On the Lebesgue nonlinear transformations

0.14 MB·

by  Nasir Ganikhodjaev

ON LEBESGUE NONLINEAR TRANSFORMATIONS
NASIRGANIKHODJAEV,MANSOORSABUROV,AND RAMAZONMUHITDINOV
6
Abstract. In this paper, we introduce a quadratic stochastic operators on the set
1
of all probability measures of a measurable space. We study the dynamics of the
0
2 Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set
[0,1]. Namely,we provethe regularity of the Lebesgue quadratic stochastic operators.
n
a Mathematics Subject Classification: 47HXX,46TXX.
J
Keywords: Quadraticoperator,measurablespace,Lebesguenonlineartransformation.
8

]
S
D
1. Introduction
.
h
Quadratic stochastic operator (in shortQSO)was firstintroduced in Bernstein’s work
t
a [1]. The QSO was considered an important source of analysis for the study of dynamical
m
properties and modeling in various fields such as biology [13, 14], physics [24], game
[
theory [3], control system [22, 23]. Such operator frequently arises in many models of

1 mathematical genetics [5, 6]. A fixed point set and an omega limiting set of quadratic
v stochastic operators defined on the finite dimensional simplex were deeply studied in
7
[8, 11] and [15, 17]. Ergodicity and chaotic dynamics of QSO on the finite dimensional
5
7 simplex were studied in the papers [3], [6, 7], [18, 21], and [25]. In [12], it was given a
1 long self-contained exposition of recent achievements and open problems in the theory
0
of quadratic stochastic operators. The analytic theory of stochastic processes generated
.
1 by quadratic operators was established in [2]. In the paper [4], the nonlinear Poisson
0
quadratic stochastic operators over the countable state space was studied. In this paper,
6
1 we shall study the dynamics of Lebesgue quadratic stochastic operators on the set of all
: Lebesgue measures of the set [0,1]. Let us first recall some notions and notations (see
v
i [2],[4]).
X
Let (X,F) be a measurable space and S(X,F) be the set of all probability measures
ar on (X,F), where X is a state space and F is σ-algebra of subsets of X. It is known that
theset S(X,F) is acompact, convex space and aform of Dirac measureδ which defined
x
by
1 if x A
δx(A) = (cid:26) 0 if x ∈/ A
∈
for any A B is extremal element of S(X,B).
Let P(∈x,y,A) :x,y X,A F be a family of functions on X X F that satisfy
{ ∈ ∈ } × ×
the following conditions:
Date: January 11, 2016.
1
2 NASIRGANIKHODJAEV,MANSOORSABUROV,ANDRAMAZONMUHITDINOV
i) P(x,y, ) S(X,F), for any fixed x,y X, that is, P(x,y, ) : F [0,1] is the proba-
bility mea·su∈re on F; ∈ · →
ii) P(x,y,A) regarded as a function of two variables x and y with fixed A F is mea-
surable function on (X X,F F); ∈
iii) P(x,y,A) = P(y,x,×A) for a⊗ny x,y X,A F.
We consider a nonlinear transformation∈(quad∈ratic stochastic operator) V : S(X,F)
S(X,F) defined by →
(Vλ)(A) = P(x,y,A)dλ(x)dλ(x), (1.1)
Z Z
X X
where λ S(X,F) is an arbitrary initial probability measure and A F is an arbitrary
∈ ∈
measurable set.
Definition 1.1. A probability measure µ on (X,F) is said to be discrete, if there exists
a countable set of elements x ,x , X, such that µ( x ) = p for i = 1,2, ,
1 2 i i
with p = 1. Then µ(X {x ,x , ···}) ⊂= 0 and for any A{ F},µ(A) = µ( x···).
i i \{ 1 2 ···} ∈ xi∈A { i}
P P
A family P(x,y,A) : x,y X,A F on arbitrary state space X, such that for any
{ ∈ ∈ }
x,y X a measure P(x,y, ) is a discrete measure, is shown in the following example.
Exa∈mple 1 Let (X,F) be·a measurable space. For any x,y X and A F, assume
∈ ∈
0 if x / A and y / A,
∈ ∈
P(x,y,A) =  1 if x A,y / A or x / A,y A,
 12 if x ∈ A an∈d y A. ∈ ∈
∈ ∈
Assume Vnλ : n = 0,1,2, is the trajectory of the initial point λ S(X,F), where
{ ···} ∈
Vn+1λ = V(Vnλ) for all n = 0,1,2, , with V0λ = λ.
···
Definition 1.2. A quadratic stochastic operator V is called a regular (weak regu-
lar), if for any initial measure λ S(X,F), the strong limit (respectively weak limit)
∈
lim Vn(λ) = µ exists.
n→∞
It is easy to verify that the quadratic stochastic operator V generated by this family
is identity transformation, that is for any measure λ S(X,F) we have Vλ = λ. In fact,
for any A F, ∈
∈
Vλ(A) = P(x,y,A)dλ(x)dλ(y) = 1 dλ(x)dλ(y)
Z Z Z Z ·
X X A A
1 1
+ dλ(x)dλ(y)+ dλ(x)dλ(y)+ 0 dλ(x)dλ(y)
ZAZAc 2 · ZAcZA 2 · ZAcZAc ·
1 1
= λ2(A)+ λ(A)(1 λ(A))+ (1 λ(A))λ(A) = λ(A),
2 − 2 −
where Ac = X A.
\
If a state space X = 1,2, ,m is a finite set and corresponding σ-algebra is a power
{ ··· }
set (X), i.e., the set of all subsets of X, then the set of all probability measures on
P
ON LEBESGUE NONLINEAR TRANSFORMATIONS 3
(X, ) has the following form:
F
m
Sm−1 = x = (x ,x , ,x ) Rm : x 0 for any i, and x = 1 (1.2)
1 2 m i i
{ ··· ∈ ≥ }
Xi=1
that is called a (m 1)-dimensional simplex.
−
In this case for any i,j X a probabilistic measure P(i,j, ) is a discrete measure with
m ∈ ·
P(ij, k ) = 1,whereP(ij, k ) P andcorrespondingqsoVhas thefollowing
k=1 { } { } ≡ ij,k
Pform
m
(Vx) = P x x (1.3)
k ij,k i j
iX,j=1
for any x Sm−1 and for all k = 1, ,m, where
∈ ···
m
a)P 0, b)P = P for all i,j,k; c) P = 1.
ij,k ij,k ji,k ij,k
≥
Xk=1
Such operator can be reinterpreted in terms of evolutionary operator of free population
and in this form it has a fair history. Note that the theory of qso with finite state space
is well developed and on the whole all well-known papers devoted to such qso [1]-[3],
[5]-[25]. In [4] the authors studied qso with infinite countable state space.
In this paper we construct the family of quadratic stochastic operators defined on the
continual state space X = [0,1) and investigate their trajectory behavior.
Definition 1.3. A transformation V is called a Lebesgue qso, if X = [0,1) and F is a
Borel σ-algebra B on [0,1).
In the next section we present a family of Lebesgue qso.
2. A construction of Lebesgue qso
Let X = [0,1) and B is a Borel σ-algebra on [0,1). For any element (x,y) X X,
∈ ×
we define a discrete probability P(x,y, ) as follows:
·
(i) for x < y assume P(x,y, x ) = p and P(x,y, y ) = p, (2.1)
{ } { }
(ii) for x = y assume P(x,x, x ) = 1, (2.2)
{ }
(iii) for x > y assume P(y,x, ) = P(x,y, ) (2.3)
· ·
where p+q = 1, with p 0 and q 0.
≥ ≥
Let V is a quadratic stochastic operator
(Vλ)(A) = P(x,y,A)dλ(x)dλ(x), (2.4)
Z Z
X X
generated by family of functions (2.1)-(2.3), where λ S(X,F) is an arbitrary initial
probability measureandA F isan arbitrary measurab∈leset. Thisoperator is anatural
∈
generalization of Volterra qso. Note that if p = q = 0.5, then the correspondingoperator
is the identity operator.
4 NASIRGANIKHODJAEV,MANSOORSABUROV,ANDRAMAZONMUHITDINOV
We show that for any initial measure λ S(X,F), there exists strong limit of the
∈
sequence Vnλ : n = 0,1,2, .
{ ···}
3. A Limit behaviour of the trajectories
In this section we study the limit behaviour of the trajectory Vnλ : n = 0,1,2,
for any initial measure λ S(X,F). { ···}
∈
3.1. A discrete initial measure λ. It is easy to verify that for any a [0,1) an
∈
extremal Dirac measure δ is a fixed point of the operator V. Since δ ( a ) = 1, then
a a
{ }
from (2.2) we have
(Vδ )( a ) = P(x,y, a )dδ (x)dδ (y) = P(a,a, a ) = 1 = δ ( a ), (3.1)
a a a a
{ } Z Z { } { } { }
X X
that is the Dirac measure δ for any a [0,1) is a fixed point.
a
∈
Let a measure λ is a convex linear combination of two Dirac measures δ and δ , i.e.,
a b
λ = αδ +(1 α)δ , where α [0,1] and a,b [0,1] with a < b. Simple algebra gives
a b
− ∈ ∈
(Vλ)( a ) = P(x,y, a )dλ(x)dλ(y) = λ(a)[λ(a)+2pλ(b)], (3.2)
{ } Z Z { }
X X
and
(Vλ)( b ) = P(x,y, b )dλ(x)dλ(y) = λ(b)[λ(b)+2qλ(a)], (3.3)
{ } Z Z { }
X X
i.e., Vλ is the convex linear of the same two Dirac measures δ and δ with
a b
Vλ = α δ +(1 α )δ ,
1 a 1 b
−
where α = α[α+2p(1 α)] and 1 α = (1 α)[1 α+2qα]. Then it is evident that
1 1
− − − −
V2λ is the convex linear of the same two Dirac measures δ and δ with
a b
V2λ = α δ +(1 α )δ ,
2 a 2 b
−
where α = α [α +2p(1 α )] and 1 α = (1 α )[1 α +2qα ]. Thus one can show
2 1 1 1 2 1 1 1
that Vnλ is the convex li−near of the s−ame two D−irac me−asures δ and δ with
a b
Vnλ =α δ +(1 α )δ ,
n a n b
−
where α = α [α +2p(1 α )] and 1 α = (1 α )[1 α +2qα ].
n n−1 n−1 n−1 n n−1 n−1 n−1
− − − −
Simple calculus gives
1 if p > 1
lim α = 2
n→∞ n (cid:26) 0 if p < 1
2
that is
δ if p > 1
lim Vnλ = a 2
n→∞ (cid:26) δb if p < 12
Let now a measure λ is a convex linear combination of three Dirac measures δ ,δ and
a b
δ , i.e., λ = αδ +βδ +(1 α β)δ , where α,β [0,1] and a,b,c [0,1] with α+β 1
c a b c
− − ∈ ∈ ≤
and a < b < c. Simple algebra gives
(Vλ)( a ) = P(x,y, a )dλ(x)dλ(y) = λ(a)[λ(a)+2p(λ(b)+λ(c)], (3.4)
{ } Z Z { }
X X
ON LEBESGUE NONLINEAR TRANSFORMATIONS 5
(Vλ)( b ) = P(x,y, a )dλ(x)dλ(y) = λ(b)[λ(b)+2qλ(a)+2pλ(c)], (3.5)
{ } Z Z { }
X X
and
(Vλ)( c ) = P(x,y, a )dλ(x)dλ(y) = λ(c)[λ(c)+2q(λ(a)+λ(b)], (3.6)
{ } Z Z { }
X X
i.e., Vλ is the convex linear of the same three Dirac measures δ ,δ and δ with
a b c
Vλ = α δ +β δ +(1 α β )δ ,
1 a 1 b 1 1 c
− −
where α = α[α + 2p(1 α)],β = β[β + 2qα + 2p(1 α β)] and 1 α β =
1 1 1 1
− − − − −
(1 α β)[1 α β+2q(1 α β]. Then it is evident that V2λ is the convex linear of
− − − − − −
the same three Dirac measures δ ,δ and δ , with V2λ = α δ +β δ +(1 α β )δ ,
a b c 2 a 2 b 2 2 c
− −
where α = α [α +2p(1 α )],β = β [β +2qα +2p(1 α β )] and 1 α β =
2 1 1 1 2 1 1 1 1 1 2 2
− − − − −
(1 α β )[1 α β +2q(1 α β ]. Thus one can show that Vnλ is the convex
1 1 1 1 1 1
− − − − − −
linear of the same three Dirac measures δ ,δ and δ with
a b c
Vnλ = α δ +β δ +(1 α β )δ ,
n a n b n n c
− −
where α =α [α +2p(1 α )],β = β [β +2qα +2p(1 α β )]
n n−1 n−1 n−1 n n−1 n−1 n−1 n−1 n−1
− − −
and 1 α β = (1 α β )[1 α β +2q(1 α β ]. As noted
n n n−1 n−1 n−1 n−1 n−1 n−1
− − − − − − − −
above if p > 1 then lim α = 1 and if p < 1 then lim (1 α β ) = 1 that is
2 n→∞ n 2 n→∞ − n− n
δ if p > 1
lim Vnλ = a 2
n→∞ (cid:26) δc if p < 12
Let a measureλ is aconvex linear combination of n Dirac measures δ ,i = 1, ,n
n { ai ··· }
i.e., λ = α δ , where α [0,1],i = 1, ,n and a [0,1],i = 1, ,n with
n i=1 i ai i ∈ ··· i ∈ ···
i=1α =P1 and a1 < a2 < ··· < an. Simple algebra gives
P
(Vλ)( a ) = P(x,y, a )dλ(x)dλ(y) = λ(a )[λ(a )+2p(1 λ(a ))], (3.7)
1 1 1 1 1
{ } Z Z { } −
X X
and
(Vλ)( a ) = P(x,y, a )dλ(x)dλ(y) = λ(a )[λ(a )+2q(1 λ(a ))], (3.8)
n n n n n
{ } Z Z { } −
X X
As shown above we have
δ if p > 1
lim Vnλ = a1 2
n→∞ (cid:26) δan if p < 12
6 NASIRGANIKHODJAEV,MANSOORSABUROV,ANDRAMAZONMUHITDINOV
3.2. A continuous initial measure λ. Let λ S(X,B) be a continuous probability
measure and A= [a,b] B be a segment in X wi∈th Ac = [0,a) (b,1). Then,
∈ ∪
b b b a b 1
Vλ(A) = 1 dλ(x)dλ(y)+ p dλ(x)dλ(y)+ q dλ(x)dλ(y)
Z Z · Z Z · Z Z ·
a a a 0 a b
a b 1 b a a
+ p dλ(x)dλ(y)+ q dλ(x)dλ(y)+ 0 dλ(x)dλ(y)
Z Z · Z Z · Z Z ·
0 a b a 0 0
a 1 1 a 1 1
+ 0 dλ(x)dλ(y)+ 0 dλ(x)dλ(y)+ 0 dλ(x)dλ(y)
Z Z · Z Z · Z Z ·
0 b b 0 b b
= λ([a,b])[λ([a,b])+2pλ([0,a))+2qλ((b,1))].
It is evident that the measure Vλ is absolutely continuous with respect to measure λ.
ThenaccordingRadon-NikodymTheorem,thereexistsnon-negativemeasurablefunction
(1)
f : X R called a density, such that
λ →
(1)
Vλ(A) = f (x)dλ(x). (3.9)
Z λ
A
The derivations of the density functions are presented as follows. For rather small seg-
ment [x,x+∆x] we have
Vλ([x,x+∆x]) = λ([x,x+∆x])[λ([x,x+∆x])+2pλ([0,x))+2qλ((x+∆x,1))] (3.10)
and
Vλ([x,x+∆x])
(1)
f (x) = lim
λ ∆x→0 λ([x,x+∆x])
= lim [λ([x,x+∆x])+2pλ([0,x))+2qλ((x+∆x,1))]
∆x→0
= 2pλ([0,x))+2qλ((x,1)).
Now consider a measure V2λ = V(Vλ). It is evident that
V2λ(A) = f(1)(x)dVλ(x). (3.11)
Z Vλ
A
and since V2λ is absolutely continuous with respect to measure λ, we have
V2λ(A) = f(2)(x)dλ(x). (3.12)
Z λ
A
According (2.6) we have
V2λ([x,x+∆x])
= Vλ([x,x+∆x])[Vλ([x,x+∆x])+2pVλ([0,x))+2qVλ((x+∆x,1))]
= λ([x,x+∆x])[λ([x,x+∆x])+2pλ([0,x))+2qλ((x+∆x,1))]
λ([x,x+∆x])[λ([x,x+∆x])+2pλ([0,x))+2qλ((x+∆x,1))]
·{
+2pλ([0,x])[λ([0,x])+2qλ((x,1))]+
+2qλ([x+∆x,1))[λ([x+∆x,1))+2pλ([0,x+∆x))
}
ON LEBESGUE NONLINEAR TRANSFORMATIONS 7
Then
(2)
f (x) = [2pλ([0,x))+2qλ([x,1))]
λ
2pλ([0,x])[λ([0,x)) +2qλ([x,1))]+2qλ([x,1))[λ([x,1))+2pλ([0,x]) .
·{ }
Similarly, one can show that a measure Vnλ is absolutely continuous with respect to λ
for any n and
Vnλ(A) = f(n)(x)dλ(x). (3.13)
Z λ
A
(n) (n−1) (n−1) (n−1)
Let g (x) = λ([0,x)) and g (x) = g (x)(g (x) + 2q(1 g (x)), for n =
λ λ λ λ − λ
(0) (1)
1,2,3, , where g (x) = x and g (x) =g (x). It is evident that
··· λ λ λ
(n) (n−1) (n−1) (n−1)
1 g (x) = (1 g (x))(1 g (x)+2pg (x)).
− λ − λ − λ λ
Then, since λ([x,1)) = 1 λ([0,x)), we have
−
f(x) = 2px+2q(1 x), (3.14)
−
(1) (1)
f (x) = f(g (x)), (3.15)
λ λ
and
(2) (1) (2)
f (x) = f(g (x)) f(g (x)). (3.16)
λ λ · λ
Using induction, one can prove that for any n we have
n
(n) (i)
f (x) = f(g (x)). (3.17)
λ λ
Yi=1
It is easy to see that f(n)(0) = (2q)n and f(n)(1) = (2p)n. Since
λ λ
1
(n)
f (x)dλ(x) = 1,
Z λ
0
one has that
(n) (n)
f (0) 0 and f (1) if p > 1/2, (3.18)
λ → λ → ∞
and
(n) (n)
f (0) and f (1) 0 if p < 1/2. (3.19)
λ → ∞ λ →
(1)
If λ = m is a usual Lebesgue measure on [0,1], then m([0,x)) = x and g (x) = x. In
m
(n)
this case one can explicitly find the functions f (x) for any n.
m
8 NASIRGANIKHODJAEV,MANSOORSABUROV,ANDRAMAZONMUHITDINOV
4. Regularity of Lebesgue qso
Now, we are aiming to study the limit behavior of the Radon-Nikodym derivatives
(n)
f () for n . Let G(x) = x(x+2q(1 x)) for p,q 0 and p+q = 1. We always
λ · → ∞ − ≥
assume that p,q = 1. One can easily check that G′(x) = f(x) and G′′(x) = f′(x) =
6 2
2(p q). Since f(x) 0 for any x [0,1], the function G : [0,1] [0,1] is increasing.
Mor−eover, the functi≥on f : [0,1] ∈R is increasing whenever p→> q (or equivalently
+
→
p > 1) and decreasing whenever p < q (or equivalently p < 1).
2 2
(n) (n−1)
It is easy to check that g (x) = G g (x) . We know that g : [0,1] [0,1],
λ λ λ →
(cid:16) (cid:17)
g (x) = λ([0,x)) is the increasing function. We suppose that the function g : [0,1]
λ λ
→
[0,1] is also differentiable. It is clear that
(g(n)(x))′ = G′ g(n−1)(x) (g(n−1)(x))′
λ λ · λ
(cid:16) (cid:17)
Consequently, g(n) : [0,1] [0,1], for all n N, are differentiable and increasing func-
λ → ∈
tions.
Proposition 4.1. Let f(n) : [0,1] R , n N be functions given by (3.17). Then, the
λ → + ∈
function f(n) is increasing whenever p > 1 and decreasing whenever p < 1.
λ 2 2
Proof. We know that
n n
′
(fλ(n)(x))′ =  f(gi)λ(x)·f′(gλ(k)(x))· gλ(k)(x) .
Xk=1 i=1Y, i6=k (cid:16) (cid:17)
 
n ′
Since f(gi) (x) 0 and g(k)(x) 0, the function f(n) is increasing whenever
λ ≥ λ ≥ λ
i=1Q, i6=k (cid:16) (cid:17)
p > 1 and decreasing whenever p < 1. This completes the proof. (cid:3)
2 2
Let p > 1 and β = 1−n1 . It is clear that 0 < β < 1 and lim β = 1. So, since
2 n n−√1 16p4 n n→∞ n
2q < 1, one has that β >2q for large n. Moreover, we have that 2p βn−1 < 1 < 1.
n n 2p
Let B = βn−2q. One has that 0 < B < 1 and lim B =1. p
n 1−2q n n→∞ n
Proposition 4.2. Let p > 1 and β ,B be given as above. The following statements
2 n n
hold true:
(i) One has that G(x) β x for any x [0,B ];
n n
≤ ∈
(ii) One has that G[0,B ]= [0,β B ] [0,B ].
n n n n
⊂
Proof. Since 0 x B = βn−2q, we have that x + 2q(1 x) β or equivalently
≤ ≤ n 1−2q − ≤ n
G(x) β x. On the other hand, since G(x) is the increasing function, we get that
n
0= G≤(0) G(x) G(B ) = β B < B . (cid:3)
n n n n
≤ ≤
Corollary 4.3. Let p > 1. One has that 0 g(i)(x) βi−1g (x), x [0,B ],i = 2,n.
2 ≤ λ ≤ n λ ∀ ∈ n
ON LEBESGUE NONLINEAR TRANSFORMATIONS 9
Theorem 4.4. Let p > 1. One has that 0 f(n)(x) ( 1 )n for any x [0,B ].
2 ≤ λ ≤ 2p ∈ n
n
(n) (i)
Proof. We know that f (x) = f(g (x)). Due to Corollary 4.3, since f is the in-
λ λ
iQ=1
creasing linear function, we then obtain for any x [0,B ] that
n
∈
n n
f(n)(x) βi−1f(g (x)) = βn−1f(g (x)) .
λ ≤ n λ (cid:18)q n λ (cid:19)
Yi=1
n n
Since 2q f(g (x)) 2p, we have that 0 f(n)(x) 2p βn−1 1 for any
≤ λ ≤ ≤ λ ≤ n ≤ 2p
x [0,B ]. This completes the proof. (cid:16) p (cid:17) (cid:16) (cid:17) (cid:3)
n
∈
Similarly, one can prove the following result.
Theorem 4.5. Let p < 1. There exists 0 < A < 1 such that lim A = 0 and 0
2 n n→∞ n ≤
f(n)(x) ( 1 )n for any x [A ,1].
λ ≤ 2q ∈ n
(n)
Hence, the sequence of functions f (x) has a tall spike at the end points of the
λ
segment [0,1] whenever p,q = 1. Consequently, the (weak) limit of this sequence of
6 2
(n)
the functions f (x) is the Dirac delta function concentrated at the end points of the
λ
segment [0,1].
Theorem 4.6. Let V be a qso generated by family of functions (2.1)-(2.3). Then for
any initial continuous measure λ S(X,B) there exists a strong limit of the sequence of
∈
measures Vkλ where
{ }
δ if p < 1
lim Vnλ = 0 2
n→∞ (cid:26) δ1 if p > 12.
Recall that if p =q = 1, then the corresponding qso is the identity transformation.
2
Corollary 4.7. The Lebesgue quadratic stochastic operator V generated by family of
functions (2.1)-(2.3) is a regular transformation.
Acknowledgement
The work has been supported by the MOHE grant FRGS14-116-0357. The third
author (R.M.) wishes to thank International Islamic University Malaysia (IIUM), where
this paper was written, for the invitation and hospitality.
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