quadratic stochastic operators and processes

8/2/2019 Quadratic Stochastic Operators and Processes

http://slidepdf.com/reader/full/quadratic-stochastic-operators-and-processes 1/57

Infinite Dimensional Analysis, Quantum Probability

and Related Topics

Vol. 14, No. 2 (2011) 279–335c World Scientific Publishing Company

DOI: 10.1142/S0219025711004365

QUADRATIC STOCHASTIC OPERATORS AND PROCESSES:

RESULTS AND OPEN PROBLEMS

RASUL GANIKHODZHAEV

Department of Mechanics and Mathematics,National University of Uzbekistan,

100174, Tashkent, Uzbekistan

FARRUKH MUKHAMEDOV

Department of Computational and Theoretical Sciences Faculty of Sciences,

International Islamic University Malaysia,

P. O. Box 141, 25710, Kuantan, Pahang, Malaysia

[email protected]

farrukh [email protected]

UTKIR ROZIKOV

Institute of Mathematics and Information Technologies,

29, Do’rmon Yo’li str., 100125, Tashkent, Uzbekistan

[email protected]

Received 11 May 2009

Revised 22 November 2010Communicated by R. Rebolledo

The history of the quadratic stochastic operators can be traced back to the work of

Bernshtein (1924). For more than 80 years, this theory has been developed and many

papers were published. In recent years it has again become of interest in connection withits numerous applications in many branches of mathematics, biology and physics. But

most results of the theory were published in non-English journals, full text of which arenot accessible. In this paper we give all necessary definitions and a brief description of

the results for three cases: (i) discrete-time dynamical systems generated by quadratic

stochastic operators; (ii) continuous-time stochastic processes generated by quadraticoperators; (iii) quantum quadratic stochastic operators and processes. Moreover, we

discuss several open problems.

Keywords: Quadratic stochastic operator; quadratic stochastic process; quantumquadratic stochastic operator; quantum quadratic stochastic process; fixed point; tra-

jectory; Volterra and non-Volterra operators; ergodic; simplex.

AMS Subject Classification: 15A63, 17D92, 34A25, 35Q92, 37N25, 37L99, 46L53, 47D07,58E07, 60G07, 60G99, 60J28, 60J35, 60K35, 81P16, 81R15, 81S25

279

http://dx.doi.org/10.1142/S0219025711004365

http://dx.doi.org/10.1142/S0219025711004365



280 R. Ganikhodzhaev, F. Mukhamedov & U. Rozikov

Contents

1. Introduction 281

2. Discrete-Time Dynamical Systems Generated by QSOs 2832.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

2.2. The Volterra operators . . . . . . . . . . . . . . . . . . . . . . . . . . 283

2.3. The permuted Volterra QSO . . . . . . . . . . . . . . . . . . . . . . 287

2.4. -Volterra QSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

2.5. Non-Volterra QSO as a combination of a Volterra and non-Volterra

operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

2.6. F-QSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

2.7. Strictly non-Volterra QSO . . . . . . . . . . . . . . . . . . . . . . . . 290

2.8. Regularity of QSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2902.9. Quadratic bistochastic operators . . . . . . . . . . . . . . . . . . . . 291

2.10. Surjective QSOs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

2.11. Construction of QSO. Finite case . . . . . . . . . . . . . . . . . . . . 292

2.12. Non-Volterra QSO generated by a product measure . . . . . . . . . . 293

2.13. Trajectories with historic behavior . . . . . . . . . . . . . . . . . . . 294

2.14. A generalization of Volterra QSO . . . . . . . . . . . . . . . . . . . . 294

2.15. Bernstein’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

2.16. Topological conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . 295

3. Quadratic Stochastic Processes 296

3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

3.2. E is a finite set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

3.3. E is a continuum set . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

3.4. Averaging of the process . . . . . . . . . . . . . . . . . . . . . . . . . 302

3.5. Simple QSPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

3.6. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

4. Quantum Quadratic Stochastic Operators 304

4.1. Quantum quadratic stochastic operators . . . . . . . . . . . . . . . . 3044.2. Quadratic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

4.3. Quantum quadratic stochastic operators on M2(C) . . . . . . . . . . 309

4.4. Dynamics of quadratic operators acting on S (M2(C)) . . . . . . . . . 311

4.5. On infinite-dimensional quadratic Volterra operators . . . . . . . . . 313

4.6. Construction of q.s.o. infinite case . . . . . . . . . . . . . . . . . . . 316

4.7. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

5. Quantum Quadratic Stochastic Processes 320

5.1. Quantum quadratic stochastic processes . . . . . . . . . . . . . . . . 3205.2. Expansion of QQSP . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

5.3. The ergodic principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

5.4. The connection between the fiberwise Markov process

and the ergodic principle . . . . . . . . . . . . . . . . . . . . . . . . . 325



Quadratic Stochastic Operators 281

5.5. Marginal Markov processes . . . . . . . . . . . . . . . . . . . . . . . 326

5.6. The regularity condition . . . . . . . . . . . . . . . . . . . . . . . . . 327

5.7. Differential equations for QQSP . . . . . . . . . . . . . . . . . . . . . 328

1. Introduction

It is known that there are many systems which are described by nonlinear operators.

One of the simplest nonlinear case is quadratic one. Quadratic dynamical systems

have been proved to be a rich source of analysis for the investigation of dynamical

properties and modeling in different domains, such as population dynamics,2,10,42

physics,81,104 economy,9 mathematics.43,49,105 On the other hand, the theory of

Markov processes is a rapidly developing field with numerous applications to many

branches of mathematics and physics. However, there are physical and biologicalsystems that cannot be described by Markov processes. One of such system is given

by quadratic stochastic operators (QSO), which are related to population genet-

ics.2 The problem of studying the behavior of trajectories of quadratic stochastic

operators was stated in Ref. 105. The limit behavior and ergodic properties of

trajectories of quadratic stochastic operators and their applications to population

genetics were studied.44,48,49 In those papers a QSO arises as follows: consider a

population consisting of m species. Let x0 = (x01, . . . , x0m) be the probability dis-

tribution (where x0i = P (i) is the probability of i, i = 1, 2, . . . , m) of species in

the initial generation, and P ij,k the probability that individuals in the ith and jthspecies interbred to produce an individual k, more precisely P ij,k is the conditional

probability P (k|i, j) that ith and jth species interbred successfully, then they pro-

duce an individual k. In this paper, we consider models of free population, i.e. there

is no difference of “sex” and in any generation, the “parents” ij are independent,

i.e. P (i, j) = P (i)P ( j) = xixj . Then the probability distribution x = (x1, . . . , xm)

(the state) of the species in the first generation can be found by the total probability

xk =

m

i,j=1

P (k

|i, j)P (i, j) =

m

i,j=1

P ij,kx0i x0j , k = 1, . . . , m . (1.1)

This means that the association x0 → x defines a map V called the evolution oper-

ator . The population evolves by starting from an arbitrary state x0, then passing

to the state x = V (x) (in the next “generation”), then to the state x = V (V (x)),

and so on. Thus, states of the population described by the following discrete-time

dynamical system

x0, x = V (x), x = V 2(x), x = V 3(x), . . . (1.2)

where V n

(x) = V (V (· · · V n

(x)) · · · ) denotes the n times iteration of V to x.




Note that V (defined by (1.1)) is a nonlinear (quadratic) operator, and it is

higher-dimensional if m ≥ 3. Higher-dimensional dynamical systems are important,

but there are relatively few dynamical phenomena that are currently understood.7

The main problem for a given dynamical system (1.2) is to describe the limitpoints of x(n)∞n=0 for arbitrary given x(0).

In Sec. 2 of this paper we shall discuss the recently obtained results on the

problem, and also give several open problems related to the theory of QSOs.

Note that Boltzmann considered the following problem in his paper “On the

connection between the second law of thermodynamics and probability theory in

heat equilibrium theorems”4: “calculate the probability from relations between the

numbers of different state distributions.” In the first part of Ref. 4, Boltzmann

investigated the simplest object, namely, a gas enclosed between absolutely elas-

tic walls. The molecules of the gas are absolutely elastic balls of the same radiusand mass. It is assumed that the speed of every molecule takes its values in a

certain finite set of numbers, for example, 0, 1/q, 2/q,. . . ,p/q (after any collision

the speed of any molecule can take its value only in this set). In Refs. 12, 95–97

Boltzmann’s model was studied in more general setting by introducing a continuous-

time dynamical system as a quadratic stochastic process. In Sec. 3, we shall describe

results and some open problems related to the continuous-time quadratic stochastic

processes.

However, such kind of operators and processes do not cover the case of quan-

tum systems. Therefore, in Refs. 19, 16, 58 quantum quadratic operators actingon a von Neumann algebra were defined and studied. Certain ergodic properties

of such operators were studied in Refs. 58 and 68. In these papers, dynamics of

quadratic operators were basically defined due to some recurrent rule which marks

a possibility to study asymptotic behaviors of such operators. However, with a

given quadratic operator, one can also define a nonlinear operator whose dynam-

ics (in non-commutative setting) is not studied yet. Note that in Ref. 51 another

construction of nonlinear quantum maps were suggested and some physical expla-

nations of such nonlinear quantum dynamics were discussed. There, it was also

indicated certain applications to quantum chaos. Recently, in Ref. 11 convergence

of ergodic averages associated with mentioned nonlinear operator are studied by

means of absolute contractions of von Neumann algebras. Actually, a nonlinear

dynamics of convolution operators is not investigated. Therefore, a complete anal-

ysis of dynamics of quantum quadratic operator is not well studied. In Sec. 4,

we discuss results obtained for quantum quadratic stochastic operators. On the

other hand, the defined quadratic stochastic processes in Sec. 2 do not encompass

quantum systems, therefore, it is natural to define quantum quadratic stochastic

processes (QQSP). Note that such systems also arise in the study of biological andchemical processes at the quantum level. Furthermore, in Sec. 5 we discuss and

formulate several known results for QQSO. All sections contain main definitions

which make the paper self-contained.




2. Discrete-Time Dynamical Systems Generated by QSOs

2.1. Definitions

The quadratic stochastic operator (QSO) is a mapping of the simplex

S m−1 =

x = (x1, . . . , xm) ∈ Rm : xi ≥ 0,

mi=1

xi = 1

(2.1)

into itself, of the form

V : xk =

mi,j=1

P ij,kxixj , k = 1, . . . , m , (2.2)

where P ij,k are coefficients of heredity and

P ij,k ≥ 0, P ij,k = P ji,k,

mk=1

P ij,k = 1, i, j, k = 1, . . . , m . (2.3)

Thus, each quadratic stochastic operator V can be uniquely defined by a cubic

matrix P = (P ij,k)ni,j,k=1 with conditions (2.3).

Note that each element x ∈ S m−1 is a probability distribution on E =

1, . . . , m.

For a given x(0) ∈ S m−1 the trajectory (orbit)

x(n), n = 0, 1, 2, . . . of x(0)

under the action of QSO (2.2) is defined by

x(n+1) = V (x(n)), where n = 0, 1, 2, . . . .

One of the main problems in mathematical biology consists in the study of the

asymptotical behavior of the trajectories. The difficulty of the problem depends on

the given matrix P. In this section we shall consider several particular cases of P

for which the above-mentioned problem is (particularly) solved.

2.2. The Volterra operators

A Volterra QSO is defined by (2.2), (2.3) and the additional assumption

P ij,k = 0, if k ∈ i, j, ∀ i,j,k ∈ E. (2.4)

The biological treatment of condition (2.4) is clear: the offspring repeats the

genotype of one of its parents.

In Ref. 28, the general form of Volterra QSO is given, i.e.

V : x = (x1, . . . , xm) ∈ S m−1 → V (x) = x = (x1, . . . , xm) ∈ S m−1.

xk = xk

1 +

mi=1

akixi

, k ∈ E, (2.5)




where

aki = 2P ik,k − 1 for i = k and aii = 0, i ∈ E. (2.6)

Moreover,

aki = −aik and |aki| ≤ 1.

Denote by A = (aij)mi,j=1 the skew-symmetric matrix with entries (2.6).

Note that the operator (2.5) is a discretization of the Lotka–Volterra model47,107

which models an interacting, competing species in population. Such a model has

received considerable attention in the fields of biology, ecology, mathematics (see,

for example, Refs. 42, 43 and 103).

Let x(n)∞n=1 be the trajectory of the point x0 ∈ S m−1 under QSO (2.5). Denote

by ω(x0) the set of limit points of the trajectory. Since

x(n)

⊂S m−1 and S m−1

is compact, it follows that ω(x0) = ∅. Obviously, if ω(x0) consists of a single point,

then the trajectory converges, and ω(x0) is a fixed point of (2.5). However, looking

ahead, we remark that convergence of the trajectories is not the typical case for the

dynamical systems (2.5). Therefore, it is of particular interest to obtain an upper

bound for ω(x0), i.e. to determine a sufficiently “small” set containing ω(x0).

Denote

int S m−1 =

x ∈ S m−1 :

m

i=1

xi > 0

, ∂S m−1 = S m−1\int S m−1.

Definition 2.1. A continuous function ϕ : S m−1 → R is called a Lyapunov function

for the dynamical system (2.5) if the limit limn→∞ ϕ(x(n)) exists for any initial

point x0.

Obviously, if limn→∞ ϕ(x(n)) = c, then ω(x0) ⊂ ϕ−1(c). Consequently, for an

upper estimate of ω(x0) we should construct the set of Lyapunov functions that is

as large as possible.

Using the theory of Lyapunov functions and tournaments in Refs. 8, 25, 28–35

and 105 the theory of QSOs (2.5) was developed. DenoteP = p = ( p1, . . . , pm) ∈ S m−1 : Ap ≥ 0.

The following results are known.

Theorem 2.2. (Refs. 28 and 35) For the Volterra QSO (2.5) the following asser-

tions hold true:

(i) The set P is non-empty ;

(ii) For the dynamical system (2.5) there exists a Lyapunov function of the form

ϕ p(x) = x p1

1 · · · x pmm , where p = ( p1, . . . , pm) ∈ P, and x = (x1, . . . , xm) ∈

S m−1;

(iii) If there is r ∈ 1, . . . , m such that aij < 0 (see (2.6)) for all i ∈ 1, . . . , r,

j ∈ r + 1, . . . , m, then ϕ(x) =m

i=r+1 xi, x ∈ S m−1 is a Lyapunov function

for QSO (2.5);




(iv) There are Lyapunov functions of the form

ϕ(x) =xixj

, i = j, x ∈ int S m−1.

Problem 2.3. Does there exist another kind of Lyapunov function for QSO (2.5)?

The next theorem is related to the set of limit points of QSO (2.5).

Theorem 2.4. (Refs. 28 and 35) For the Volterra QSO (2.5) the following asser-

tions hold true:

(i) If x(0) ∈ int S m−1 is not a fixed point (i.e. V (x(0)) = x(0)), then ω(x0) ⊂∂S m−1.

(ii) The set ω(x0) either consists of a single point or is infinite.

(iii) If QSO (2.5) has an isolated fixed point x∗

∈ int S m−1

, then for any initial point x(0) ∈ int S m−1\x∗ the trajectory x(n) does not converge.

(iv) The “negative” trajectories V −n(x), n ≥ 0 always converge.

The formulated theorems have the following biological interpretations:

(a) The evolution begins in a neighborhood of one of the equilibrium states of the

population (stable fixed point).

(b) As a rule, the population does not tend to an equilibrium state with the passage

of time (non-stable fixed point).

(c) The “future” of the population is unstable, since certain species turn out to beon the verge of extinction with the passage of time.

(d) The “past” of such biological systems can be uniquely reproduced.

A skew-symmetric matrix A is called transversal if all even order leading (princi-

pal) minors are nonzero. A Volterra QSO V is called transversal if the corresponding

skew-symmetric matrix A is transversal.31,35,77

Problem 2.5. Define a concept of transversality for arbitrary QSO , and find nec-

essary and sufficient conditions on a matrix P = (P ij,k) of a QSO under which the

QSO is a transversal.

Note that if a Volterra QSO is transversal, then the set of fixed points X =

x ∈ S m−1 : V (x) = x, is a finite set.35

Let U ≡ U X be a neighborhood of the set X and x(n) be an arbitrary trajec-

tory. Denote

nU = | j = 1, . . . , n : x(j) ∈ U |,where |M | denotes the number of elements in M .

Then it is known thatlimn→∞

nU n

= 1,

i.e. the trajectory a large part of the time will stay in the neighborhood of the fixed

points.




Denote U = U 1 ∪ U 2 ∪ · · · ∪ U t, where U i, i = 1, . . . , t is the neighborhood of the

fixed point xi.

Thus, the trajectory first visits the neighborhood of a fixed point xn1 then it

visits the neighborhood of a fixed point xn2

, and so on.

The sequence n1, n2, . . . is called the itinerary (route-march) of the trajectory

x(0), x(1), . . . . Since the set of fixed points is a finite set, the numbers n1, n2, . . . will

repeat.

Problem 2.6. Is there a trajectory with a periodic itinerary ?

On the basis of numerical calculations, Ulam105 conjectured that ergodic theo-

rem holds for any QSO V , that is, the limit limn→∞ C n(V k(x)k≥0) exists for any

x ∈ S m−1, where

C n(V k(x)k≥0) =1

n

n−1k=0

V k(x). (2.7)

In Ref. 108 Zakharevich proved that this conjecture, in general, is false. In Ref. 25

it was considered the following class of Volterra QSOs V : S 2 → S 2

x = x(1 + ay − bz),

y = y(1 − ax + cz),

z = z(1 + bx

−cy),

(2.8)

where a,b,c ∈ [−1, 1]. Note that such class contains Zakharevich’s example as a

particular case (i.e. a = b = c = 1). Note that certain extension of Zakharevich’s

example was considered in Ref. 106.

Theorem 2.7. (Ref. 25) If the parameters a,b,c for the Volterra QSO (2.8) have

the same sign and each is nonzero, then the ergodic theorem will fail for this

operator.

Problem 2.8. Find necessary and sufficient conditions on the matrix A of a

Volterra QSO under which the ergodic theorem is true on S m−1, m ≥ 2.

Note that Theorem 2.7 states that under suitable conditions, the first Cesaro

mean C n(V k(x)k≥0) of the trajectory of the Volterra QSO (2.8) does not con-

verge. But we may consider the second Cesaro mean of the first Cesaro mean, i.e.

C n(C m(V k(x)k≥0)m≥0). We are interested in the following question: does the

second Cesaro mean converge, whereas the first one diverges? If it is not so, we will

continue to consider next Cesaro means.

In Ref. 92 one of the nice properties of the Volterra QSO (2.8) has been

established.

Theorem 2.9. (Ref. 92) Let the condition of Theorem 2.7 be satisfied. Then any

order of the Cesaro mean of the trajectory of the Volterra QSO (2.8) does not

converge.




This theorem implies that the set of limiting points ω(x) of the trajectory

V n(x) of the Volterra QSO (2.8) has unusual structure.

Problem 2.10. Describe the set of all limiting points of the trajectory of the

Volterra QSO (2.8), whereas the condition of Theorem 2.7 is satisfied.

2.3. The permuted Volterra QSO

Let τ be a cyclic permutation on the set of indices 1, 2, . . . , m and let V be a Volterra

QSO. Define a QSO V τ by

V τ : xτ (j) = xj 1 +

m

k=1

ajkxk , j = 1, . . . , m , (2.9)

where ajk is defined in (2.6) (see Refs. 37, 35, 33 and 32).

Note that QSO V τ is a non-Volterra QSO iff τ = id.

Theorem 2.11. (Ref. 35) For any quadratic automorphism W : S m−1 → S m−1,

there exist a permutation τ and a Volterra QSO V such that W = V τ .

Corollary 2.12. The set of all quadratic automorphisms of the simplex S m−1 can

be geometrically presented as the union of m! nonintersecting cubes of dimension

m(m−1)2 .

In Refs. 75 and 89 the behavior of the trajectories of a class of non-Volterra

automorphisms of S 2 has been studied.

Problem 2.13. Investigate the asymptotic behavior of the trajectories of the oper-

ators V τ (automorphisms) for an arbitrary permutation τ .

Let us observe that any linear operator A : S m−1 → S m−1 can be considered

as a particular case of quadratic operator. Indeed, due to x∈

S m−1 we havemk=1 xk = 1, hence

Ax =

mi,j=1

a1ixixj , . . . ,

mi,j=1

amixixj

.

It is known that the k-periodic point of A is a fixed point of the linear operator

Ak. Hence, in order to find all periodic points of some linear operator, we need

to find fixed points of another linear operator. One of the nice properties of linear

operators is that a number of its isolated fixed points is at most one. Indeed, assumethat an operator A : S m−1 → S m−1 has two isolated fixed points x and y. Then for

any λ ∈ [0, 1] the point λx + (1 − λ)y is a fixed point of A, which contradicts to

assumption. Similarly, if a linear operator A has isolated k-periodic points, then the

number of its isolated k-periodic points is exactly k. However, in a quadratic case,




the situation is difficult. In Refs. 8, 37 and 91 it was obtained certain estimations

to the number of periodic points of (2.9) when τ = id.

Problem 2.14. Assume τ

= id, and QSO (2.9) has an isolated k-periodic points.

Is the number of k-periodic points exactly k? In particular case, if QSO (2.9) hasan isolated fixed point , then is the number of isolated fixed points exactly one?

2.4. -Volterra QSO

Fix ∈ E and assume that elements P ij,k of the matrix P satisfy

P ij,k = 0 if k ∈ i, j for any k ∈ 1, . . . , , i, j ∈ E ; (2.10)

P ij,k > 0 for at least one pair (i, j), i

= k, j

= k if k

∈ + 1, . . . , m

.

(2.11)

Definition 2.15. (Refs. 86 and 87) For any fixed ∈ E , the QSO defined by (2.2),

(2.3), (2.10) and (2.11) is called -Volterra QSO.

Denote by V the set of all -Volterra QSOs.

Remark 2.16. Here we stress the following:

(1) The condition (2.11) guarantees that

V 1

∩ V 2 =

∅for any 1

= 2.

(2) Note that -Volterra QSO is Volterra if and only if = m.(3) By Theorem 2.4 we know that there is no periodic trajectory for Volterra QSO.

But for -Volterra QSO there are such trajectories (see Proposition 2.17 below).

Let ei = (δ1i, δ2i, . . . , δmi) ∈ S m−1, i = 1, . . . , m be the vertices of S m−1, where

δij is the Kronecker delta.

Proposition 2.17. (Refs. 86 and 87) The following assertions hold true:

(i) For any set I s =

ei1 , . . . , eis

⊂ e+1, . . . , em

, s

≤m, there exists a family

V (I s) ⊂ V such that I s is an s-cycle for every V ∈ V (I s).(ii) For any I 1, . . . , I q ⊂ + 1, . . . , m such that I i ∩ I j = ∅ (i = j,i,j = 1, . . . , q),

there exists a family V (I 1, . . . , I q) ⊂ V such that ei, i ∈ I j ( j = 1, . . . , q) is

a |I j |-cycle for every V ∈ V (I 1, . . . , I s).

Problem 2.18. Find the set of all periodic trajectories of a given -Volterra QSO.

In Refs. 75, 86 and 87 the trajectories of a class of 1-Volterra and 2-Volterra

QSOs have been investigated.

Problem 2.19. Develop the theory of dynamical systems generated by a -Volterra

QSO. Find its Lyapunov functions, the set of limit points of its trajectories etc.

Note that in Ref. 22 a quasi-Volterra QSO was considered, such an operator is

a particular case of -Volterra QSO.




2.5. Non-Volterra QSO as a combination of a Volterra and

non-Volterra operators

In Ref. 26 the following family of QSOs V λ : S 2 → S 2 : V λ = λV 0 + (1 − λ)V 1, 0 ≤λ ≤ 1 was considered, where V 0(x) = (x21 + 2x1x2, x22 + 2x2x3, x23 + 2x1x3) is aVolterra QSO and V 1(x) = (x21+2x2x3, x2

2+2x1x3, x23+2x1x2) is a non-Volterra one.

Note that the behavior of the trajectories of V 0 is very irregular (see Refs. 49

and 108). It has fixed points M 0 = ( 13 , 13 , 13), e1, e2, e3. The point M 0 is repelling

and ei, i = 1, 2, 3 are saddle points. These four points are also fixed points for V 1but M 0 is an attracting point for V 1. Thus, properties of V λ change depending on

the parameter λ. In Ref. 26 some examples of invariant curves and the set of limit

points of the trajectories of V λ are given.

Problem 2.20. For two arbitrary QSOs V 1 and V 2 connect the properties of V λ =λV 1 + (1 − λ)V 2, λ ∈ [0, 1] with properties of V 1 and V 2.

Problem 2.21. Describe the values of λ for which the operator V λ has n-periodic

points (n ∈ N).

2.6. F-QSO

Consider E 0 = E ∪ 0 = 0, 1, . . . , m. Fix a set F ⊂ E . This set is called the set

of “females” and the set M = E \F is called the set of “males”. The element 0 willplay the role of an “empty-body.”

We define coefficients P ij,k of the matrix P as follows:

P ij,k =

1, if k = 0, i, j ∈ F ∪ 0 or i, j ∈ M ∪ 0;

0, if k = 0, i, j ∈ F ∪ 0 or i, j ∈ M ∪ 0;

≥ 0, if i ∈ F, j ∈ M, ∀ k.

(2.12)

The biological interpretation of the coefficients (2.12) is obvious: the “child” k

can be born only if its parents are taken from different classes F and M . Generally,P ij,0 can be strictly positive for i ∈ F and j ∈ M , which corresponds, for example,

to the case in which “female” i with “male” j cannot have a “child,” because one

of them is ill or both are.

Definition 2.22. For any fixed F ⊂ E , the QSO defined by (2.2), (2.3) and (2.12)

is called the F -quadratic stochastic operator (F -QSO).

Remark 2.23. Let us note that:

(1) For any F -QSO we have P ii,0 = 1, for every i = 0, therefore such QSO is

non-Volterra.

(2) For m = 1 there is a unique F -QSO (independently of F = 1 and F = ∅)

which is constant, i.e. V (x) = (1, 0) for any x ∈ S 1.




Theorem 2.24. (Ref. 88) Any F -QSO has a unique fixed point (1, 0, . . . , 0) (with

m zeros). Besides, for any x0 ∈ S m, the trajectory x(n) tends to this fixed point

exponentially rapidly.

Problem 2.25. Consider a partition ξ = E 1, . . . , E q of E, i.e. E = E 1 ∪ · · · ∪E q, E i ∩ E j = ∅, i = j. Assume P ij,k = 0 if i, j ∈ E p, for p = 1, . . . , q. Call

the corresponding operator a ξ-QSO. Is an analogue of Theorem 2.24 true for any

ξ-QSO ?

2.7. Strictly non-Volterra QSO

Recently in Ref. 89 a new class of non-Volterra QSOs have been introduced. Such

QSO is called strictly non-Volterra and is defined as follows:

P ij,k = 0 if k ∈ i, j, ∀ i,j,k ∈ E. (2.13)

One can easily check that the strictly non-Volterra operators exist only for m ≥ 3.

An arbitrary strictly non-Volterra QSO defined on S 2 (i.e. m = 3) has the form:

x = αy2 + cz2 + 2yz,

y = ax2 + dz2 + 2xz,

z = bx2 + βy2 + 2xy,

(2.14)

where

a,b,c,d,α,β ≥ 0, a + b = c + d = α + β = 1. (2.15)

Theorem 2.26. (Ref. 89) The following assertions hold true:

(i) For any values of parameters a,b,c,d,α,β with (2.15) the operator (2.14) has

a unique fixed point. Moreover , such a fixed point is not attractive.

(ii) The QSO (2.14) has 2-cycles and 3-cycles depending on the parameters (2.15).

Problem 2.27. Is Theorem 2.26 true for m ≥ 4?

2.8. Regularity of QSO

A QSO V : S m−1 → S m−1 is called regular if any of its trajectories converges to a

point a ∈ S m−1. One can see that any regular QSO has a unique fixed point, i.e. a

is that fixed point. In Ref. 40 the authors consider an arbitrary QSO V : S m−1 →S m−1 with matrix P = (P ij,k) and studied the problem of finding the smallest αmsuch that the condition P ij,k > αm implies the regularity of V .


(i) If P ij,k > 12m , then V is regular.

(ii) α2 = 12 (3 − √

7).

Problem 2.29. Find exact values of αm for any m ≥ 3.




2.9. Quadratic bistochastic operators

Let x ∈ S m−1. Denote by x↓ the point x↓ = (x[1], . . . , x[m]) ∈ S m−1, where

x[1] ≥ · · · ≥ x[m] are the coordinates of x in non-increasing order.

If x, y ∈ S m−1 andki=1

x[i] ≤ki=1

y[i], k = 1, . . . , m ,

then we say that y majorizes x and write x ≺ y.

As is known,53 x ≺ y iff there is a doubly stochastic (bistochastic) matrix B

such that x = By. Therefore, if B is a bistochastic matrix, then Bx ≺ x for any

point x ∈ S m−1.

Reference 29 considered a more general definition.

Definition 2.30. An arbitrary continuous operator V : S m−1 → S m−1 satisfying

the condition

V (x) ≺ x, x ∈ S m−1 (2.16)

is called a bistochastic operator.

Theorem 2.31. (Refs. 29 and 41) If V : S m−1 → S m−1 is a bistochastic operator ,

then the coefficientsP ij,k

satisfy the conditions:

mi,j=1

P ij,k = m, ∀ k = 1, . . . , m; (2.17)

mj=1

P ij,k ≥ 1

2, ∀ i, k = 1, . . . , m; (2.18)

i,j∈I P ij,k ≤ t, ∀ t, k = 1, . . . , m , (2.19)

where I = i1, . . . , it is an arbitrary subset of 1, . . . , m containing t elements.

Conversely , if (2.19) holds, for a QSO , then it is a bistochastic.

Let B be the set of all bistochastic quadratic operators acting in S m−1. The set

B can be regarded as a polyhedron in an m(m2−1)2 -dimensional space. Let ExtrB

be the set of extreme points of B.

Theorem 2.32. (Refs. 36 and 29) If V ∈ ExtrB, then

P ii,k = 0 or 1;

P ij,k = 0,1

2or 1, for i = j.

Note that the converse assertion of the theorem is false.29




In Ref. 4, Birkhoff characterized the set of extreme doubly stochastic matrices.

Namely his result states as follows: the set of extreme points of the set of m × m

doubly stochastic matrices coincides with the set of all permutations matrices. Sim-

ilarly, one can ask:

Problem 2.33. Describe the set of extreme points of the set of bistochastic QSO.

In Ref. 41 a subclass, called quasi-linear operators, of bistochastic QSO has been

described. Further investigations of Birkhoff’s problem for bistochastic QSO have

been studied in Ref. 38. In general, Birkhoff’s problem still remains open.

Next theorem asserts about limiting behavior of bistochastic QSO.

Theorem 2.34. (Ref. 29) Let V : S m−1

→S m−1 be a bistochastic operator , then

for any x ∈ S m−1 the Cesaro mean C n(V k(x)k≥0) converges.

Problem 2.35. Is there a regular bistochastic QSO ?

2.10. Surjective QSOs

In Refs. 23 and 54 a description of surjective QSOs defined on S m−1 for m = 2, 3, 4

and classification of extreme points of the set of such operators are given.

Problem 2.36. Describe the set of all surjective QSOs defined on S m−1 for any m ≥ 5.

2.11. Construction of QSO. Finite case

In Refs. 13 and 24 a constructive description of P (i.e. QSO) is given. The con-

struction depends on cardinality of E , namely two cases: (i) E is finite, (ii) E is

a continual set, were separately considered. Note that for the second case one of

the key problems is to determine the set of coefficients of heredity which is already

infinite-dimensional; the second problem is to investigate the quadratic operator

which corresponds to this set of coefficients. By the construction the operator V

depends on a probability measure µ being defined on a measurable space (E, F ).

Recall the construction for finite E = 1, . . . , m.

Let G = (Λ, L) be a finite graph without loops and multiple edges, where Λ is

the set of vertices and L is the set of edges of the graph.

Furthermore, let Φ be a finite set, called the set of alleles (in problems of sta-

tistical mechanics, Φ is called the range of spin). The function σ : Λ → Φ is called

a cell (in mechanics it is called configuration). Denote by Ω the set of all cells, thisset corresponds to E . Let S (Λ, Φ) be the set of all probability measures defined on

the finite set Ω.

Let Λi, i = 1, . . . , q be the set of maximal connected subgraphs (components)

of the graph G. For any M ⊂ Λ and σ ∈ Ω denote σ(M ) = σ(x) : x ∈ M . Fix two




cells σ1, σ2 ∈ Ω, and put

Ω(G, σ1, σ2) = σ ∈ Ω : σ(Λi) = σ1(Λi) or σ(Λi) = σ2(Λi) for all i = 1, . . . , m.

Now let µ

∈S (Λ, Φ) be a probability measure defined on Ω such that µ(σ) > 0

for any cell σ ∈ Ω; i.e. µ is a Gibbs measure with some potential.82 The hereditycoefficients P σ1σ2,σ are defined as

P σ1σ2,σ =

µ(σ)

µ(Ω(G, σ1, σ2)), if σ ∈ Ω(G, σ1, σ2),

0 otherwise.

(2.20)

Obviously, P σ1σ2,σ ≥ 0, P σ1σ2,σ = P σ2σ1,σ and

σ∈Ω P σ1σ2,σ = 1 for all σ1, σ2 ∈ Ω.

The QSO V ≡ V µ acting on the simplex S (Λ, Φ) and determined by coefficients

(2.20) is defined as follows: for an arbitrary measure λ∈

S (Λ, Φ), the measure

V (λ) = λ ∈ S (Λ, Φ) is defined by the equality

λ(σ) =

σ1,σ2∈Ω

P σ1σ2,σλ(σ1)λ(σ2) (2.21)

for any cell σ ∈ Ω.

Theorem 2.37. (Ref. 13) The QSO (2.21) is Volterra if and only if the graph G

is connected.

Thus, if Φ, G and µ are given, then we can constuct a QSO corresponding to

these objects. In Refs. 13 and 55 several examples of Φ, G and µ are consideredand the trajectories of corresponding QSOs are studied.

Note that the construction above does not give all possible QSOs. So the fol-

lowing problem is interesting.

Problem 2.38. Describe the class of QSOs which can be obtained by the

construction.

2.12. Non-Volterra QSO generated by a product measure

In Ref. 85 it was shown that if µ is the product of probability measures being defined

on each maximal connected subgraphs of G, then corresponding non-Volterra oper-

ator can be reduced to q number (where q is the number of maximal connected

subgraphs of G) of Volterra operators defined on the maximal connected subgraphs.

Let G = (Λ, L) be a finite graph and Λi, i = 1, . . . , q the set of all maximal

connected subgraphs of G. Denote by Ωi = ΦΛi the set of all configurations defined

on Λi, i = 1, . . . , q . Let µi be a probability measure defined on Ωi, such that

µi(σ) > 0 for any σ ∈ Ωi, i = 1, . . . , q .

Consider a probability measure µ on Ω = Ω1 × · · · × Ωq defined by

µ(σ) =

qi=1

µi(σi), (2.22)

where σ = (σ1, . . . , σq), with σi ∈ Ωi, i = 1, . . . , q .




According to Theorem 2.37 if q = 1, then the QSO constructed on G is a Volterra

QSO.

Theorem 2.39. (Ref. 85) The QSO constructed by the construction (2.21) with

respect to measure (2.22) is reducible to q separate Volterra QSOs.

This result allows us to study a wide class of non-Volterra operators in the

framework of the well-known theory of Volterra quadratic stochastic operators.

Problem 2.40. Describe the set of all non-Volterra QSOs which are reducible to

several Volterra QSOs.

Problem 2.41. Find a measure µ different from (2.22) such that the non-Volterra

QSO corresponding to µ can be investigated in the framework of a well known theory

of QSOs.

2.13. Trajectories with historic behavior

The problem which we shall discuss here is a particular case of the problem stated

in Ref. 101.

Consider a QSO V : S m−1 → S m−1. We say that a trajectory x, V (x),

V 2(x), . . . has historic behavior if for some continuous function f : S m−1 → R

the average

limn→∞

1

n + 1

ni=0

f (V i(x))

does not exist.

If this limit does not exist, it follows that “partial averages” 1n+1

ni=0 f (V i(x))

keep changing considerably so that their values give information about the epoch

to which n belongs: they have a history.101

Problem 2.42. Find a class of QSOs such that the set of initial states which give

rise to trajectories with historic behavior has positive Lebesgue measure.

A similar problem was discussed by Ruelle in Ref. 90.

2.14. A generalization of Volterra QSO

Consider QSO (2.2), (2.3) with the additional condition

P ij,k = aikbjk , ∀ i,j,k ∈ E, (2.23)

where aik, bjk ∈ R are entries of matrices A = (aik) and B = (bjk) such thatconditions (2.3) are satisfied for the coefficients (2.23).

Then the QSO V corresponding to the coefficients (2.23) has the form

xk = (V (x))k = (A(x))k · (B(x))k, (2.24)




where

(A(x))k =

m

i=1

aikxi, (B(x))k =

m

j=1

bjkxj .

Note that if A (or B) is the identity matrix, then operator (2.24) is a Volterra QSO.

Problem 2.43. Develop a theory of QSOs defined by (2.24).

Note that this problem was already solved in Ref. 84. It is worth to note that in

Ref. 39 it was concerned with another generalization of Volterra QSO. It has been

established as analogies of Theorems 2.2 and 2.4.

2.15. Bernstein’s problem

The Bernstein problem49,50 is related to a fundamental statement of population

genetics, the so-called stationarity principle. This principle holds provided that the

Mendel law is assumed, but it is consistent with other mechanisms of heredity. An

adequate mathematical problem is as follows. QSO V is a Bernstein mapping if

V 2 = V . This property is just the stationarity principle. This property is known as

Hardy–Weinberg law.43 The problem is to describe all Bernstein mappings explic-

itly. The case m ≤ 2 is mathematically trivial and biologically not interesting.

Bernstein2 solved the above problem for the case m = 3 and obtained some results

for m ≥ 4. In works by Lyubich (see e.g., Refs. 49 and 50) the Bernstein problemwas solved for all m under the regularity assumption. The regularity means that

V (x) depends only on the values f (x), where f runs over all invariant linear forms.

In investigations by Lyubich,49 the algebra AV with the structure constants P ij,kplayed a very important role. Since V (x) = x2, the Bernstein property of V is

equivalent to the identity

(x2)2 = s2(x)x2.

This identity means that AV is a Bernstein algebra with respect to the algebra

homomorphism s : AV → R. The mapping V is regular iff the identity

x2y = s(x)xy

holds in the algebra AV , by definition, this identity means that AV is regular.

Problem 2.44. Describe all QSOs which satisfy V r(x) = V (x) for any x ∈ S m−1

and some r ≥ 2.

2.16. Topological conjugacy

Let V 1 : S m−1 → S m−1 and V 2 : S m−1 → S m−1 be two QSOs with coefficients P (1)ij,k

and P (2)ij,k, respectively. Recall that V 1 and V 2 are said to be topologically conjugate

if there exists a homeomorphism h : S m−1 → S m−1 such that, h V 1 = V 2 h. The

homeomorphism h is called a topological conjugacy .




Mappings which are topologically conjugate are completely equivalent in terms

of their dynamics.7

Definition 2.45. A polynomial f (P ij,k) is called an indicator if from the topolog-

ically conjugateness of V 1 and V 2 it follows that

αf ≤ f (P (1)ij,k) ≤ β f and αf ≤ f (P

(2)ij,k) ≤ β f ,

where αf , β f ∈ R.

Definition 2.46. A system f 1, . . . , f t of indicators is called complete if from

αf n ≤ f n(P (1)ij,k) ≤ β f n and αf n ≤ f n(P

(2)ij,k) ≤ β f n ,

for any n = 1, . . . , t it follows the topologically conjugateness of V 1 and V 2.

A minimal complete system of indicators is called a basis.

Problem 2.47. Does there exist a finite complete system of indicators? Find the

basis of the system of indicators.

There are several approaches to solve Problem 2.47. In Ref. 77 it has been

introduced a notion of homotopic equivalence of two Volterra QSOs. A criterion of

homotopic equivalence of such kind of operators was established.

Problem 2.48. If two Volterra QSOs are homotopic equivalent , then are they topo-logically conjugate?

Note that if the last problem has a positive solution, then by means of a criterion

in Ref. 77, Problem 2.47 can be easily solved in the class of Volterra QSO.

3. Quadratic Stochastic Processes

In this section we shall describe quadratic stochastic processes, with continuous

time, which are related with quadratic operators as well as Markov process with

linear operators, this section is based on Refs. 12, 24, 76, 95–97.

3.1. Definitions

Let (E, F ) be a measurable space and let M be the set of all probability measures

on (E, F ). Let there be given a family of functions P (s,x,y,t,A) defined for

t − s ≥ 1 for all x and y ∈ E and an arbitrary measurable set A ∈ F , and assume

that the family of functions P (s,x,y,t,A) : x, y ∈ E, A ∈ F , s , t ∈ R+, t − s ≥ 1satisfies the following conditions:

(i) P (s,x,y,t,A) = P (s,y,x,t,A) for any x, y ∈ E and A ∈ F ;(ii) P (s,x,y,t, ) ∈ M for any fixed x, y ∈ E ;

(iii) P (s,x,y,t,A) as a function of x and y is measurable on (E × E, F ⊗ F ) for

any A ∈ F ;




(iv) (Analogue of the Chapman–Kolmogorov equation) for the initial measure µ0 ∈M and arbitrary s , τ , t ∈ R+ such that t − τ ≥ 1 and τ − s ≥ 1, we have either

(iv)A

P (s,x,y,t,A) = E

E

P (s,x,y,τ,du)P (τ,u,v,t,A)µτ (dv),

where measure µτ on (E, F ) is defined by

µτ (B) =

E

E

P (0, x , y , τ , B)µ0(dx)µ0(dy),

for any B ∈ F , or

(iv)B

P (s,x,y,t,A) = E

E

E

E

P (s,x,z,τ,du)P (s,y,v,τ,dw)

· P (τ,u,w,t,A)µs(dz)µs(dw).

Then the process defined by the functions P (s,x,y,t,A) is called a

quadratic stochastic process (QSP) of type (A) if (iv)A holds and a

quadratic stochastic process of type (B) if (iv)B holds.

In this definition P (s,x,y,t,A) is called the transition probability which is the

probability of the following event: if x and y in E interact at time s, then oneof the elements of the set A ∈ F will be realized at time t. The realization

of interaction in physical, chemical, and biological phenomena requires some

time. We assume that the maximum of these values of time is equal to 1 (see

Boltzmann’s model4 or the biological models in Ref. 49). Hence, P (s,x,y,t,A)

is defined for t − s ≥ 1.

One can also assume the following:

(v) P (t,x,y,t + 1, A) = P (0,x,y, 1, A) for all t ≥ 1.The condition (v) can be considered as a homogeneity of the process for the

duration of time unity. From the condition it does not follow the homogeneity

of the process in general.

Thus, QSPs can be divided to three classes:

(I) homogeneous, i.e. P (s,x,y,t,A) depends only on t − s for all s and t with

t − s ≥ 1;

(II) homogeneous in duration of time unity , i.e. which satisfy the condition (v).

(III) non-homogeneous which does not belong in class (II).

In short by QSP we mean a QSP of class (II) and by P (s,x,y,t, ·) we denote

the transition probability of a QSP of type (A) and by P (s,x,y,t, ·) we denote the

transition probability of a QSP of type (B).




3.2. E is a finite set

First we shall give three examples of different type of QSPs.

Example 3.1. (Ref. 97) Let E =

1, 2

and (x, 1−

x) be an initial distribution on

E , 0 ≤ x ≤ 1. Consider the following system of transition probabilities:

P [s,t]11,1 =

(1 − 2)t−s

2t−s−1[(2t−s−1 − 1)(1 − 2)sx + 1];

P [s,t]12,1 = P

[s,t]21,1 =

(1 − 2)t−s

2t−s−1

(2t−s−1 − 1)(1 − 2)sx +

1

2

;

P [s,t]22,1 =

(1 − 2)t−s

2t−s−1(2t−s−1 − 1)x;

P [s,t]ij,2 = 1 − P [s,t]ij,1 , i, j = 1, 2.

For ∈ [0, 1/2] these transition probabilities generate a QSP which is of type (A)

and type (B) simultaneously. In this case we have

x(t)1 = (1 − 2)tx, x

(t)2 = 1 − (1 − 2)tx.

For = 0 this QSP is homogeneous; for = 0 it is from class (II).

Example 3.2. (Ref. 97) Let E = 1, 2, 3 and (x1, x2, 1 − x1 − x2) be an initial

distribution on E , where x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ 1. Consider the following systemof transition probabilities:

P [s,t]11,1 = 2t−s +

2t−s−1 − 1

2t−s−1x(t+1)1 ;

P [s,t]12,1 = P

[s,t]13,1 = t−s +

2t−s−1 − 1

2t−s−1x(t+1)1 ;

P [s,t]22,1 = P

[s,t]23,1 = P

[s,t]33,1 =

2t−s−1 − 1

2t−s−1x(t+1)1 ;

P [s,t]11,2 = P

[s,t]13,2 = P

[s,t]33,2 = 2t−s−1 − 1

2t−s−1x(t+1)2 ;

P [s,t]22,2 = 2t−s +

2t−s−1 − 1

2t−s−1x(t+1)2 ;

P [s,t]12,2 = P

[s,t]23,2 = t−s +

2t−s−1 − 1

2t−s−1x(t+1)2 ;

P [s,t]ij,3 = 1 − P

[s,t]ij,1 − P

[s,t]ij,2 , i, j = 1, 2, 3,

where x(t)1 = (2)tx1, x(t)2 = (2)tx2. For ∈ [0, 1/2] these transition probabilitiesgenerate a QSP which is of type (A), but is not of type (B).

Example 3.3. (Ref. 97) Let E = 1, 2, 3 and (x1, x2, 1 − x1 − x2) be an initial

distribution on E , where x1 ≥ 0, x2 ≥ 0, x1 + x2 ≤ 1. Consider the following system




of transition probabilities:

P [s,t]11,1 = 2t−s +

2t−s−1 − 1

2t−s−1x(t)1 ;

P [s,t]12,1 = P

[s,t]13,1 = t−s +

2t−s−1 − 1

2t−s−1x(t)1 ;

P [s,t]22,1 = P

[s,t]23,1 = P

[s,t]33,1 =

2t−s−1 − 1

2t−s−1x(t)1 ;

P [s,t]11,2 = P

[s,t]13,2 = P

[s,t]33,2 =

2t−s−1 − 1

2t−s−1x(t)2 ;

P [s,t]22,2 = 2t−s +

2t−s−1 − 1

2t−s−1

x(t)2 ;

P [s,t]12,2 = P

[s,t]23,21 = t−s +

2t−s−1 − 1

2t−s−1x(t)2 ;

P [s,t]ij,3 = 1 − P

[s,t]ij,1 − P

[s,t]ij,2 , i, j = 1, 2, 3,

where x(t)1 = (2)tx1, x

(t)2 = (2)tx2. For ∈ [0, 1/2], these transition probabilities

generate a QSP which is of type (B), but is not of type (A).

Let E =

1, 2, . . . , n

and

x(t)1 , x

(t)2 , . . . , x

(t)n

be a distribution on E at time t.

Denote

P [s,t]ij,k = P (s,i,j,t,k).

In Refs. 95–97 by analogue of the known results of Kolmogorov46 the following

systems of differential equations are obtained:

For QSP of type (A):

∂P [s,t]ij,k

∂t =

nm,l=1

aml,k(t)x(t−1)

l P [s,t−1]

ij,k ,

∂P [s,t]ij,k

∂s= −

nm,l=1

aij,m(s + 1)x(s+1)l P

[s+1,t]ml,k ,

x(t)k =

ni,j=1

aij,k(t)x(t−1)i x

(t−1)j ,

(3.1)

where

aml,k(t) = limδ→0+0

P [t−1,t+δ]ml,k − P

[t−1,t]ml,k

δ,

the existence of the limit is assumed.




For QSP of type (B):

∂ P [s,t]ij,k

∂t=

m,l,r,qalq,k(t)x(s)m x(s)r P

[s,t−1]im,l P

[s,t−1]jr,q

∂ P [s,t]ij,k

∂s=

m,l,r,q

P im,lP jr,q

d

dsx(s)m x(s)

r − (P im,lajr,q(s + 1)

+ P jr,q aim,l(s + 1))x(s)m x(s)r

P [s+1,t]lq,k .

(3.2)

Problem 3.4. Find conditions on parameters of the systems of Eqs. (3.1) and (3.2)

under which these systems have unique solution.

Here we shall describe some solutions of (3.1) for a homogeneous QSP. Thefollowing lemma is useful.

Lemma 3.5. (Ref. 76) If P [s,t]ij,k is a homogeneous QSP then for each k =

1, 2, . . . , n and any t ≥ 2 we have

x(t)k = x

(2)k .

Using Lemma 3.5 for a homogeneous QSP from Eq. (3.1) we get

∂P (t)ij,k

∂t =

n

m,l=1 aml,kx

(t−1)

l P

(t−1)

ij,m , 2 ≤ t < 3

∂P (t)ij,k

∂s=

nm,l=1

aml,kx(2)l P

(t−1)ij,m , t ≥ 3.

(3.3)

Note that in Ref. 56 the existence and uniqueness of a solution of equations

of type (3.3) is proven. But it is not clear that this solution defines a QSP. The

following calculation shows that for some boundary conditions the solution of (3.3)

defines a QSP but for some other boundary condition it does not define a QSP.

Let us consider Eq. (3.3) at n = 2 and with the following boundary condition

P (t)ij,k =

ϕ(t), if k = 1,

1 − ϕ(t), if k = 2,t ∈ [1, 2], (3.4)

where 0 ≤ ϕ(t) ≤ 1 is a continuous function.

Theorem 3.6. (Ref. 76) For n = 2 and the boundary condition (3.4), Eq. (3.3)

has the unique solution :

P (t)ij,k =

ϕ(t), if t ∈ [1, 2],ϕ(2), if t ≥ 2,

k = 1,1 − ϕ(t), if t ∈ [1, 2],

1 − ϕ(2), if t ≥ 2,k = 2.

(3.5)




Moreover , this solution defines a QSP if and only if ϕ(t) ≡ 1 or ϕ(t) ≡ 0 for all

t ≥ 1.

Note that any solution (with arbitrary boundary condition) of equations derived

in Ref. 46 defines a Markov process. But Theorem 3.6 shows that some solution of (3.3) (for a boundary condition) does not define a QSP.

Problem 3.7. Find all boundary conditions under which the solution of system

(3.3) defines a QSP.

3.3. E is a continuum set

We first give the following example:

Example 3.8. (Ref. 97) Let (E, F ) be a measurable space, where E is a continuum

set and let µ0 be an initial measure. Put

δx(A) =

1, if x ∈ A,

0, if x /∈ A.

Consider the following transition probability:

P (s,x,y,t,A) =1

2t−s−1 δx(A) + δy(A)

2+ (2t−s−1 − 1)µ0(A) ,

defined for t − s ≥ 1 and for all x, y ∈ E . This generates a QSP of type (A) and

(B) simultaneously.

Let E = R. Then for QSPs of type (A) and (B), one can define the following

functions of distributions:

F (s,x,y,t,z) = P (s,x,y,t,Az), F (s,x,y,t,z) = P (s,x,y,t,Az),

where Az = (−∞, z], z ∈ R. If f and f are density functions of the distributions

F and˜

F respectively, then Eqs. (iv)A and (iv)B can be written as follows: fors < τ < t such that τ − s ≥ 1 and t − τ ≥ 1,

(iv)A

f (s,x,y,t,z) =

∞−∞

f (s,x,y,τ,u)f (τ,u,v,t,z)duµτ (dv)

and

(iv)B

f (s,x,y,t,z) = ∞−∞

f (s,x,u,τ,v)f (s , y , w, τ , h)

· f (τ,v,h,t,z)dvdhµs(du)µσ(dw),

where du is the Lebesgue measure on R.




For f and f we get the following integro-differential equations:

For QSP of type (A):

∂f (s,x,y,t,z)

∂t=

a(t,u,v,z)f (s,x,y,t − 1, u)duµt−1(dv),

∂f (s,x,y,t,z)

∂s= −

a(s + 1, x , y , u)f (s + 1, u, v, t , z)duµs+1(dv),

(3.6)

For QSP of type (B):

∂ f (s,x,y,t,z)

∂t = a(t,v,h,z)f (s,x,u,t − 1, v)

· f (s,y,w,t − 1, h)dvdhµs(du)µs(dw),

∂ f (s,x,y,t,z)

∂s=

f (0,x,u, 1, v)f (0, y , w, 1, h)

d

ds(µs(du)µs(dw)

− a(s + 1, y , w , h)f (0,x,u, 1, v) + a(s + 1, x , u, v)f (0, y , w, 1, h))

· µs(du)µs(dw)f (s + 1, v, h , t , z)dvdh.

(3.7)

Problem 3.9. Find conditions on parameters of the systems of Eqs. (3.6) and (3.7)

under which these systems have unique solution.

3.4. Averaging of the process

The following theorem is useful for simplification of systems (3.1)–(3.7):

Theorem 3.10. (Ref. 12) Let P (s,x,y,t,A) (respectively , P (s,x,y,t,A)) be the

translation probability function , defining QSP of type (A) (respectively , (B )) and

µ0 ∈ M is an initial distribution. Then the averaging Q(s,x,t,A):

Q(s,x,t,A) =

E

P (s,x,y,t,A)µs(dy)

respectively, Q(s,x,t,A) = E

P (s,x,y,t,A)µs(dy)is a family of transition probabilities of a Markov process with the same initial

distribution µ0.




Applying Theorem 3.10, Eqs. (3.1) and (3.2) can be written as

∂P [s,t]ij,k

∂t=

n

l=1

Al,k(t)P [s,t]ij,l ,

∂P [s,t]ij,k

∂s=m,l,r

Arm(s + 1)x(s+1)l P

[s+1,t]ml,k ,

x(t)k =

nm=1

Amk(t)x(t)m .

(3.8)

∂ P [s,t]ij,k

∂t

= m,l,r,q,u,v

(Aqv(t

−1)P lv,k + Alu(t

−1)P uq,k)x(s)

m x(s)r P

[s,t−1]im,l P

[s,t−1]jr,q

∂ P [s,t]ij,k

∂s= −

nl=1

(Ajl(s)P [s,t]im,l + Ail(s)P

[s,t]lj,k ),

(3.9)

where Aij(t) and Aij(t) are defined from the following equalities:

aij,k(t) =

nm=1

P ij,mAmk(t),

aij,k(t) =

nl=1

(P lj,kAil(t) + P il,kAjl(t)).

The first equations of (3.6) and (3.7) become as follows:

∂f (s,x,y,t,z)

∂t= N (t, z)f (s,x,y,t,z)

+ A(t, z)∂f (s,x,y,t,z)

∂z+ B2(t, z)

∂ 2f (s,x,y,t,z)

∂z2, (3.10)

∂ f (s,x,y,t,z)

∂s = −˜

A(s,x,z)

∂ f (s,x,y,t,z)

∂x −˜

A(s,y,z)

∂ f (s,x,y,t,z)

∂y

− B2(s,x,z)∂ 2f (s,x,y,t,z)

∂x2− B2(s,y,z)

∂ 2f (s,x,y,t,z)

∂y2.

(3.11)

Problem 3.11. Find conditions on parameters of Eqs. (3.10) and (3.11) under

which these equations have unique solution.

3.5. Simple QSPsA QSP is called simple if it has type (A) as well as type (B).

Theorem 3.12. (Ref. 97) The analytical theory of simple QSPs is analogical to

the analytical theory of Markov processes.




Consider the following density function

f (s,x,y,t,z) = 2t−s−1exp(− 4t−s−1

22(t−s)−1−1(z − x+y

2t−s )2)

(22(t−s)−1

−1)π

. (3.12)

Proposition 3.13. (Ref. 15) The QSP corresponding to function (3.12) is a dif-

fusive process.

The forward and backward equations for this process are as follows:

∂f (s,x,y,t,z)

∂t= ln 2

f (s,x,y,t,z) + z

∂f (s,x,y,t,z)

∂z

+ (1 + z2)∂ 2f (s,x,y,t,z)

∂z2 , (3.13)

∂ f (s,x,y,t,z)

∂s= ln 2

x

∂ f (s,x,y,t,z)

∂x+ y

∂ f (s,x,y,t,z)

∂y

− ln 2

∂ 2f (s,x,y,t,z)

∂x2+

∂ 2f (s,x,y,t,z)

∂y2

. (3.14)

3.6. Remarks

According to Theorem 3.10 every QSP defines a Markov process. Hence, whenstate space E is finite, then given QSP in Ref. 98 Markov chain associated with

QSP has been constructed. For such a Markov process, Central Limit theorem was

established. Moreover, in Ref. 21 singularity and absolute continuity of such Markov

chains were studied.

4. Quantum Quadratic Stochastic Operators

In this section we are going to study quantum analogous of quadratic stochastic

operators. Dynamics of such kind of operators will be studied as well. Note that inRef. 51 another construction of nonlinear quantum maps were suggested and some

physical explanations of such nonlinear quantum dynamics were discussed. There,

it was also indicated certain applications to quantum chaos. On the other hand,

very recently, in Ref. 11 convergence of ergodic averages associated with mentioned

nonlinear operator are studied by means of absolute contractions of von Neumann

algebras. Actually, it is not investigated nonlinear dynamics of quadratic operators.

This section is based on Refs. 58, 59, 62, 63, 69, 71–74.

4.1. Quantum quadratic stochastic operators

Let B(H ) be the algebra of all bounded linear operators on a separable complex

Hilbert space H . Let M ⊂ B(H ) be a von Neumann algebra with unit 1. By

M + we denote the set of all positive elements of M . By M ∗ and M ∗, respectively,




we denote predual and dual spaces of M . The σ(M, M ∗)-topology on M is called

the ultraweak topology. By S (M ) (respectively, S 1(M )) we denote the set of all

ultraweak (respectively, norm) continuous states on M . It is well known102 that a

state is normal if and only if it is an ultraweak continuous. Now recall some notionsfrom tensor product of Banach spaces.

A linear map α : M → M is called *-morphism (respectively, a positive), if

α(x∗) = α(x)∗ for all x ∈ M (respectively, α(M +) ⊂ M +). A linear map T : M →N between von Neumann algebras M and N is said to be completely positive if

T n := T ⊗ 1Mn:Mn(M ) → Mn(N ) is positive for each n = 1, 2, . . . . It is well

known that the completely positivity can be formulated as follows: for any two

collections a1, . . . , an ∈ M and b1, . . . , bn ∈ N the following relation

ni,j=1

b∗i T (a∗i aj)bj ≥ 0. (4.1)

It is well known (cf. Ref. 80) that supn T n = T (1) for completely positive maps. It

is clear that completely positivity of T implies positivity one. In general, converse,

it is not true.

A positive (respectively, completely positive) linear map T : M → M with T 1 =

1 is called Markov operator (M.o.) (respectively, unital completely positive (ucp)

map).

Let M be a von Neumann algebra. Recall that weak (operator) closure of alge-braic tensor product M M in B(H ⊗ H ) is denoted by M ⊗ M , and it is called

tensor product of M into itself. For detail we refer the reader to Refs. 5 and 102.

By S (M ⊗ M ) we denote the set of all normal states on M ⊗ M . Let U : M ⊗M → M ⊗ M be a linear operator such that U (x ⊗ y) = y ⊗ x for all x, y ∈ M .

Definition 4.1. A linear operator P : M → M ⊗ M is said to be quantum quadratic

stochastic operator (q.q.s.o.) if it satisfies the following conditions:

(i) P 1M = 1M ⊗M , where 1M and 1M ⊗M are units of algebras M and M ⊗ M respectively;

(ii) P (M +) ⊂ (M ⊗ M )+;

(iii) U P x = P x for every x ∈ M .

Note that if q.q.s.o. satisfies some extra conditions (for example, coassociativity),

then such an operator generates a compact quantum group.110

By QΣ(M ) we denote the set of all q.q.s.o. on M . Let us equip these sets with

a weak topology by the following seminorms

pϕ,x(P ) = |ϕ(P x)|, ϕ ∈ M ∗ ⊗α∗0M ∗, x ∈ M,

where α∗0 is the dual norm to the smallest C ∗-crossnorm α0 on M ⊗ M (see Sec. 1.22

of Ref. 93).




Let ϕ ∈ S (M ) be a fixed state. We define the conditional expectation operator

E ϕ : M ⊗ M → M on elements a ⊗ b, a, b ∈ M by

E ϕ(a

⊗b) = ϕ(a)b (4.2)

and extend it by linearity and continuity to M ⊗ M . Clearly, such an operator is

completely positive and E ϕ1M ⊗M = 1M .

Theorem 4.2. (Refs. 59 and 63) The set QΣ(M ) is weak compact.

4.2. Quadratic operators

Each q.q.s.o. P defines a conjugate operator P ∗ : (M

⊗M )∗

→M ∗ by

P ∗(f )(x) = f (P x), f ∈ (M ⊗ M )∗, x ∈ M. (4.3)

One can define an operator V P by

V P (ϕ) = P ∗(ϕ ⊗ ϕ), ϕ ∈ S 1(M ), (4.4)

which is called a quadratic operator (q.o.). Thanks to conditions (i), (ii) of

Definition 4.1 the operator V P maps S 1(M ) into itself. In some literature oper-

ator V P is called quadratic convolution (see, for example, Ref. 11).

Problem 4.3. Let P be an extremal point of QΣ(M ). Would the corresponding

q.o. V P be a bijection of S 1(M )?

Problem 4.4. Describe the set of all bijective q.o. V P of S 1(M ).

Now by QΣu(M ) denote the set of all ultraweak continuous q.q.s.o. Then

for each q.q.s.o. P ∈ QΣu(M ) one can consider conjugate operator P ∗. Due

to ultraweak continuity P ∗ maps S (M ⊗ M ) to S (M ). Therefore, by V P we

denote the restriction of P ∗

to S (M ⊗ M ), and it is called conjugate quadraticoperator (c.q.o.). Further for the shortness instead of V P (ϕ ⊗ ψ) we will write

V P (ϕ, ψ), where ϕ, ψ ∈ S (M ). Note that the equality (iii) in Definition 4.1 implies

that

V P (ϕ, ψ) = V P (ψ, ϕ). (4.5)

It is clear that the set S (M ) is invariant w.r.t. V P , for every P ∈ QΣu(M ).

Example 4.5. Here we give how linear operator and q.q.s.o. related with each

other. Let T : M → M be a Markov operator. Define a linear operator P : M →M ⊗ M as follows:

P T x =T x ⊗ 1 + 1⊗ T x

2, x ∈ M. (4.6)




It is clear that P T is q.q.s.o. Then associated c.q.o. and q.o. have the following form

respectively:

V P T (ϕ, ψ)(x) =1

2

(ϕ + ψ)(T x),

V P T (ϕ)(x) = ϕ(T x), x ∈ M,(4.7)

for every ϕ, ψ ∈ S (M ). Thus linear operator can be viewed as a particular case of

q.q.s.o. If T is the identity operator, then from (4.7) we can find that the associated

q.o. would also be the identity operator of S (M ).

Proposition 4.6. (Ref. 59) Each q.q.s.o. P ∈ QΣu(M ) defines a linear operator

T : M ∗ → Σ(M ) by

T (ϕ)(x) = E ϕ(P x), ϕ ∈ M ∗, x ∈ M. (4.8)Moreover , for every ϕ, ψ ∈ M ∗ holds

T ∗(ϕ)ψ = T ∗(ψ)ϕ, T (ϕ) ≤ 2ϕ1, (4.9)

where T ∗(ϕ)ψ(a) = ψ(T (ϕ)(a)). In addition ,

V P (ϕ ⊗ ψ) = T ∗(ϕ)ψ, ∀ ϕ, ψ ∈ M ∗. (4.10)

Note that a similar result can be proved for arbitrary q.q.s.o. (i.e. without

ultraweak continuity).

One can see that the trajectory ϕ(n) = V nP (ϕ) of ϕ ∈ S 1(M ) under the action

of V P can be written as follows:

ϕ(n) = T ∗(ϕ(n−1))T ∗(ϕ(n−2)) · · · T ∗(ϕ)ϕ. (4.11)

A quadratic operator V P is called Abelian if the equation T ∗(ϕ)T ∗(ψ) =

T ∗(ψ)T ∗(ϕ) holds for all ϕ, ψ ∈ S 1(M ).

For an Abelian quadratic operator using (4.11) one finds

ϕ(n) = T 2n−1∗ (ϕ)ϕ. (4.12)

In Ref. 109, the formula (4.12) allowed us to use properties of Markov operators

acting on finite-dimensional spaces and proved the following:

Theorem 4.7. (Ref. 109) Let M is a finite-dimensional von Neumann algebra ,

and V P be an Abelian quadratic operator on S 1(M ). Let ϕ(n) be the trajectory

of an arbitrary point ϕ ∈ S 1(M ) w.r.t. V P . By ω(ϕ) we denote the set of limiting

points of ϕ(n). Then

(i) ω(ϕ) = ϕ1, . . . , ϕs is a finite set ;

(ii) the sequence of means

1

n + 1

n+1k=1

ϕ(k) (4.13)

converges as n → ∞.




Problem 4.8. For an Abelian quadratic operator given on arbitrary von Neumann

algebra , investigate properties of ω(ϕ) and the convergence of (4.13). Note that , in

this setting , one can consider several kinds of convergence such as weak convergence,

strong etc.

Problem 4.9. Classify all Abelian quadratic operators.

A linear map V : M ∗⊗α∗0M ∗ → M ∗ is called conjugate quadratic operator if the

following conditions hold

(i) V (S (M ⊗ M )) ⊂ S (M );

(ii) V (ϕ ⊗ ψ) = V (ψ ⊗ ϕ), ∀ ϕ, ψ ∈ M ∗.

By QΣV (M ) we denote the set of all conjugate quadratic operators.

Proposition 4.10. Every V ∈ QΣV (M ) uniquely defines a q.q.s.o. P ∈ QΣu(M ).

Let P ∈ QΣ(M ) and consider the corresponding q.o. V P on S 1(M ).

Definition 4.11. A q.o. V P is called

(i) asymptotically stable if there exists a state µ ∈ S 1(M ) such that for any ϕ ∈S 1(M ) one has

limn→∞

V nP (ϕ) − µ1 = 0, (4.14)

where by · 1 we denote the norm on M ∗;(ii) weak asymptotically stable if there exists a state µ ∈ S 1(M ) such that for any

ϕ ∈ S 1(M ) and a ∈ M one has

limn→∞

V nP (ϕ)(a) = µ(a). (4.15)

It is clear that asymptotical stability implies weak asymptotical stability, but

in general, the converse is not true. Note that if we consider a q.o. V P T associated

with a Markov operator (see (4.7)), then the introduced notions, i.e. asymptotical

stability and weak asymptotical stability, coincides with complete mixing and weak

mixing, respectively, of Markov operator T (see Ref. 1). Therefore, one can find

many examples of such kind of operators.

Problem 4.12. It would be better to find a weak asymptotically stable q.o., which

is not asymptotically stable and not generated by Markov one.

By S (respectively, S ) we denote the set of all functionals g : M + → R+ such

that

g(x + y) ≤ g(x) + g(y), (respectively, g(x + y) ≥ g(x) + g(y)), ∀ x, y ∈ M +,

g(λx) = λg(x), for all λ ∈ R+, x ∈ M +,

g(1) = 1.

Let us put S = S ∪ S . The set S we endow with the topology of pointwise conver-

gence. By C w(S 1(M ), S ) we denote the set of all weak continuous operators from




S 1(M ) to S , i.e. f ∈ C w(S 1(M ), S ) if a net xα ∗-weak converges to x in S 1(M ),

then f (xα) converges in S . Similarly, by C (S 1(M ), S ) we denote the set of all strong

continuous operators from S 1(M ) to S , i.e. f ∈ C (S 1(M ), S ) if a sequence xn norm

converges to x in S 1(M ), then f (xα) converges in˜S .

Definition 4.13. A q.o. V P is called η-ergodic (respectively, weak η-ergodic if for

f ∈ C (S 1(M ), S ) (respectively, f ∈ C w(S 1(M ), S )), the equality f (V P (ϕ))(x) =

f (ϕ)(x) for every ϕ ∈ S 1(M ) and x ∈ M +, implies that f (ϕ) does not depend on

ϕ, i.e. there is δf ∈ S such that f (ϕ) = δf for all ϕ ∈ S 1(M ).

One has the following

Theorem 4.14. (Ref. 58) Let P ∈ QΣ(M ) and V P be the associated q.o. Then for

the assertions

(i) Q.o. V P is asymptotically stable;

(ii) Q.o. V P is η-ergodic;

(iii) Q.o. V P is weak η-ergodic;

(iv) Q.o. V P is weak asymptotically stable;

the following implications holds true: (i) ⇒ (ii) ⇒ (iii) ⇔ (iv).

Corollary 4.15. If M acts on a finite-dimensional Hilbert space, then all three

conditions of the theorem are equivalent.

Problem 4.16. Investigate the reverse implications in Theorem 4.14.

4.3. Quantum quadratic stochastic operators on M2(C)

Consider an algebra of 2 × 2 complex matrices M2(C). It is known (see Ref. 5) that

the identity and Pauli matrices 1, σ1, σ2, σ3 form a basis for M2(C), where

σ1 = 0 1

1 0 , σ2 = 0 −i

i 0 , σ3 = 1 0

0

−1 .

In this basis every matrix x ∈ M2(C) can be written as x = w01 + ω · σ with

w0 ∈ C, ω ∈ C3. As well as any state ϕ ∈ S (M2(C)) can be represented by

ϕ(x) = w0 + ω · f , (4.16)

here x = w01 + ω · σ and f = (f 1, f 2, f 3) ∈ R3 such that f ≤ 1. Therefore, in the

sequel we will identify a state with a vector f .

Now let P :M2(C) →M2(C)⊗M2(C) be a q.q.s.o. Now we are going to represent

P in the basis of M2(C) ⊗M2(C). Due to P 1 = 1⊗1, and the considered algebras

are finite-dimensional, for us it is enough to present P only on σ1, σ2, σ3. So usingDefinition 4.1 we obtain

P (σk) = bk1 +

3u,v=1

buv,kσu ⊗ σu +

3u=1

bu,k(σu ⊗ 1 + 1⊗ σu). (4.17)




Due to positivity of P we have P (σk) = P (σk)∗, therefore all coefficients

bk, bu,k and buv,k are real.

Problem 4.17. Find necessary and sufficient conditions for the positivity of P in

terms of the coefficients.

Remark 4.18. Note that Proposition 4.21 below provides a necessary condition

for the positivity of P .

Now consider conjugate quadratic operator V P related to P . Taking into account

(4.16) from (4.3) and (4.17) we infer that

V P (f , p)(σk) = bk +

3

u,v=1

buv,kf u pv +

3

u=1

bu,k(f u + pu). (4.18)

Therefore, the associated quadratic operator has a form

V P (f )(σk) = bk +3

u,v=1

buv,kf uf v + 23

u=1

bu,kf u. (4.19)

Note that according to (4.16), V P (f ) defines a state iff V P (f ) ≤ 1.

Example 4.19. Consider q.q.s.o. P T defined by (4.6). From (4.17) one gets

P T (σk) = b(T )k 1 +

3u=1

b(T )u,k(σu ⊗ 1 + 1⊗ σu)

and the corresponding q.o. has the form

V T (f )(σk) = b(T )k + 2

3u=1

b(T )u,kf u.

Example 4.20. Let us consider commutative quadratic stochastic operator(q.s.o.), which is defined by the cubic matrix pij,k with properties

pij,k ≥ 0, pij,k = pji,k, pij,1 + pij,2 = 1, ∀ i,j,k ∈ 1, 2. (4.20)

Define an operator P :C2 → C2 ⊗ C2 by

(P(x))i,j =2

k=1

pij,kxk, i, j ∈ 1, 2, (4.21)

where x = (x1, x2). Here as usual C2 = x = (x1, x2) ∈ C2 : x = max|x1|, |x2|.

By DM2(C) we denote the commutative subalgebra of M2(C) generated by

1 and σ3. It is obvious that DM2(C) can be identified with C2, and further we

will use this identification. Let E :M2(C) → DM2(C) be the canonical conditional




expectation. Now define another operator P P :M2(C) → DM2(C) ⊗ DM2(C) by

P P(x) = P(E (x)), x ∈ M2(C). (4.22)

From (4.21) and the properties of the conditional expectation one concludes thatP P is a q.q.s.o. Now we rewrite it in the form (4.17). From (4.22) and (4.21) we get

P P(σ1) = P P(σ2) = 0 and

P P(σ3) =1

2( p11,1 + 2 p12,1 + p22,1 − 2)1 +

1

2( p11,1 − p22,2)(σ3 ⊗ 1 + 1⊗ σ3)

+1

2( p11,1 − 2 p12,1 + p22,1)σ3 ⊗ σ3.

Hence for the corresponding q.o. we have V P(f )(σ1) = V P(f )(σ2) = 0 and

V P(f )(σ3) = 12

( p11,1 + 2 p12,1 + p22,1 − 2) + ( p11,1 − p22,2)f 3

+1

2( p11,1 − 2 p12,1 + p22,1)f 23 , (4.23)

as before f = (f 1, f 2, f 3) ∈ R3.

In general, a description of positive operators is one of the main problems of

quantum information. In the literature most tractable maps are positive and trace-

preserving ones, since such maps arise naturally in quantum information theory

(see Ref. 79). Therefore, in the sequel we shall restrict ourselves to q.q.s.o. which

is trace-invariant, i.e. Tr ⊗ Tr (P (x)) = Tr (x) for every x ∈ M2(C). For the sake of

simplicity, in the sequel we are going to investigate the non-commutative quadratic

term of (4.17), i.e. we will put bu,k = 0 for all u, k.

Then from (4.17) one gets

P (σk) =

3u,v=1

buv,kσu ⊗ σu, (4.24)

and q.o. V P reads as follows:

V P (f )(σk) =3

u,v=1

buv,kf uf v. (4.25)

This suggests us to consider a nonlinear operator V : S → S defined by

V (f )k =

3i,j=1

bij,kf if j , k = 1, 2, 3, (4.26)

where f = (f 1, f 2, f 3) ∈ S . Furthermore, we are going to study dynamics of V .

4.4. Dynamics of quadratic operators acting on S (M2(C))

To study dynamics of q.o. on M2(C), it is enough to investigate it on the basis

elements. Therefore, we are going to investigate one on vectors f ∈ R3 such that




f ≤ 1. Therefore, denote

S = p = ( p1, p2, p3) ∈ R3 : p21 + p22 + p23 ≤ 1.

Proposition 4.21. (Ref. 74) Let P :M2(C) →M2(C)⊗M2(C) be a linear operator given by (4.24). Then V P (·, ·) is well defined in S (M2(C)), i.e. for any ϕ, ψ ∈S (M2(C)) one has V P (ϕ, ψ) ∈ S (M2(C)), if and only if one holds

3k=1

3

i,j=1

bij,kf i pj

2

≤ 1 for all f , p ∈ S. (4.27)

Remark 4.22. We stress that the condition (4.27) does not imply the positivity

of P , and hence the corresponding V P may not be a conjugate quadratic operator

(see Proposition 4.10). A corresponding example has been provided in Ref. 72.

One can see that if the following holds

3i,j,k=1

|bij,k|2 ≤ 1, (4.28)

then (4.27) is satisfied.

Denote

αk =

3j=1

3i=1

|bij,k|2

+

3i=1

3j=1

|bij,k|2

, α =

3k=1

α2k.

Theorem 4.23. (Ref. 74) If α < 1 then V is a contraction , hence (0, 0, 0) is a

unique stable fixed point.

Note that the condition α < 1 in Theorem 4.23 is too strong, therefore, it would

be interesting to find weaker conditions than the provided one.

Put

δk =

3i,j=1

|bij,k|, k = 1, 2, 3.

Theorem 4.24. (Ref. 74) Assume that δk ≤ 1 for every k = 1, 2, 3. If there is

k0 ∈ 1, 2, 3 such that δk0 < 1 and for each k = 1, 2, 3 one can find i0 ∈ 1, 2, 3with |bi0,k0,k| + |bk0,i0,k| = 0, then (0, 0, 0) is a unique stable fixed point , i.e. for

every f ∈ S one has V n(f ) → (0, 0, 0) as n → ∞.

Problem 4.25. Develop and investigate dynamics of quadratic operators on S associated with (4.19). Is there a chaotic q.o.?

In Ref. 100 certain kind of quadratic operator of triangle was investigated

through the topological conjugacy between quadratic operator acting on S .




Problem 4.26. Is there topological conjugacy between q.s.o. and quantum

quadratic operators?

4.5. On infinite-dimensional quadratic Volterra operators

In this subsection, we follow Ref. 73 and consider Volterra operators on infinite-

dimensional space.

In this subsection we consider a case when the von Neumann algebra M is an

infinite-dimensional commutative discrete algebra, i.e.

M = ∞ =

x = (xn) : xn ∈ R, x = sup

n∈N|xi|

,

then the set of all normal functionals defined on ∞ coincides with

1 =

x = xn : x1 =

∞k=1

|xk| < ∞

,

(i.e. 1 is a pre-dual space to ∞) and S (∞) with

S =

x = (xn) ∈ 1 : xi ≥ 0,

∞n=1

xn = 1

.

It is known83 thatS

= convh(ExtrS

), where Extr(S

) is the set of extremal

points of S , and convh(A) stands for the convex hall of a set A.

Any extremal point ϕ of S has the following form:

ϕ = (0, 0, . . . , 1 n

, 0, . . .),

for some n ∈ N. Such element is denoted by e(n).

Theorem 4.27. (Ref. 73) Every c.q.o. V defines an infinite-dimensional matrix

( pij,k)i,j,k∈N such that

pij,k ≥ 0, pij,k = pji,k,

∞k=1

pij,k = 1, i, j ∈ N. (4.29)

Conversely , every such matrix defines c.q.o. V as follows:

(V (x, y))k =∞

i,j=1

pij,kxiyj , k ∈ N, x = (xi), y = (yi) ∈ S . (4.30)

We note that in this case q.o. V defined by (4.4) has the following form:

(V (x))k =

∞i,j=1

pij,kxixj , k ∈ N, x = (xi) ∈ S . (4.31)

The constructed matrix ( pij,k)i,j,k∈N is called determining matrix of q.o. V .




One can see that the corresponding q.q.s.o. has the following form:

(P (x))ij =

∞

k=1

pij,kxk, i, j ∈ N, x = (xn) ∈ ∞.

Remark 4.28. Note that if T : ∞ → ∞ is a positive identity preserving oper-

ator. Then it is easy to see that such an operator can be represented as infinite-

dimensional stochastic matrix ( pij)i,j∈N, i.e. pij ≥ 0,∞

j=1 pij = 1 for every i, j ∈ N.

Remark 4.29. It is known that the set S is not compact in norm topology of

1, even in σ(1, ∞)-topology. This is the difference between finite- and infinite-

dimensional cases. In finite-dimensional case every q.o. V : S n−1 → S n−1 has at

least one fixed point. In the infinite-dimensional setting, not every q.o. has fixed

points. Indeed, define a q.o. V acting as follows:V (ϕ1, ϕ2, . . . , ϕn, . . .) = (0, ϕ1, ϕ2, . . . , ϕn, . . .),

where (ϕn) ∈ S . It is easy to see that this operator has no fixed points belonging

to S .Recall that a convex set C ⊂ S is called face, if λx + (1 − λ)y ∈ C , where x, y ∈

S , λ ∈ (0, 1), implies that x, y ∈ C . For ϕ, ψ ∈ S denote Γ(ϕ, ψ) = λϕ + (1 − λ)ψ :

λ ∈ [0, 1].

Definition 4.30. An operator V defined by (4.31) is called Volterra operator if V (ϕ, ψ) ∈ Γ(ϕ, ψ) is valid, for every ϕ, ψ ∈ Extr(S ).

By QV we denote the set of all quadratic operators defined on S , and the set of

all Volterra operators is denoted by V .Proposition 4.31. (Ref. 73) Let V ∈ QV be a q.o. Then V is Volterra if and only

if its determining matrix ( pij,k) satisfies the following property :

pij,k = 0, if k /∈ i, j. (4.32)

Theorem 4.32. (Ref. 73) Let V ∈ QV be a q.o. Then V is Volterra operator if and only if it can be represented as follows:

(V (x))k = xk

1 +

∞i=1

akixi

, k ∈ N, (4.33)

where aki = −aik, |aki| ≤ 1 for every k, i ∈ N.

Theorem 4.33. (Ref. 73) Let V ∈ V be a Volterra operator , then it is a bijection

of

S .

Remark 4.34. Note that in Ref. 78 we have introduced so-called M -Volterra oper-

ator, which is a generalization of Volterra one, and shown that bijectivity such kind

of nonlinear operators on S . In finite-dimensional setting such kind operators were

investigated in Ref. 39.




Now endow QV with a topology which is defined by the following system of

seminorms:

pϕ,ψ,k(V ) = |(V (ϕ, ψ))k|, V ∈ QV ,where ϕ, ψ ∈ S and k ∈ N. This topology is called weak topology and is denoted

by τ w.

A net V v of quadratic operators converges to V with respect to the defined

topology if for every ϕ, ψ ∈ S and k ∈ N,

(V v(ϕ, ψ))k → (V (ϕ, ψ))k

is valid.

Since V ∈ QV , therefore on V we consider the induced topology by QV . Note

that due to Theorem 4.2 the set of all quantum quadratic stochastic operatorsdefined on semi-finite von Neumann algebra, without normality condition, forms a

weak compact convex set. In the present situation the mentioned result cannot be

applied, since the q.q.s.o. under consideration are normal. In general, the set of all

normal q.q.s.o. is not weakly compact (see Ref. 73).

A q.o. V ∈ QV is called pure if for every ϕ, ψ ∈ Extr(S ) the relation holds

V (ϕ, ψ) ∈ ExtrΓ(ϕ, ψ) = ϕ, ψ.

It is clear that pure q.o. are Volterra.


(i) The set V is weakly convex compact ;

(ii) The set V is convex. Moreover , V is an extreme point of V if and only if it is

pure.

Now we give some limit theorems concerning trajectories of Volterra operators.

Let V : S → S be a Volterra operator. Then according to Theorem 4.32 it has the

form (4.33). Denote

Q =

y ∈ S : ∞i=1

akiyi ≤ 0, k ∈ N

.

It is clear that Q is convex subset of S .Proposition 4.36. (Ref. 73) For every Volterra operator V, one has Q ⊂ Fix(V ).

Theorem 4.37. (Ref. 73) Let V be a Volterra operator such that Q = ∅. Suppose

x0 ∈ riS (i.e. x0i > 0, ∀ i ∈ N) such that V x0 = x0 and the limit limn→∞ V nx0

exists. Then limn→∞ V nx0

∈Q.

Remark 4.38. It is known28 that the set Q is not empty for any Volterra operator

in finite-dimensional setting. But unfortunately, in infinite-dimensional case, Q may

be empty. In Ref. 73 some examples of Volterra q.o. for which Q is empty and non-

empty, respectively, are provided.




Now we are going to give a sufficient condition for V which ensures that the set

Q is not empty. Let V : S → S be a Volterra operator which has the form (4.33). Let

A = (aki) be the corresponding skew-symmetric matrix. Further, we assume that A

acts on 1. A matrix A is called finite-dimensional if A(1) is finite-dimensional.

A is called finitely generated if there are a sequence of finite-dimensional matrices

An such that supnAn < ∞ and

A = A1 ⊕ A2 ⊕ · · · ⊕ An ⊕ · · · .

Proposition 4.39. (Ref. 73) If a skew-symmetric matrix A = (aki), corresponding

to a Volterra operator (see (4.33)), is finitely generated. Then the set Q is not empty.

In Ref. 73 a construction of infinite-dimensional Volterra operators by meansof consistent sequence of finite-dimensional Volterra operators has been studied.

In our opinion such a construction allows us to investigate limiting behavior of

infinite-dimensional Volterra operators.

Problem 4.40. Investigate dynamics of infinite-dimensional Volterra operators.

4.6. Construction of q.s.o. infinite case

Construction of q.s.o. described in Sec. 2.11 corresponds to the case when E is

finite. But the construction does not work when E is not finite. In this subsection

we shall consider infinite set E . Let G = (Λ, L) be a countable graph. For a finite

set Φ denote by Ω the set of all functions σ : Λ → Φ. Let S (Ω, Φ) be the set of all

probability measures defined on (Ω,F), where F is the standard σ-algebra generated

by the finite-dimensional cylindrical set. Let µ be a measure on (Ω,F) such that

µ(B) > 0 for any finite-dimensional cylindrical set B ∈ F. Note that only Gibbs

measures have such a property.82

Fix a finite connected subset M ⊂ Λ. We say that σ ∈ Ω and ϕ ∈ Ω areequivalent if σ(x) = ϕ(x) for any x ∈ M , i.e. σ(M ) = ϕ(M ). Let ξ = Ωi, i =

1, 2, . . . , |Φ||M |, be the partition of Ω generated by this equivalent relation, where

Ωi contains all equivalent elements. Here, as before, | · | denotes the cardinality of

a set.

Denote

bij,k =

1 if i = j = k

µ(Ωk)

µ(Ωi) + µ(Ωj) , if k = i or k = j, i = j

0 otherwise,

(4.34)

where i,j,k ∈ 1, 2, . . . , |Φ||M |.




Assume

pσ1σ2,σ = bij,k if σ1 ∈ Ωi, σ2 ∈ Ωj , σ ∈ Ωk. (4.35)

Then the heredity coefficients (which do not depend on time) P (s, σ1, σ2, t , A) ≡P (σ1, σ2, A) (σ1, σ2 ∈ Ω, A ∈ F) are defined as

P (σ1, σ2, A) = Z (σ1, σ2)

A

pσ1σ2,σdµ(σ)

= Z (σ1, σ2)

·

µ(Ωi)µ(A ∩ Ωi) + µ(Ωj)µ(A ∩ Ωj), if σ1 ∈ Ωi, σ2 ∈ Ωj , i = j

µ(A ∩ Ωi) if σ1, σ2 ∈ Ωi.

(4.36)

where

Z (σ1, σ2) =

1

µ2(Ωi) + µ2(Ωj), if σ1 ∈ Ωi, σ2 ∈ Ωj , i = j

1

µ(Ωi)if σ1, σ2 ∈ Ωi.

The q.s.o. V acting on the set S (Ω, Φ) and determined by coefficients (4.36) is

defined as follows: for an arbitrary measure λ

∈S (Ω, Φ), the measure λ = V λ is

λ(A) = Ω

Ω

P (σ1, σ2, A)dλ(σ1)dλ(σ2)

=

|Φ||M|i=1

µ(A ∩ Ωi)

µ(Ωi)λ2(Ωi)

+ 2

1≤i<j≤|Φ||M|

µ(Ωi)µ(A ∩ Ωi) + µ(Ωj)µ(A ∩ Ωj)

µ2(Ωi) + µ2(Ωj)λ(Ωi)λ(Ωj). (4.37)

Now we shall describe the behavior of trajectories of the operator (4.37).By (4.37) we have

λ(Ωi) = λ2(Ωi) + 2

mj=1j=i

µ2(Ωi)

µ2(Ωi) + µ2(Ωj)λ(Ωi)λ(Ωj), (4.38)

where m = |Φ||M |.

Denoting λi = λ(Ωi), aij =µ2(Ωi)−µ

2(Ωj)µ2(Ωi)+µ2(Ωj)

from (4.38) we get

λi = λi

1 +

mj=1j=i

aijλj

, i = 1, . . . , m . (4.39)

It is easy to see that aij = −aji and |aij | < 1.




Note that the nth iteration, i.e. λ(n) = V (n)λ(0) of the operator (4.37) can be

written as

λ(n)(A) =

m

i=1

µ(A ∩ Ωi)

µ(Ωi)

(λ(n−1)i )2

+ 2

1≤i<j≤m


µ2(Ωi) + µ2(Ωj)λ(n−1)i λ

(n−1)j , (4.40)

where λ(n)j , j = 1, . . . , m are coordinates of the trajectory of operator (4.39) for the

given λ.

Thus in order to study the trajectory of operator (4.37) it is sufficient to know

the behavior of trajectories of the operator (4.39).

After a suitable renumbering of the classes Ωi we can assume thatµ(Ω1) = µ(Ω2) = · · · = µ(Ωr) > µ(Ωr+1) ≥ µ(Ωr+1) ≥ · · · ≥ µ(Ωm) > 0,

(4.41)

for some integer r ≥ 1.

Theorem 4.41. (Ref. 24) Assume for a given measure µ (4.41) holds, then for any

λ = (λ1, . . . , λm) ∈ S m−1 we have

limn→∞

λ(n) = (λ∗1, λ∗2, . . . , λ∗r , 0, 0, . . . , 0), (4.42)

where λ∗i = λiPrj=1 λj

.

If r = m, then V 2 = V i.e. V is stationary QSO.

Theorem 4.42. (Ref. 24) For any λ ∈ S (Ω, Φ) the trajectory λ(n)(A) of operator

(4.37) has the following limit

λ(A) = limn→∞

λ(n)(A)

=r

i=1

µ(A ∩ Ωi)

µ(Ωi)(λ∗i )2

+ 2

1≤i<j≤r


µ2(Ωi) + µ2(Ωj)λ∗i λ∗j , (4.43)

where λ∗ defined in Theorem 4.41 and λi = λ(Ωi).

Application to the Potts model. In this subsection we shall apply Theorem 4.42

to the Potts model on Z d. In this case Ω is the set of all configurations (cells)

σ : Z d → Φ = 1, . . . , q.

The (formal) Hamiltonian of the Potts model isH (σ) = −J

x,y∈Zd

x−y=1

δσ(x)σ(y),

where J ∈ R. (See Ref. 99 for details and definitions of Gibbs measure.)




It is well known (see Ref. 99, Theorem 2.3) that for the Potts model with q ≥ 2

there exists critical temperature T c such that for any T < T c there are q distinct

extreme Gibbs measures µi, i = 1, . . . , q . For T low enough, each measure µi is a

small deviation of the constant configuration σ

(i)

≡ i, i = 1, . . . , q . Thus condition(4.41) is satisfied for any measure µi with r = 1, i.e. µi(Ωi) > µi(Ωj) for any j = i,

here Ωi is the set of all configurations σ ∈ Ω such that σ(x) = i for any x ∈ M,

where M is a fixed (as above) subset of Z d.

As a corollary of Theorem 4.42 we get

Theorem 4.43. (Ref. 24) Any trajectory of q.s.o. (4.37) corresponding to measure

µi of the Potts model has the following limit :

limn→∞λ

(n)

(A) =

µi(A

∩Ωi)

µi(Ωi) , i = 1, . . . , q . (4.44)

Remark 4.44. (1) The R.H.S. of (4.44) is the conditional probability µi(A|Ωi).

Note (see Ref. 99) that µi(Ωi) → 1 if T → 0. Thus for T low enough, trajectory of

any measure λ ∈ S (Λ, Φ) with respect to q.s.o. constructed by µi tends to µi.

(2) Note that by the construction the heredity coefficients (4.36) depend on fixed

M . Consider an increasing sequence of connected, finite sets M 1 ⊂ M 2 ⊂ · · · ⊂M n ⊂ · · · such that ∪nM n = Z d. Fix two configurations σ1 and σ2 and consider

sequence of boundary conditions: σ1(Z

d

\M k) or σ2(Z

d

\M k), k = 1, 2, . . . . Thesesequences of subsets M k and boundary conditions (for the Potts model) define a

sequence of conditional Gibbs distributions µk on Ω (see Refs. 82 and 99). For any

measurable set A we denote by P (n)(σ1, σ2, A) the heredity coefficients (4.36) which

is constructed by M n and µn. An interesting problem is to describe the set of all

configurations σ1, σ2 such that the following limit exist

P (σ1, σ2, A) = limn→∞

P (n)(σ1, σ2, A). (4.45)

Note that if σ1 and σ2 are equal almost sure (i.e. the set

x∈

Z d : σ1(x)= σ2(x)

is finite) then the limit (4.45) exists. It is easy to see that the limit does not exist

if d = 1, and

σ1 = . . . , 1, 1, 2, 2, σ1(−1) = 1, σ1(0) = 1, σ1(1) = 1, 2, 2, 1, 1, . . .;

σ2 = . . . , 2, 2, 1, 1, 2, σ2(−1) = 2, σ2(0) = 1, σ2(1) = 2, 2, 1, 1, 2, . . ..

This example can be easily extended to case d > 1.

Problem 4.45. Find the set of configurations (σ1, σ2) for which the limit (4.45)

does not exist.

We have seen that if E is a continual set then one can associate the Gibbs

measure µ by a Hamiltonian H (defined on E ) and temperature T > 0.82 It is

known that depending on the Hamiltonian and the values of the temperature, the




measure µ can be non-unique. In this case there is a phase transition of the physical

system with the Hamiltonian H .

Problem 4.46. (by Ganikhodjaev) How will the thermodynamics (the phase

transition ) affect the behavior of the trajectories of a q.s.o. corresponding to a Gibbsmeasure of the Hamiltonian H ?

4.7. Remarks

Note that given q.q.s.o. on C ∗-algebra, one can define a dynamical system of q.q.s.o.s

via certain recurrent formula. Namely, let P be q.q.s.o. on a C ∗-algebra A, and ϕ

be an initial state on A. Define

P (n+1)

= P (n)

E ϕnP (1)

,where ϕn = ϕ⊗ ϕ P (n), P (1) = P . As before, E ϕ denote the canonical conditional

expectation from A ⊗ A to A. Here A ⊗ A denotes the completion of A A with

respect to the minimal C ∗-crossnorm. In Refs. 60, 61, 64, 65 and 67 uniform ergodic

theorems, weighted ergodic theorems, an analog of Blum–Hamson theorem, and

mixing properties of such kinds of dynamical systems were investigated.

Note in Ref. 52 several kinds of recurrent dynamics of cubic stochastic matrices

were investigated.

5. Quantum Quadratic Stochastic Processes

Let us note that QSPs defined in the previous section describe the physical systems.

However, they do not encompass quantum systems, it is natural to define quantum

quadratic stochastic processes (QQSP). Note that such systems also arise in the

study of biological and chemical processes at the quantum level. This section is

based on Refs. 14–18, 58–62, 71–79.

5.1. Quantum quadratic stochastic processes

Let M be a von Neumann algebra acting on a Hilbert space H . The set of all

continuous (respectively, ultraweak continuous) functionals on M is denoted by

M∗ (respectively, M∗), and put M∗,+ = M∗ ∩ M∗+, here M∗

+ denotes the set

of all positive linear functionals. By M ⊗ M we denote tensor product of M into

itself. The sets S and S 2 denote the set of all normal states on M and M ⊗ M,

respectively. As before by U we denote a linear operator U : M⊗M→M⊗M such

that U (x ⊗ y) = y ⊗ x for all x, y ∈ M. In what follows, as before, given a state

ϕ ∈ S , E ϕ denotes the conditional expectation from M ⊗ M to M. It is known102

that such an operator is completely positive. We refer the reader to Refs. 93 and

102, for more details on von Neumann algebras.

Now consider a family of linear operators P s,t : M → M ⊗ M, s , t ∈ R+,

t − s ≥ 1.




Definition 5.1. We say that a pair (P s,t, ω0), where ω0 ∈ S is an initial state,

forms a quantum quadratic stochastic process (QQSP), if every operator P s,t is

ultra weakly continuous and the following conditions hold:

(i) Each operator P s,t is a unital (i.e. preserves the identity operators) completelypositive mapping with UP s,t = P s,t;

(ii) An analogue of Kolmogorov–Chapman equation is satisfied: for initial state

ω0 ∈ S and arbitrary numbers s , τ , t ∈ R+ with τ − s ≥ 1, t − τ ≥ 1 one has

either

(ii)A P s,tx = P s,τ (E ω(P τ,tx)), x ∈ Mor

(ii)B P s,tx = E ωsP s,τ ⊗ E ωsP s,τ (P τ,tx), x ∈ M,

where ωτ (x) = ω0 ⊗ ω0(P 0,τ x), x ∈ M.

If QQSP satisfies one of the fundamental equations either (ii)A or (ii)B , then

we say that QQSP has type (A) or type (B ), respectively. Since the interaction

in physical, chemical and biological phenomena requires some time, we take this

interval for the unit of time (see Ref. 97). Therefore P s,t is defined for t − s ≥ 1.

Note that if we take instead of time s, t the discrete set N0 = N ∪ 0, then QQSP

is called discrete QQSP (DQQSP).

Remark 5.2. By using the QQSP, we can specify a law of interaction of states.

For ϕ, ψ ∈ S , we set

V s,t(ϕ, ψ)(x) = ϕ ⊗ ψ(P s,tx), x ∈ M.

This equality gives a rule according to which the state V s,t(ϕ, ψ) appears at time t

as a result of the interaction of states ϕ and ψ at time s. From the physical point

of view, the interaction of states can be explained as follows: Consider two physical

systems separated by a barrier and assume that one of these systems is in the stateϕ and the other one is in the state ψ. Upon the removal of the barrier, the new

physical system is in the state ϕ ⊗ ψ and as a result of the action of the operator

P s,t, a new state is formed. This state is exactly the result of the interaction of the

states ϕ and ψ.

Remark 5.3. If the algebra M is Abelian, that is, M = L∞(X, B), then the

QQSP coincides with the quadratic stochastic process.

Here are some examples of QQSPs.

Example 5.4. All commutative quadratic processes are QQSPs.

Example 5.5. (Ref. 18) Let M be a von Neumann algebra and let T : M → M be

an ultraweak continuous Markov operator. Assume that a state ω0 ∈ S is invariant




under T , that is ω0(T x) = ω0(x) for all x ∈ M. Then

P k,nx =1

2(T k,nx ⊗ 1 + 1 ⊗ T k,nx),

where

T k,nx =1

2n−k−1(T n−kx + (2n−k−1 − 1)ω0(x)1), x ∈ M, n > k,

k ∈ N0, n ∈ N,

is a DQQSP with the initial state ω0.

Example 5.6. (Ref. 18) Let T k,n : k < n, k ∈ N0, n ∈ N be a family of Markov

operators on

Mand let ω0

∈S be a state. Let

P k,nx =1

2(T k,nx ⊗ 1 + 1 ⊗ T k,nx), x ∈ M.

If the family T k,n satisfies the condition

T k,nx =1

2(T k,mT m,nx + ω0(T 0,mT m,nx)1), x ∈ M,

k < m < n, k ∈ N0, m , n ∈ N, and T k,k+1 = T 0,1 for all k ∈ N, then P k,n is

a DQQSP of type (A) with the initial state ω0. If the family T k,n satisfies the

condition

T k,nx =1

2(T k,mT m,nx + ω0(T 0,kT k,mT m,nx)1), x ∈ M,

k < m < n,k ∈ N0, m , n ∈ N, and T k,k+1 = T 0,1 for all k ∈ N, then P k,n is a

DQQSP of type (B) with the initial state ω0.

As in classic case, QQSPs can be divided into the following cases:

(I) homogeneous, i.e. P s,t depends only on t − s for any s ∈ R+ and n ∈ R+ such

that t ≥ s + 1;(II) homogeneous in duration of time unity , i.e. P t,t+1 = P 0,1 for all t ≥ 1;

(III) non-homogeneous which does not belong to class (II).

5.2. Expansion of QQSP

In this subsection we give an expansion of QQSP into fiberwise Markov processes.

Recall that by α∗0 we denote the norm dual to the smallest C ∗-crossnorm α0 on

M ⊗ M. Now let us consider a family of maps

H s,t :

M∗

⊗α∗0

M∗

→ M∗, s , t

∈R+, t − s ≥ 1 such that H s,t(· ⊗ ·) is a bilinear function on M∗ and the followingconditions hold:

(i) H s,t(S 2) ⊂ S ,

(ii) H s,t(ϕ ⊗ ψ) = H s,t(ψ ⊗ ϕ) for all ϕ, ψ ∈ S ,




(iii) one of the following equations holds for the initial state ω0 ∈ S and all numbers

s , τ , t ∈ N+ with τ − s ≥ 1 and t − τ ≥ 1:

(iii)a H s,t(ϕ) = H τ,t(H s,τ (ϕ)⊗τ ), ϕ ∈ S 2;

(iii)b H s,t(ϕ ⊗ ψ) = H τ,t(H s,τ (ωs ⊗ ϕ) ⊗ H s,τ (ωs ⊗ ψ)), ϕ , ψ ∈ S , whereωs(x) = H 0,s(ω0 ⊗ ω0)(x).

Such a family is denoted by (H s,t, ω0) and called a quadratic process (QP) of

type (A) (respectively, type (B)) when Eq. (iii)a (respectively, (iii)b) holds.

Note that every QQSP (P s,t, ω0) determines an associated QS(V s,t, ω0) given by

V s,t(ϕ)(a) = ϕ(P s,ta), ϕ ∈ M∗ ⊗α∗0M∗ → M∗, a ∈ M. (5.1)

It is clear that the QP (V s,t, ω0) is of type (A) or (B) if the QQSP(P s,t, ω0) has

the corresponding type.Given a QP, can one find a QQSP such that (5.1) holds? The answer is given

by the following proposition.

Proposition 5.7. (Ref. 68) Every QP (H s,t, ω0) determines a QQSP (P s,t, ω0) of

the corresponding type such that

H s,t(ϕ)(x) = ϕ(P s,tx), ϕ ∈ M∗ ⊗α∗0M∗, x ∈ M. (5.2)

Let T = T

s,t

(ϕ) : M → M : s, t ∈ R+, t − s ≥ 1, ϕ ∈ M∗ be a family of ultraweak continuous linear maps on a von Neumann algebra M.

Definition 5.8. A pair (T , ω0), where ω0 ∈ S is an initial state, is called a fiberwise

Markov process (FMP) if T s,t(·) is a linear function on M∗ for any fixed s, t and

the following conditions hold:

(i) T s,t(ϕ)1M = 1M for ϕ ∈ S ;

(ii) T s,t(ϕ) is completely positive for every ϕ ∈ M+;

(iii) one of the following equations holds, for the initial state ω0

∈S and arbitrary

s , τ , t ∈ R+ with τ − s ≥ 1 and t − τ ≥ 1:

(iii)A T s,t(ϕ) = T s,τ (ϕ)(H τ,t(ωτ )), ϕ ∈ S ;

(iii)B T s,t(ϕ) = T s,τ (ωs)T τ,t(T s,τ ∗ (ωs)ϕ), ϕ ∈ S ,

where ωτ (x) = ω0(T 0,τ (ω0)x), x ∈ M, and (T s,t∗ (ωs)ϕ)(x) = ϕ(T s,t(ωs)x), x ∈ M.

We say that the FMP(T , ω0) is of type (A) (respectively, type (B)) if the fun-

damental equation (iii)A (respectively, (iii)B) holds.

Theorem 5.9. (Ref. 68) Every QQSP (P s,t, ω0) uniquely determines an

FMP (T p, ω0) of the corresponding type given by

T s,t(ϕ)x = E ϕ(P s,tx), ϕ ∈ M∗, x ∈ M. (5.3)




The following equations hold for all ϕ, ψ ∈ S :

T s,t∗ (ϕ)ψ = T s,t∗ (ψ)ϕ, T s,t(ϕ) ≤ 1. (5.4)

Moreover ,

V s,t(ϕ ⊗ ψ) = T s,t∗ (ϕ)ψ, ∀ ϕ, ψ ∈ S.

The fiberwise Markov process (T p, ω0) defined by (5.3) is called the expansion

of the QQSP into a fiberwise Markov process. One can naturally ask the following

question: given an FMP, can we find an QQSP whose expansion into FMP coincides

with the given one?

An FMP is said to be symmetric if condition (5.4) holds.

Theorem 5.10. (Ref. 68) Every symmetric FMP (

T , ω0) determines

(i) a QP (H s,t, ω0) of the corresponding type such that

H s,t(ϕ ⊗ ψ) = T s,t∗ (ϕ)ψ, ∀ ϕ, ψ ∈ M∗;

(ii) a QQSP (P s,t, ω0) of the corresponding type such that

T s,t(ϕ)x = E ϕ(P s,tx), x ∈ M, ϕ ∈ M∗.

5.3. The ergodic principle

In this subsection we state the necessary and sufficient conditions for the ergodic

principle to be valid for QQSPs.

Let (P s,t, ω0) be a QQSP. Then by P s,t∗ we denote the linear operator, mapping

from (M ⊗ M)∗ into M∗, given by

P s,t∗ (ϕ)(x) = ϕ(P s,tx), ϕ ∈ (M ⊗ M)∗, x ∈ M.

Definition 5.11. A QQSP(P s,t, ω0) is said to satisfy the ergodic principle, if for

every ϕ, ψ ∈ S 2 and s ∈ R+

limt→∞

P s,t∗ ϕ − P s,t∗ ψ1 = 0,

where · 1 is the norm on M∗.

If the von Neumann algebra M is Abelian, then the definition of the ergodic

principle can be stated as follows96:

limt→∞

|P (s,x,y,t,A) − P (s,u,v,t,A)| = 0,

for any x,y,u,v ∈ E and A ∈ F .Let us note that Kolmogorov firstly introduced the concept of an ergodic prin-

ciple for Markov processes (see, for example, Ref. 46). For quadratic stochastic

processes this notion was introduced and studied in Refs. 96 and 94.

We say that a QQSP(P s,t, ω0) satisfies condition (A1) (respectively, condition

(A1) uniformly) on N ⊂ S 2 if there is λ ∈ [0, 1) such that for every pair ϕ, ψ ∈ N




and s ∈ R+ we have

P s,t∗ ϕ − P s,t∗ ψ1 ≤ λϕ − ψ1, (respectively, for all ϕ, ψ ∈ N )

and for at least one t0 = t0(s,ϕ,ψ).

Theorem 5.12. (Ref. 19) Let (P s,t, ω0) be a QQSP on a von Neumann algebra

M. The following conditions are equivalent :

(i) (P s,t, ω0) satisfies the ergodic principle;

(ii) (P s,t, ω0) satisfies condition (A1) on S 2;

(iii) (P s,t, ω0) satisfies condition (A1) on a dense subset F of S 2.

Remark 5.13. Note that, in commutative setting, ergodic principle was investi-

gated in Refs. 12, 14, 27 and 96.

5.4. The connection between the fiberwise Markov process

and the ergodic principle

In this subsection we formulate some results relating ergodic principle for QQSP

and FMP.


M and let (T , ω0) be its expansion into an FMP. Then the following conditions are

equivalent :

(i) the QQSP (P s,t, ω0) satisfies the ergodic principle;

(ii) for all σ,ϕ,ψ ∈ S and s ∈ R+ we have

limt→∞

T s,t∗ (σ)ϕ − T s,t∗ (σ)ψ1 = 0;

(iii) for all σ,ϕ,ψ ∈ R (where R is a dense subset of S ) and s ∈ R+ we have

limt→∞

T s,t∗ (σ)ϕ − T s,t∗ (σ)ψ1 = 0.

An FMP(T , ω0) is said to satisfy condition (E ) if there is a number λ ∈ [0, 1)

such that, given any σ,ϕ,ψ ∈ S and s ∈ R+ we have

T s,t∗ (σ)ϕ − T s,t∗ (σ)ψ1 ≤ λϕ − ψ1,

for at least one t = t(ϕ,ψ,σ,s) ∈ R+.

Theorem 5.15. (Ref. 68) Let (P s,t

, ω0) be a QQSP on a von Neumann algebra M and let (T , ω0) be its expansion into an FMP. Then the following conditions are

equivalent.

(i) The QQSP (P s,t, ω0) satisfies the ergodic principle;

(ii) The FMP (T , ω0) satisfies condition (E ).




Define a new process Qs,t : M → M by putting

Qs,tx = T s,t(ωs)(x), x ∈ M.

From Definition 5.8 one can see that Qs,t is a non-stationary Markov process.

Note that this a non-commutative analogous of Theorem 3.10. One can see that(see Ref. 19), if QQSP P s,t satisfies the ergodic principle, then the Markov process

Qs,t also satisfies this principle. However, the converse assertion was not previously

known, even in a commutative setting. Now using Theorem 5.14 we find the converse

assertion is also true.


M, and let Qs,t be the corresponding Markov process. Then the following conditions

are equivalent.

(i) The QQSP (P s,t, ω0) satisfies the ergodic principle;

(ii) The Markov process Qs,t satisfies the ergodic principle.

5.5. Marginal Markov processes

From the previous subsection one can see that each QQSP defines a Markov process,

but that Markov process cannot uniquely determine the QQSP. In this subsection

we are interested in the reconstruction result. Namely, we shall show that two

Markov processes (i.e. with above properties) can uniquely determine the given

q.q.s.p.Let M be a von Neumann algebra. Consider Qs,t : M → M and H s,t : M ⊗

M → M ⊗ M be two Markov processes with an initial state ω0 ∈ S . Denote

ϕt(x) = ω0(Q0,tx), ψt(x) = ω0 ⊗ ω0(H 0,t(x ⊗ 1)).

Assume that the given processes satisfy the following conditions:

(i) UH s,t = H s,t;

(ii) E ψsH s,t = Qs,tE ϕt ;

(iii) H

s,t

x = H

s,t

(E ψt(x) ⊗1

) for all x ∈ M ⊗ M.First note that if we take x = 1⊗ x in (iii) then we get

H s,t(1⊗ x) = H s,t(E ψt(1⊗ x) ⊗ 1)

= H s,t(ψt(x)1⊗ 1)

= ψt(x)1⊗ 1. (5.5)

Now from (ii) and (5.5), we have

E ψsH s,t(1

⊗x) = E ψs(ψt(x)1

⊗1)

= ψt(x)1

= Qs,tE ϕt(1⊗ x)

= ϕt(x)1.

This means that ϕt = ψt, therefore in the sequel we denote ωt := ϕt = ψt.




Now we are ready to formulate the result.

Theorem 5.17. (Ref. 70) Let Qs,t and H s,t be two Markov processes with

(i)–(iii) and initial state ω0

∈S . Then by the equality P s,tx = H s,t(x

⊗1) one

defines a QQSP (P s,t, ω0) of type (A). Moreover , one has

(a) P s,t = H s,τ P τ,t for any s , τ , t ∈ R+ with τ − s ≥ 1, t − θ ≥ 1,

(b) Qs,t = E ωsP s,t.

Theorem 5.18. (Ref. 70) Let Qs,t be a Markov process and Hs,t be another

process with initial state ω0 ∈ S, which satisfy (i)–(iii) and

Hs,t = Qs,τ ⊗ Qs,τ Hτ,t (5.6)

for any s , τ , t

∈R+ with τ

−s

≥1, t

−τ

≥1. Then by the equality P s,tx =

Hs,t(x ⊗ 1) one defines a QQSP (P s,t, ω0) of type (B ). Moreover , one has Qs,t =E ωsP s,t.

These two Qs,t and H s,t (respectively, Hs,t) Markov processes are called

marginal Markov processes associated with QQSP(P s,t, ω0).

Theorem 5.19. (Ref. 70) Let (P s,t, ω0) be a QQSP on M and let Qs,t, H s,tbe its marginal processes. Then the following conditions are equivalent :

(i) (P s,t, ω0) satisfies the ergodic principle;

(ii) Qs,t satisfies the ergodic principle;(iii) H s,t satisfies the ergodic principle.

5.6. The regularity condition

In this subsection describe the regularity of discrete quantum quadratic stochastic

processes.19

Definition 5.20. A DQQSP(P k,n, ω0) on M is said to satisfy the regularity (expo-

nential regularity) condition if there is a state µ1

∈S such that

limn→∞

P k,nϕ − µ11 = 0

for any ϕ ∈ S 2 and k ∈ N0 (respectively, P k,n∗ ϕ − µ11 ≤ d exp(−bn) for any

ϕ ∈ S 2 and n ≥ n0 with some n0 ∈ N, where d, b > 0).

In the case when the von Neumann algebra is commutative, the regularity con-

dition can be stated as follows: there is a probability (that is, normalized) measure

µ1 on (E, F ) such that

limn→∞ |P (k,x,y,n,A) − µ1(A)| = 0for any x, y ∈ E , A ∈ F , and k ∈ N 0.

Lemma 5.21. (Ref. 19) Assume that (P k,n, ω0) is a DQQSP on the von Neumann

algebra M. Then ω2 = ωn for any n ≥ 2.




Theorem 5.12 and Lemma 5.21 imply that the following theorem holds.

Theorem 5.22. (Ref. 19) Assume that (P k,n, ω0) is a homogeneous DQQSP on

the von Neumann algebra M

. Then the following conditions are equivalent :

(i) (P k,n, ω0) is a regular process (respectively , exponentially regular );

(ii) (P k,n, ω0) satisfies condition (A1) (respectively , uniformly ) on S 2;

(iii) (P k,n, ω0) satisfies condition (A1) (respectively , uniformly ) on a dense subset

F of S 2.

A DQQSP(P k,n, ω0) satisfies condition (A2) (uniformly) on N ∈ S 2 if one can

find a λ ∈ (0, 1] and a µ ∈ S such that for any ϕ ∈ N and k ∈ N0 one can find

a sequence

τ kn

n≥k+1

⊂ M+∗ and a number n0

∈N that satisfy the following

conditions:

(i) τ kn1 → 0 (respectively, supϕ∈N τ kn1 → 0) as n → ∞,

(ii) P k,n∗ ϕ + τ kn ≥ λµ for all n ≥ n0.

Theorem 5.23. (Refs. 19 and 66) Assume that (P k,n, ω0) is a homogeneous

DQQSP on the von Neumann algebra M. Then the following conditions are

equivalent :

(i) (P k,n, ω0) is regular (respectively , exponentially regular ),(ii) (P k,n, ω0) satisfies condition (A2) (respectively , uniformly ) on S 2,

(iii) (P k,n, ω0) satisfies condition (A2) (respectively , uniformly ) on a dense subset

K whose convex hull is dense in S 2.

5.7. Differential equations for QQSP

Consider a QQSP (P s,t, ω0) on a von Neumann algebra M from class (II). We shall

assume that P s,t

is continuous and differentiable with respect to s and t.Denote

A(t)x = limδ→0+

(P t−1,t+δ − P 0,1)x

δ

and the set of all x ∈ M for which the last limit exists is denoted by D(A).

Theorem 5.24. (Ref. 20) Let (P s,t, ω0) be a QQSP of type (A) from class (II) on

a von Neumann algebra

M. Then it satisfies the following differential equations

∂P s,tx

∂t= (P s,t−1 ⊗ P 0,t−1∗ (ω0 ⊗ ω0))(A(t)x), x ∈ D(A) (5.7)

∂P s,tx

∂s= −(A(s + 1) ⊗ P 0,s+1

∗ (ω0 ⊗ ω0))(P s+1,tx). (5.8)




Now assume that a QQSP is homogeneous, i.e. from class (I), then due to

Lemma 5.21, Eq. (5.7) reduces to

∂P tx

∂t= (P t−1

⊗P t−1∗ (ω0

⊗ω0))(Ax), 2 < t

≤3

∂P tx

∂t= (P t−1 ⊗ ω2)(Ax), t ≥ 3.

(5.9)

One of the important problems is to find the process P t from Eq. (5.9) with a given

boundary condition:

P t = Qt, 1 ≤ t ≤ 2,

where Qt is a given process, such that dQt

dt|t=1 = A.

Let us consider a simple boundary value problem:

∂P tx

∂t= (P t−1 ⊗ P t−1∗ (ω0 ⊗ ω0))(Ax), 2 ≤ t

P tx = Qtx, 1 ≤ t ≤ 2,

(5.10)

where

Qtx =1

2(β (t)x ⊗ 1 + 1 ⊗ β (t)x), 1 ≤ t ≤ 2,

β (t)x =1

2t−1(T tx + (2t−1

−1)ω

0(x)1), x

∈ M,

and T t is a completely positive Markov semigroup, i.e. the following conditions

hold: for every t ∈ R+:

(i) T t is continuous differentiable (in the sense of norm);

(ii) ω0(T tx) = ω0(x), x ∈ M,

and A is defined as follows:

Ax =dQtx

dt t=1

=1

2(BT x ⊗ 1 + 1⊗ BT x − ln2(T x ⊗ 1 + 1⊗ T x) + 2ln 2ω0(x)1),

where B is the generator of the semigroup T t.

Theorem 5.25. (Ref. 20) For the problem (5.10) there exists unique solution which

has the following form :

P tx =1

2(β (t)x ⊗ 1 + 1⊗ β (t)x), 1 ≤ t,

β (t)x =1

2t−1(T tx + (2t−1 − 1)ω0(x)1), x ∈ M.

Problem 5.26. Find conditions to A(t) which ensures the existence of solutions

of Eqs. (5.7) and (5.8).




Acknowledgments

This work was done within the scheme of Junior Associate at the ICTP, Trieste,

Italy, and F.M. and U.R. thank ICTP for providing financial support and all facili-

ties for his several visits to ICTP during 2005–2010. U.R. also supported by TWASResearch Grant No.: 09-009 RG/MATHS/AS−I–UNESCO FR:3240230333. The

authors (R.G. and F.M.) acknowledge the MOSTI grants 01-01-08-SF0079 and

CLB10-04. We are grateful to both referees for their suggestions which improved

the style and contents of the paper.

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