Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 14, No. 2 (2011) 279– 335 c 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, T ashkent, Uzb ekistanFARRUKH 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, T ashkent, Uzbekistan[email protected]Received 11 May 2009 Revised 22 November 2010 Communicated by R. Rebolledo The history of the quadratic stochastic operators can be traced back to the work ofBernshtein (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 with its 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 are not accessible. In this paper we give all necessary definitions and a brief description ofthe results for three cases: (i) discrete-time dynamical systems generated by quadratic stoc hasti c operat ors; (ii) con tin uous-time stochasti c processe s gen erated by quad ratic operat ors; (iii) quantu m quadratic stochastic operat ors and processes. Moreover, we discuss several open problems. Keywords : Qua dratic stochastic ope rat or; quadratic stochas tic process; qua ntum quadratic 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
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8/2/2019 Quadratic Stochastic Operators and Processes
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
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.
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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.
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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,
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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.
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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.
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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 .
Theorem 2.28. (Ref. 40) The following assertions hold true:
(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.
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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
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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 .
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(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).
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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,
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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).
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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
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 .
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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.
Theorem 4.35. (Ref. 73) The following assertions hold true:
(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.
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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 |.
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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
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
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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.
Theorem 5.16. (Ref. 68) Let (P s,t, ω0) be a QQSP on a von Neumann algebra
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.
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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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