+𝐵²𝜇+𝑒^−𝜆2𝜆^𝑘₂𝜇𝐵𝜇,𝑉𝑘!2𝐴₂𝜇(𝑘)=^∞∞_𝑖=0𝑗=0𝑃_{𝑖𝑗,𝑘}(𝑖)𝑗=𝑉𝜇𝑉𝜇_{∞𝑚,𝑛=0}𝑃_{2𝑚,2𝑛,𝑘}𝑉𝜇(2𝑚)𝑉𝜇(2𝑛)+𝑃_{2𝑚+1,2𝑛+1,𝑘}()(+𝑉𝜇2𝑚+1𝑉𝜇2𝑛+1)_{∞𝑚,𝑛=0}𝑃_{2𝑚+1,2𝑛,𝑘}𝑉𝜇(2𝑚+1)𝑉𝜇(2𝑛)+𝑃_{2𝑚,2𝑛+1,𝑘}𝑉𝜇(2𝑚)𝑉𝜇(2𝑛+1)=𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝑉𝜇+𝐵²𝑉𝜇+𝑒^−𝜆2𝜆^𝑘₂𝐵.𝑘!2𝐴𝑉𝜇𝑉𝜇 (16)

By simple calculations, we have 𝐴𝑉𝜇=1+𝑒^−2𝜆12𝐴²𝜇+𝐵²𝜇+1+𝑒^−2𝜆22𝜇𝐵𝜇,𝐵=2𝐴𝑉𝜇1−𝑒^−2𝜆12𝐴²𝜇+𝐵²𝜇+1−𝑒^−2𝜆222𝐴𝜇𝐵𝜇. (17)

Thus, by using induction on the sequence 𝑉^𝑛(𝜇), we produce the following recurrent equation: 𝑉^𝑛+1𝜇(𝑘)=𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇+𝑒^−𝜆2𝜆^𝑘₂𝑉𝑘!2𝐴^𝑛𝜇𝐵𝑉^𝑛𝜇, (18) where 𝑛=0,1,…, Besides, for parameters 𝐴(𝑉^𝑛𝜇) and 𝐵(𝑉^𝑛𝜇), we have the following recurrent equations: 𝐴𝑉^𝑛+1𝜇=1+𝑒^−2𝜆12𝐴²𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇+1+𝑒^−2𝜆222𝐴𝑉^𝑛𝜇𝐵𝑉^𝑛𝜇,𝐵𝑉^𝑛+1𝜇=1−𝑒^−2𝜆12𝐴²𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇+1−𝑒^−2𝜆222𝐴𝑉^𝑛𝜇𝐵𝑉^𝑛𝜇. (19)

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

Since 𝐴(𝑉^𝑛𝜇)+𝐵(𝑉^𝑛𝜇)=1, where 𝐴(𝑉^𝑛𝜇)≥0 and 𝐵(𝑉^𝑛𝜇)≥0, the recurrent equations (19) are rewritten as follows: 𝑥^′=𝐴𝜆₁𝑥²+𝑦²+2𝐴𝜆₂𝑦𝑥𝑦,_′=𝐵𝜆₁𝑥²+𝑦²+2𝐵𝜆₂𝑥𝑦 (20) with 𝑥≥0, 𝑦≥0, and 𝑥+𝑦=1.

Solving the following quadratic equation 𝑥=𝐴𝜆₁𝑥²+(1−𝑥)²+2𝐴𝜆₂𝑥(1−𝑥), (21) 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.

(a) When 𝜆1=0.8 and 𝜆2=0.4

(b) When 𝜆1=0.4 and 𝜆2=1.4

Figure 1 

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

Thus, for any initial measure 𝜇, we have lim_𝑛→∞𝐴𝑉^𝑛𝜇=𝑥^∗,lim_𝑛→∞𝐵𝑉^𝑛𝜇=𝑦^∗. (22) Then, passing to limit in (18), for any singleton 𝑘, we have lim_𝑛→∞𝑉^𝑛+1𝜇(𝑘)=lim_𝑛→∞𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇+𝑒^−𝜆2𝜆^𝑘₂𝑉𝑘!2𝐴^𝑛𝜇𝐵𝑉^𝑛𝜇=𝑒^−𝜆1𝜆^𝑘₁𝑥𝑘!^∗2+𝑦^∗2+𝑒^−𝜆2𝜆^𝑘₂𝑘!2𝑥^∗𝑦^∗=𝑥^∗2+𝑦^∗2𝑃_𝜆1(𝑘)+2𝑥^∗𝑦^∗𝑃_𝜆2(𝑘). (23)

Thus, for any initial measure 𝜇, the strong limit of the sequence 𝑉^𝑛𝜇 exists and is equal to the convex linear combination lim_𝑛→∞𝑉^𝑛𝜇(𝑘)=𝑥^∗2+𝑦^∗2𝑃_𝜆1(𝑘)+2𝑥^∗𝑦^∗𝑃_𝜆2(𝑘), (24) of two Poisson measures 𝑃_𝜆1 and 𝑃_𝜆2. It is evident that Fix(𝑉)=(𝑥^∗2+𝑦^∗2)𝑃_𝜆1(𝑘)+2𝑥^∗𝑦^∗𝑃_𝜆2(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 𝑃_{𝑖𝑗,𝑘}=𝑒^−𝜆1𝜆^𝑘₁(),𝑒𝑘!if𝑖+𝑗=0mod3_−𝜆2𝜆^𝑘₂(),𝑒𝑘!if𝑖+𝑗=1mod3_−𝜆3𝜆^𝑘₃().𝑘!if𝑖+𝑗=2mod3 (25) For any initial measure 𝜇∈𝑆(𝑋,ℱ), let 𝐴𝜇=^∞_𝑛=0𝜇(3𝑛),𝐵𝜇=^∞_𝑛=0𝜇(),𝐶𝜇=3𝑛+1_∞𝑛=0𝜇(3𝑛+2), (26) where 𝐴(𝜇)+𝐵(𝜇)+𝐶(𝜇)=1. It is easy to show that, for Poisson distribution 𝑃_𝜆 with parameter 𝜆, we have 𝐴(𝜆)=1+2𝑒^−(3/2)𝜆cos√3/2𝜆3,𝐵(𝜆)=1−2𝑒^−(3/2)𝜆cos√3/2𝜆+𝜋/33,𝐶(𝜆)=1−2𝑒^−(3/2)𝜆cos√3/2𝜆−𝜋/33. (27) Then, for any measure 𝜇∈𝑆(𝑋,ℱ), we have 𝑉𝜇(𝑘)=^∞∞_𝑖=0𝑗=0𝑃_{𝑖𝑗,𝑘}𝜇(𝑖)𝜇𝑗=^∞_𝑚,𝑛=0𝑃_{3𝑚,3𝑛,𝑘}𝜇(3𝑚)𝜇(3𝑛)+𝑃_{3𝑚+1,3𝑛+2,𝑘}𝜇(3𝑚+1)𝜇(3𝑛+2)+𝑃_{3𝑚+2,3𝑛+1,𝑘}𝜇()𝜇(+3𝑚+23𝑛+1)_{∞𝑚,𝑛=0}𝑃_{3𝑚+1,3𝑛,𝑘}𝜇(3𝑚+1)𝜇(3𝑛)+𝑃_{3𝑚,3𝑛+1,𝑘}𝜇(3𝑚)𝜇(3𝑛+1)+𝑃_{3𝑚+2,3𝑛+1,𝑘}𝜇()𝜇(+3𝑚+23𝑛+1)_{∞𝑚,𝑛=0}𝑃_{3𝑚+2,3𝑛,𝑘}𝜇(3𝑚+2)𝜇(3𝑛)+𝑃_{3𝑚,3𝑛+2,𝑘}𝜇(3𝑚)𝜇(3𝑛+2)+𝑃_{3𝑚+1,3𝑛+1,𝑘}𝜇(3𝑚+1)𝜇(3𝑛+1)=𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝜇+2𝐵𝜇𝐶𝜇+𝑒^−𝜆2𝜆^𝑘₂𝜇𝐵𝜇𝑘!2𝐴+𝐶²𝜇+𝑒^−𝜆3𝜆^𝑘₃𝜇𝐶𝜇𝑘!2𝐴+𝐵²𝜇,𝑉²𝜇(𝑘)=^∞∞_𝑖=0𝑗=0𝑃_{𝑖𝑗,𝑘}(𝑖)𝑗=𝑉𝜇𝑉𝜇_{∞𝑚,𝑛=0}𝑃_{3𝑚,3𝑛,𝑘}𝑉𝜇(3𝑚)𝑉𝜇(3𝑛)+𝑃_{3𝑚+1,3𝑛+2,𝑘}𝑉𝜇(3𝑚+1)𝑉𝜇(3𝑛+2)+𝑃_{3𝑚+2,3𝑛+1,𝑘}()(+𝑉𝜇3𝑚+2𝑉𝜇3𝑛+1)_{∞𝑚,𝑛=0}𝑃_{3𝑚+1,3𝑛,𝑘}𝑉𝜇(3𝑚+1)𝑉𝜇(3𝑛)+𝑃_{3𝑚,3𝑛+1,𝑘}𝑉𝜇(3𝑚)𝑉𝜇(3𝑛+1)+𝑃_{3𝑚+2,3𝑛+2,𝑘}()(+𝑉𝜇3𝑚+2𝑉𝜇3𝑛+2)_{∞𝑚,𝑛=0}𝑃_{3𝑚+2,3𝑛,𝑘}𝑉𝜇(3𝑚+2)𝑉𝜇(3𝑛)+𝑃_{3𝑚,3𝑛+2,𝑘}𝑉𝜇(3𝑚)𝑉𝜇(3𝑛+2)+𝑃_{3𝑚+1,3𝑛+1,𝑘}𝑉𝜇(3𝑚+1)𝑉𝜇(3𝑛+1)=𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝑉𝜇+𝐵𝑉𝜇𝐶𝑉𝜇+𝑒^−𝜆2𝜆^𝑘₂𝐵𝑘!2𝐴𝑉𝜇𝑉𝜇+𝐶²𝑉𝜇+𝑒^−𝜆3𝜆^𝑘₃𝐶𝑘!2𝐴𝑉𝜇𝑉𝜇+𝐵²𝑉𝜇. (28) By simple calculations, we have 𝐴𝑉𝜇=𝐴𝜆₁𝐴²𝜇𝜇𝐶𝜇𝜆+2𝐵+𝐴₂2𝐴𝜇𝐵𝜇+𝐶²𝜇+𝐴𝜆₃2𝐴𝜇𝐶𝜇+𝐵²𝜇,𝐵𝑉𝜇=𝐵𝜆₁𝐴²𝜇𝜇𝐶𝜇𝜆+2𝐵+𝐵₂2𝐴𝜇𝐵𝜇+𝐶²𝜇+𝐵𝜆₃2𝐴𝜇𝐶𝜇+𝐵²𝜇,𝐶𝑉𝜇=𝐶𝜆₁𝐴²𝜇𝜇𝐶𝜇𝜆+2𝐵+𝐶₂2𝐴𝜇𝐵𝜇+𝐶²𝜇+𝐶𝜆₃2𝐴𝜇𝐶𝜇+𝐵²𝜇. (29)

Thus, by using induction on sequence 𝑉^𝑛(𝜇), we produce the following recurrent equation: 𝑉^𝑛+1𝜇(𝑘)=𝑒^−𝜆1𝜆^𝑘₁𝐴𝑘!²𝑉^𝑛𝜇+2𝐵𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝑒^−𝜆2𝜆^𝑘₂𝑉𝑘!2𝐴^𝑛𝜇𝐵𝑉^𝑛𝜇+𝐶²𝑉^𝑛𝜇+𝑒^−𝜆3𝜆^𝑘₃𝑉𝑘!2𝐴^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇, (30) where 𝑛=0,1,…. Besides, for parameters 𝐴(𝑉^𝑛𝜇), 𝐵(𝑉^𝑛𝜇) and 𝐶(𝑉^𝑛𝜇), we have the following recurrent equations: 𝐴𝑉^𝑛+1𝜇=𝐴𝜆₁𝐴²𝑉^𝑛𝜇+2𝐵𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐴𝜆₂2𝐴𝑉^𝑛𝜇𝐵𝑉^𝑛𝜇+𝐶²𝑉^𝑛𝜇+𝐴𝜆₃2𝐴𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇,𝐵𝑉^𝑛+1𝜇=𝐵𝜆₁𝐴²𝑉^𝑛𝜇+2𝐵𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐵𝜆₂2𝐴𝑉^𝑛𝜇𝐵𝑉^𝑛𝜇+𝐶²𝑉^𝑛𝜇+𝐵𝜆₃2𝐴𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇,𝐶𝑉^𝑛+1𝜇=𝐶𝜆₁𝐴²𝑉^𝑛𝜇+2𝐵𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐶𝜆₂2𝐴𝑉^𝑛𝜇𝐵𝑉^𝑛𝜇+𝐶²𝑉^𝑛𝜇+𝐶𝜆₃2𝐴𝑉^𝑛𝜇𝐶𝑉^𝑛𝜇+𝐵²𝑉^𝑛𝜇. (31)

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

Since 𝐴(𝑉^𝑛𝜇)+𝐵(𝑉^𝑛𝜇)+𝐶(𝑉^𝑛𝜇)=1, where 𝐴(𝑉^𝑛𝜇)≥0, 𝐵(𝑉^𝑛𝜇)≥0, and 𝐶(𝑉^𝑛𝜇)≥0, the recurrent equations (31) are rewritten as follows: 𝑥^′=𝐴𝜆₁𝑥²+𝐴𝜆₃𝑦²+𝐴𝜆₂𝑧²+2𝐴𝜆₂𝑥𝑦+𝐴𝜆₃𝑥𝑧+𝐴𝜆₁𝑦𝑦𝑧,_′=𝐵𝜆₁𝑥²+𝐵𝜆₃𝑦²+𝐵𝜆₂𝑧²+2𝐵𝜆₂𝑥𝑦+𝐵𝜆₃𝑥𝑧+𝐵𝜆₁𝑧𝑦𝑧,_′=𝐶𝜆₁𝑥²+𝐶𝜆₃𝑦²+𝐶𝜆₂𝑧²+2𝐶𝜆₂𝑥𝑦+𝐶𝜆₃𝑥𝑧+𝐶𝜆₁𝑦𝑧, (32) 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, 𝜆₂≤2, and some fixed 𝜆₃ are shown in Figure 2. In this diagram, blue color corresponds to the converges of the trajectory.

(a) Diagram when 𝜆3=0.01

(b) Diagram when 𝜆3=100

Figure 2 

Limit behavior of the dynamical system (31) 0<𝜆1, 𝜆2≤2 and some fixed values 𝜆3.

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 𝑘∈𝑋 the limit of the sequence 𝑉^𝑛𝜇(𝑘) exists and equals lim_𝑛→∞𝑉^𝑛+1𝜇(𝑘)=𝑒^−𝜆1𝜆^𝑘₁𝑥𝑘!^∗2+2𝑦^∗𝑧^∗+𝑒^−𝜆2𝜆^𝑘₂𝑘!2𝑥^∗𝑦^∗+𝑧^∗2+𝑒^−𝜆3𝜆^𝑘₃𝑘!2𝑥^∗𝑧^∗+𝑦^∗2. (33) Thus, the strong limit of the sequence 𝑉^𝑛𝜇 exists and equals convex linear combination lim_𝑛→∞𝑉^𝑛+1𝜇=𝑥^∗2+2𝑦^∗𝑧^∗𝑃_𝜆1+2𝑥^∗𝑦^∗+𝑧^∗2𝑃_𝜆2+2𝑥^∗𝑧^∗+𝑦^∗2𝑃_𝜆3, (34) of three Poisson measures 𝑃_𝜆1, 𝑃_𝜆2, and 𝑃_𝜆3. It is evident that Fix(𝑉)=(𝑥^∗2+2𝑦^∗𝑧^∗)𝑃_𝜆1+(2𝑥^∗𝑦^∗+𝑧^∗2)𝑃_𝜆2+(2𝑥^∗𝑧^∗+𝑦^∗2)𝑃_𝜆3.

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 𝜆_𝑖 is less than or equal to three. The Poisson quadratic stochastic operators with any finitely many different parameters 𝜆_𝑖 and countably many different parameters 𝜆_𝑖 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.