On Non-Linear Markov Operators: surjectivity vs orthogonal preserving property

TL;DR

This paper investigates nonlinear Markov operators, specifically polynomial stochastic operators, establishing that their surjectivity is equivalent to possessing an orthogonal preserving property.

Contribution

It introduces the concept of orthogonal preserving polynomial stochastic operators and proves their surjectivity is equivalent to this property.

Findings

Surjectivity of nonlinear Markov operators is equivalent to orthogonal preserving property.

Introduces orthogonal preserving polynomial stochastic operators.

Provides theoretical foundation linking surjectivity and orthogonal preservation.

Abstract

In the present paper, we consider nonlinear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of nonlinear Markov operators is equivalent to their orthogonal preserving property.

Equations

$$S^{m-1}=\left\{\text{x}\in {\mathbb{R}}^m:x_i\ge 0,\forall i\in I_m,\sum _{i=1}^m{x}_i=1\right\}.$$

$$supp(\mathbf{x})=\left\{i\in I_m:x_i\ne 0\right\},null(\mathbf{x})=\{i\in \mathbb{N}:x_i=0\}.$$

$$int{\Gamma }_A=\{\mathbf{x}\in \Gamma :x_i>0,\forall i\in A\}$$

$$P_{i_1\cdots i_l,\bullet }=(P_{i_1\cdots i_l,1},\dots ,P_{i_1\cdots i_l,m}),$$

$$\mathfrak{B}(\mathbf{x})=\sum _{i_1=1}^m\cdots \sum _{i_l=1}^m{x}_{i_1}\cdots x_{i_l}{P}_{i_{[i,l]},\bullet },\mathbf{x}\in S^{m-1}.$$

$$P_{i_{[1,l]},\bullet }=P_{\pi (i_{[1,l]}),\bullet }$$

$$V(\mathbf{x}{)}_k=\sum _{i,j=1}^m{p}_{ij,k}{x}_i{x}_j,k\in I_m.$$

$${\tilde{P}}_{i_1,\dots ,i_{\ell -1},k}=\frac{1}{n}\sum _{\genfrac{}{}{0ex}{}{\text{combinations}\{i_s{\}}_{s=1}^{\ell -1}}{\text{over}\ell -2\text{places}}}{P}_{i_1,\dots ,i_{\ell -2},k}$$

$$\mathfrak{B}(\mathbf{x})$$

$$\displaystyle\left\{\begin{array}[]{ll}P_{{\mathbf{i}},\bullet}={{\mathbf{e}}}_{i},\ P_{i_{[1,l]}\in\{j_{1},j_{2}\},\bullet}\in conv\{{\mathbf{e}}_{j_{1}},{\mathbf{e}}_{j_{2}}\},\cdots,\\[5.69054pt] P_{i_{[1,l]}\in\{j_{1},\dots j_{l}\},\bullet}\in conv\{{\mathbf{e}}_{j_{1}},\dots,{\mathbf{e}}_{j_{l}}\}\\ \end{array}\right.$$

$$\bigcap _{i_1,\dots ,i_l\in B}null(P_{i_{[1,l],\bullet }})$$

$$\mathfrak{B}(\mathbf{x}{)}_1$$

$$\mathfrak{B}(\mathbf{x}{)}_2$$

$$f(x)=x^l+\left(\genfrac{}{}{0ex}{}{l}{1}\right)P_{1\cdots 12,1}{x}^{l-1}(1-x)+\cdots +\left(\genfrac{}{}{0ex}{}{l}{l-(l-k)}\right)P_{\underset{l-k}{\underbrace{1\cdots 1}}\underset{k}{\underbrace{2\cdots 2}},1}{x}^{l-k}(1-x{)}^k+\cdots +(\genfrac{}{}{0ex}{}{l}{1})P_{1\underset{l-1}{\underbrace{2\cdots 2}},1}x(1-x{)}^{l-1}.$$

$$P_{\underset{l}{\underbrace{i\cdots i}},\bullet }={\mathbf{e}}_i$$

$${\mathbf{x}}^{(n)}=\left(\frac{1}{m-1},\dots ,\frac{1}{m-1},\underset{n^{th}term}{\underbrace{0}},\frac{1}{m-1},\dots ,\frac{1}{m-1}\right)$$

$$\mathfrak{B}({\mathbf{x}}^{(n)}{)}_n$$

$$P_{i_{[1,l]},n}=0\text{ if }{i}_1\ne n,\dots ,i_l\ne n$$

$$supp({\mathfrak{B}}^{-1}({\mathbf{e}}_i))=\bigcup _{\mathbf{x}\in {\mathfrak{B}}^{-1}({\mathbf{e}}_i)}supp(\mathbf{x}),{\Gamma }_{supp({\mathfrak{B}}^{-1}({\mathbf{e}}_i))}=conv\{{\mathbf{e}}_j{\}}_{j\in supp({\mathfrak{B}}^{-1}({\mathbf{e}}_i))}$$

$$\{{\mathbf{e}}_j{\}}_{j\in supp({\mathfrak{B}}^{-1}({\mathbf{e}}_i))}\subset {\mathfrak{B}}^{-1}({\mathbf{e}}_i)\text{ for any }i\in I_m.$$

$$\mathfrak{B}(int{\Gamma }_{supp(\mathbf{x})})\subset int{\Gamma }_{{\mathbf{e}}_{i_1}\cdots {\mathbf{e}}_{i_k}}$$

$$supp(\mathbf{x})=\{i_1,\dots ,i_k\}\text{ for any }\mathbf{x}\in V^{-1}(\mathbf{y}).$$

$$\mathfrak{B}(\mathbf{x}{)}_k$$

Full text

Key words: polynomial stochastic operator; surjective; orthogonal preserving;

1. Introduction

Recently, nonlinear Markov chains are intensively studied by many scientists (see [6] for recent review). A process described by a nonlinear Markov operator is a discrete time stochastic process whose transitions may depend on both the current state and the present distribution of the process. The power of nonlinear Markov operator as a modeling tool and its range of applications are immense, and include non-equilibrium statistical mechanics, evolutionary biology (replicator dynamics), population and disease dynamics (Lotka-Volterra and epidemic models) and the dynamics of economic and social systems (replicator dynamics and games).

The simplest nonlinear Markov chain is described by a quadratic stochastic operator (QSO) which is associated with a cubic stochastic matrix. This kind of operator arises in the problem of describing the evolution of biological populations [7]. The notion of QSO was first introduced by Bernstein [2] and the theory of QSOs was developed in many works (see for example [5, 7, 14]). In [3, 8], it is given along self-contained exposition of the recent achievements and open problems in the theory of the QSOs.

Letting $I_{m}=\{1,\dots,m\}$, a straightforward calculation shows that if a stochastic linear operator $L:S^{m-1}\rightarrow S^{m-1}$ is surjective (here $S^{m-1}$ is the set of all probability distributions on $I_{m}$), then, for each $i\in I_{m}$, there exists a $j\in I_{m}$ such that $L^{-1}({\mathbf{e}}_{i})={\mathbf{e}}_{j}$, where $L^{-1}({\mathbf{e}}_{i})$ is the preimage of the vertex ${\mathbf{e}}_{i}$ of the simplex $S^{m-1}$. Unfortunately, this is not the case when we consider nonlinear case. On the other hand, the surjectivity of a nonlinear operator is strongly tied up with nonlinear optimization problems [1]. The criteria for the surjectivity of QSOs was given in [13]. The obtained criteria together with results of [11] implies that a QSO is surjective if and only if it is orthogonal preserving. In [9] we have check this property for cubic stochastic operators, and described all surjective cubic stochastic operators on two-dimensional simples. Hence, it is natural to study the same implications for general non-linear Markov operators.

In this paper, we introduce a notion of orthogonal preservness for nonlinear Markov operators, and show that the surjectivity of this kind of operators is equivalent to their orthogonal preserving property.

2. Preliminaries

Let us recall some necessary notations. Let $I_{m}=\{1,\dots,m\}$. The complement of a set $A\subset I_{m}$ is denoted by $A^{c}=I_{m}\backslash A$. By $\{{\mathbf{e}}_{i}\}_{i\in I_{m}}$ we denote the standard basis in ${\mathbb{R}}^{m}$. Throughout this paper we consider the simplex as

[TABLE]

An element of the simplex $S^{m-1}$ is called a stochastic vector.

For a every ${\mathbf{x}}\in S^{m-1}$ we set

[TABLE]

We define the facet $\Gamma_{A}$ of the simplex $S^{m-1}$ by setting $\Gamma_{A}=conv\{{\mathbf{e}}_{i}\}_{i\in A}$, here $conv(B)$ stands for the convex hull of a set $B$. Let

[TABLE]

be the relative interior of $\Gamma_{A}$.

Let $\mathcal{P}=(P_{i_{1}\dots,i_{k}})^{m}_{i_{1},...,i_{k}=1}$ be a $k$-order $m$-dimensional hypermatrix. We define the following vectors

[TABLE]

for any $i_{1},\dots,i_{k}\in I_{m}$. For the sake of simplicity we use $i_{[1,l]}=i_{1},\dots,i_{l}$ for index. A hypermatrix $\mathcal{P}$ is called stochastic if each vector $P_{i_{[1,l]},\bullet}$ is stochastic for any $i_{1},\dots,i_{l}\in I_{m}$.

Let $\mathcal{P}$ be a stochastic hypermatrix, then it defines a nonlinear Markov operator (or polynomial stochastic operator (PSO)) $\mathfrak{B}:S^{m-1}\rightarrow S^{m-1}$ as follows

[TABLE]

Throughout this paper, without loss of generality, we assume that

[TABLE]

for any $i_{1},\dots,i_{l}\in I_{m}$ and any permutation ${\pi}$ of the indices.

Remark 2.1*.*

We notice that if the hypermatrix $\mathcal{P}$ is given by the cubic matrix $(p_{ij,k})$, then the associated PSO reduces to the quadratic stochastic operator (QSO) given by

[TABLE]

Remark 2.2*.*

We stress that a PSO ${\mathfrak{B}}_{0}$ associated with a stochastic hypermatrix $\mathcal{P}$ of the $\ell-1$-order can be considered as a particular case of PSO associated with a stochastic hypermatrix of $\ell$-order. Indeed, assume that $(P_{i_{[1,i_{\ell-1}]}})$ is an $\ell-1$-order stochastic hypermatrix. Now define $\ell$-order hypermatrix by

[TABLE]

Then direct calculations show that

[TABLE]

This means that ${\mathfrak{B}}_{0}$ is a particular case of ${\mathfrak{B}}$. Using the provided technique, one can show that any QSO is also particular case of PSO. Therefore, methods used for QSO may not be valid in general setting.

We recall ${\mathbf{x}}\in S^{m-1}$ is orthogonal or singular to ${\mathbf{y}}\in S^{m-1}$ (${\mathbf{x}}\perp{\mathbf{y}}$) if and only if $supp({\mathbf{x}})\cap supp({\mathbf{y}})=\emptyset$. It is clear that ${\mathbf{x}}\perp{\mathbf{y}}$ if and only if $x_{k}\cdot y_{k}=0$ for all $k\in I_{m}$ whenever ${\mathbf{x}},{\mathbf{y}}\in S^{m-1}$.

Definition 2.3**.**

A PSO ${\mathfrak{B}}$ is called orthogonal preserving (OP PSO) if for any ${\mathbf{x}},{\mathbf{y}}\in S^{m-1}$ with ${\mathbf{x}}\perp{\mathbf{y}}$ implies ${\mathfrak{B}}({\mathbf{x}})\perp{\mathfrak{B}}({\mathbf{y}})$.

An absorbing state plays an important role in the theory of the classical (linear) Markov chains. Analogously, in [12] it has been introduced the concept of absorbing sets for nonlinear Markov chains.

Definition 2.4**.**

A subset $A\subset I_{m}$ is called absorbing if $A^{c}=\bigcap\limits_{i_{[1,l]}\in A}null(P_{i_{[i,l]},\bullet})$

It was proven in [12, 13] the following results:

Proposition 2.5**.**

The following statements hold:

• (i)

$supp({\mathfrak{B}}({\mathbf{x}}))=\bigcup\limits_{i_{[1,l]}\in supp({\mathbf{x}})}supp(P_{i_{[1,l]},\bullet})$**

• (ii)

$null({\mathfrak{B}}({\mathbf{x}}))=\bigcap\limits_{i_{[1,l]}\in supp({\mathbf{x}})}null(P_{i_{[1,l]},\bullet})$**

• (iii)

${\mathfrak{B}}(int\Gamma_{A})\subset int\Gamma_{B}\textmd{ where }B=\bigcup\limits_{i_{[1,l]}\in A}supp(P_{i_{[1,l]},\bullet})$**

• (iv)

${\mathfrak{B}}(int\Gamma_{A})\subset int\Gamma_{B}\textmd{ if and only if }{\mathfrak{B}}({\mathbf{x}}^{(0)})\in int\Gamma_{B}\textmd{ for some }{\mathbf{x}}^{(0)}\in int\Gamma_{A}$**

Proposition 2.6**.**

Let $A\subset I_{m}$ be a subset. The following statements are equivalent:

• (i)

The set $A$ is absorbing

• (ii)

${\mathfrak{B}}(int\Gamma_{A})\subset int\Gamma_{A}$**

• (iii)

${\mathfrak{B}}({\mathbf{x}}^{(0)})\in int\Gamma_{A}\textmd{ for some }{\mathbf{x}}^{(0)}\in int\Gamma_{A}$**

3. Surjectivity and Orthogonal Preservness of PSOs

In this section, we prove the main result of the whole paper. Namely, we will establish that the surjectivity of PSO is equivalent to its OP property. First, we need some auxiliary facts.

Proposition 3.1**.**

If any subset $A\subset I_{m}$ with $|A|\leq l$ is absorbing, then all subsets of $I_{m}$ are absorbing.

Proof.

Using the assumption one can check that for any $i,j_{0}\in I_{m}$ we have

[TABLE]

where $j_{k}\neq j_{n}$, $k,n\in I_{l}$, ${\mathbf{i}}=\underbrace{i\cdots i}_{l}$. To see this, we consider a set $A^{(k)}=\{j_{1},\dots,j_{k}\}$ with $k\leq l$. Clearly, if $n\notin A^{(k)}$, then $P_{i_{[1,l]}\in\{j_{[1,k]}\},n}=0$. This yields (3.3).

Keeping in mind that the coefficients $\{P_{i_{[1,l+1]}}\}$ satisfy (2.2), and due to the fact (3.3) one had $null(P_{i_{[1,l]},\bullet})\supset I_{m}\backslash\{i_{1},\dots,i_{l}\}$, therefore for any $B\subset I_{m}$, one finds

[TABLE]

This means that $B$ is absorbing. This completes the proof. ∎

Proposition 3.2**.**

If any subset $A\subset I_{m}$ with $|A|\leq l$ is absorbing, then the associated PSO ${\mathfrak{B}}:S^{m-1}\rightarrow S^{m-1}$ is surjective.

Proof.

According to Propositions 2.6 and 3.1 we infer that the associated PSO ${\mathfrak{B}}:S^{m-1}\rightarrow S^{m-1}$ maps each facet of the simplex into itself. To show the operator ${\mathfrak{B}}$ is surjective, we use mathematical induction with respect to the dimension of the simplex. In the case of $m=2$, we can write ${\mathfrak{B}}$ (see to (3.3)) as follows:

[TABLE]

where ${\mathbf{x}}=(x_{1},x_{2})\in S^{1}$. It is enough for us to study ${\mathfrak{B}}({\mathbf{x}})_{1}$ because of ${\mathfrak{B}}({\mathbf{x}})_{1}+{\mathfrak{B}}({\mathbf{x}})_{2}=1$. Let

[TABLE]

One can see that $f(x)\leq 1$ and continuous on interval $[0,1]$. Due to $f(0)=0$ and $f(1)=1$, one concludes that $f(x)$ is surjective over interval $[0,1]$, hence it implies the surjectivity of ${\mathfrak{B}}({\mathbf{x}})$. Thus, the statement is true for $m=2$. Furthermore, we assume that the statement holds for $m\leq n-1$, and we will prove it for $m=n$. From the assumption, if we restrict the mapping of ${\mathfrak{B}}$ to the facet, then the mapping is surjective i.e., ${\mathfrak{B}}:\partial S^{n-1}\rightarrow\partial S^{n-1}$ is surjective. Now, consider ${\mathbf{y}}\in intS^{n-1}$. Here, surjectivity means that the set ${\mathfrak{B}}^{-1}({\mathbf{y}})$ is nonempty. To prove this statement, we use contradiction by supposing the set ${\mathfrak{B}}^{-1}({\mathbf{y}})$ is empty. We define a mapping $g:S^{n-1}\backslash\{{\mathbf{y}}\}\rightarrow\partial S^{n-1}$ which maps every point ${\mathbf{z}}\in S^{n-1}\backslash\{{\mathbf{y}}\}$ to the intersection point of the ray starting from ${\mathbf{z}}$ in the direction of ${\mathbf{y}}$ with the boundary of the simplex. It is easy to check that the mapping $\mathcal{F}:S^{n-1}\rightarrow S^{n-1},\ \mathcal{F}=g\circ{\mathfrak{B}}$ does not have any fixed point. However, this contradicts to the Brouwer fixed point theorem. This completes the proof. ∎

Theorem 3.3**.**

Let ${\mathfrak{B}}$ be a PSO that maps from $S^{m-1}$ into itself such that ${\mathfrak{B}}({\mathbf{e}}_{i})={\mathbf{e}}_{i}$ for all $i\in I_{m}$. Then the following statements are equivalent:

• (i)

${\mathfrak{B}}$* is orthogonal preserving;*

• (ii)

${\mathfrak{B}}$* is surjective;*

• (iii)

${\mathfrak{B}}$* satisfies the following conditions:*

• (1)

${\mathfrak{B}}^{-1}({\mathbf{e}}_{i})={\mathbf{e}}_{i}\textmd{ for any }i\in I_{m}$**

• (2)

${\mathfrak{B}}^{-1}(int\Gamma_{{\mathbf{e}}_{i_{1}}{\mathbf{e}}_{i_{2}}})=int\Gamma_{{\mathbf{e}}_{i_{1}}{\mathbf{e}}_{i_{2}}}\textmd{ for any }i_{1},i_{2}\in I_{m},$* *

⋮

• ($l$)

${\mathfrak{B}}^{-1}(int\Gamma_{{\mathbf{e}}_{i_{1}}\cdots{\mathbf{e}}_{i_{l}}})=int\Gamma_{{\mathbf{e}}_{i_{1}}\cdots{\mathbf{e}}_{i_{l}}}\textmd{ for any }i,j\in I_{m},$**

where $\Gamma_{{\mathbf{e}}_{i_{1}}\cdots{\mathbf{e}}_{i_{l}}}=conv\{{\mathbf{e}}_{i_{1}},\dots,{\mathbf{e}}_{i_{l}}\}$.

Proof.

To prove the theorem, we will establish the following implications: $(i)\Rightarrow(ii)\Rightarrow(iii)\Rightarrow(ii)\Rightarrow(i)$.

$(i)\Rightarrow(ii)$. Let ${\mathfrak{B}}$ be an orthogonal preserving PSO. Due to the assumption (i.e., ${\mathfrak{B}}({\mathbf{e}}_{i})={\mathbf{e}}_{i}$), one has

[TABLE]

Now, choose

[TABLE]

and ${\mathbf{e}}_{n}$, where $n\in I_{m}$. Clearly ${\mathbf{x}}^{(n)}$ is orthogonal to ${\mathbf{e}}_{n}$. Using the definition of PSO, we have

[TABLE]

From the orthogonal preservness of ${\mathfrak{B}}$, we infer that ${\mathfrak{B}}({\mathbf{x}}^{(n)})$ is orthogonal to ${\mathfrak{B}}({\mathbf{e}}_{n})$, whence

[TABLE]

This yields (3.3), therefore all subsets $A\subset I_{m}$ with $|A|\leq l$ are absorbing. According to Proposition 3.2 we obtain that ${\mathfrak{B}}$ is surjective.

$(ii)\Rightarrow(iii)$. Assume that ${\mathfrak{B}}$ is surjective and let ${\mathfrak{B}}^{-1}({\mathbf{e}}_{i})$ be the preimage of ${\mathbf{e}}_{i}$. We set

[TABLE]

Due to Proposition 2.5 one gets ${\mathfrak{B}}(\Gamma_{supp({\mathfrak{B}}^{-1}({\mathbf{e}}_{i}))})={\mathbf{e}}_{i}$. Consequently,

[TABLE]

It implies that $|supp({\mathfrak{B}}^{-1}({\mathbf{e}}_{i}))=1|$, which means that only ${\mathbf{e}}_{i}$ that maps to ${\mathbf{e}}_{i}$ hence we obtain (iii)(1).

Further, let $k\in\{2,\dots,l\}$. Take ${\mathbf{y}}\in int\Gamma_{{\mathbf{e}}_{i_{1}}\cdots{\mathbf{e}}_{i_{k}}}$ and let ${\mathbf{x}}\in{\mathfrak{B}}^{-1}({\mathbf{y}})$. Using Proposition 2.5 we have

[TABLE]

In fact, we have

[TABLE]

If not, then there exists $k^{\prime}\in supp({\mathbf{x}})\backslash\{i_{1},\dots,i_{k}\}\neq\emptyset$. Then $V({\mathbf{e}}_{k^{\prime}})\in V(int\Gamma_{supp({\mathbf{x}})})\subset\Gamma_{{\mathbf{e}}_{i_{1}}\cdots{\mathbf{e}}_{i_{k}}}$, which is a contradiction. Therefore we obtain (iii)($k$) for any $k\in\{2,\dots,l\}$.

$(iii)\Rightarrow(ii)$. This implication immediately follows from Proposition 3.2.

$(ii)\Rightarrow(i)$ Due to the surjectivity of ${\mathfrak{B}}$ and condition ${\mathfrak{B}}({\mathbf{e}}_{i})={\mathbf{e}}_{i}$ one gets any subset $A\subset I_{m}$ with $|A|\leq l$ is absorbing. It follows from (3.3) that

[TABLE]

for any ${\mathbf{x}}\in S^{m-1}$ and $k\in I_{m}$. Next, take any two orthogonal vectors ${\mathbf{x}}=(x_{1},\dots,x_{m})$ and ${\mathbf{y}}=(y_{1},\dots,y_{m})$ in the simplex $S^{m-1}$. This means that for any fix $k\in I_{m}$ $x_{k}=0$ or $y_{k}=0$. Therefore, from (3.4) we infer that ${\mathfrak{B}}({\mathbf{x}})_{k}\cdot{\mathfrak{B}}({\mathbf{y}})_{k}=0$ which yields the orthogonality ${\mathfrak{B}}({\mathbf{x}})$ and ${\mathfrak{B}}({\mathbf{y}})$. This completes the proof. ∎

Immediately, from the last theorem one concludes the following corollary.

Corollary 3.4**.**

Let ${\mathfrak{B}}$ be a CSO that maps from $S^{m-1}$ into itself. Then the following statements are equivalent:

• (i)

${\mathfrak{B}}$* is orthogonal preserving;*

• (ii)

${\mathfrak{B}}$* is surjective;*

• (iii)

${\mathfrak{B}}$* satisfy the following conditions:*

• (1)

${\mathfrak{B}}^{-1}({\mathbf{e}}_{i})={\mathbf{e}}_{\pi(i)}\textmd{ for any }i\in I_{m}$,

• (2)

${\mathfrak{B}}^{-1}(int\Gamma_{{\mathbf{e}}_{i_{1}}{\mathbf{e}}_{i_{2}}})=int\Gamma_{{\mathbf{e}}_{\pi(i_{1})}{\mathbf{e}}_{\pi(i_{2})}}\textmd{ for any }i_{1},i_{2}\in I_{m},$* *

⋮

• ($l$)

${\mathfrak{B}}^{-1}(int\Gamma_{{\mathbf{e}}_{i_{1}}\cdots{\mathbf{e}}_{i_{l}}})=int\Gamma_{{\mathbf{e}}_{\pi(i_{1})}\cdots{\mathbf{e}}_{\pi(i_{l})}}\textmd{ for any }i,j\in I_{m},$**

for some permutation $\pi$ of $I_{m}$.

Remark 3.5*.*

It is known [13] that if $V$ is a surjective QSO, then it is bijection. Therefore, we formulate the following conjecture.

Conjecture 3.6**.**

Any surjective PSO is bijection.

We point out that some sufficient conditions for the bijectivity of PSO of the form (3.4) has been provided in [10].