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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА.
_ СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
2018, Т. 160, кн. 2 С. 384-391
ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)
UDK 519.2
SHIFT-INVARIANT MEASURES ON INFINITE-DIMENSIONAL SPACES: INTEGRABLE FUNCTIONS AND RANDOM WALKS
V.Zh. Sakbaev, D.V. Zavadsky
Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia
Abstract
Averaging of random shift operators on a space of the square integrable by shift-invariant measure complex-valued functions on linear topological spaces has been studied. The case of the space has been considered as an example.
A shift-invariant measure on the space, which was constructed by Caratheodory's scheme, is a-additive, but not a-finite. Furthermore, various approximations of measurable sets have been investigated. One-parameter groups of shifts along constant vector fields in the
space and semigroups of shifts to a random vector, the distribution of which is given by a collection of the Gaussian measures, have been discussed. A criterion of strong continuity for a semigroup of shifts along a constant vector field has been established.
Conditions for collection of the Gaussian measures, which guarantee the semigroup property and strong continuity of averaged one-parameter collection of linear operators, have been defined.
Keywords: strongly continuous semigroups, averaging of operator semigroups, shift-invariant measures, square integrable functions
Introduction
In this paper, we investigate semigroups of shift operators on a Hilbert space L2 (lB(lo),A) of complex square integrable functions by a shift-invariant and a -additive measure A on a space lo. Data on various constructions of shift-invariant measures on linear topological spaces and possible connections between the measures are presented in [1-4].
We consider semigroups indexed by parameter t G R+ = [0, of bounded operators Ah, which are defined in the following way: Athf (x) = f (x + th), where h G lo, f G L2(lB(lo),A). We establish a criterion of strong continuity for operator semigroups Ath . We prove that an operator semigroup Ath is strongly continuous if and only if h G li.
Then, we construct a class of Gaussian measures on a space lo, which are concentrated on a space li - a domain of strong continuity for operator semigroups Ath .
Our next step is to average operator semigroups Ah by a random vector h G lo, whose distribution is given by the previously constructed Gaussian measures. As a result, we obtain an operator set Ht, t G R+, which is a strongly continuous contraction operator semigroup. The obtained results show the applicability of the approach to study random shift operators in the Hilbert spaces developed in [3, 4], as well as to the study of random shift operators in linear topological spaces.
1. Supporting results
In this section, we introduce supporting constructions and results in order to define and operate with strongly continuous operator semigroups on the spaces of integrable functions.
Let us denote Borel a -algebras corresponding to the topology of pointwise convergence on spaces R™ and l™ as B(R™) and B(l™), respectively, and a Borel a-algebra corresponding to the standard topology on an arbitrary subspace K C Rk - as B(K), where k G N.
Lemma 1. Let Xi, and X2 be the Frechet spaces. Then B(Xi x X2) = B(Xi) x B(X2), where B(X1 x X2), B(X1), and B(X2) denote Borel a-algebras on X1, X2 , and X3, respectively.
Proof. X\ x X2 is a Frechet space. Let us define a subset L C (X\ x X2)* in the following way: h G L if and only if for all x\, x2 G Xi x X2 : h(xi; x2) = li(xi), where l1 G Xf; or for all xi, x2 G X\ x X2 : h(x\; x2) = l2(x2), where l2 G X2 . A set L separates points of a space X\ x X2 . That is why, according to the paragraph A.3.7. from [5], we have that B(X1 x X2) C B(X1) x B(X2). Furthermore, B(Xt) x B(X2) C B{X1 x X2). Hence, B(X1 x X2) = B{Xi) x B(X2). □
Let us denote a shift-invariant and a -additive measure on S as A, where S is a a -algebra on a space R(the precise definition of S and A may be found in [2]). Then, the following statements hold.
Lemma 2. For all B G B(R),n G N : Rn x B x R™ G S.
n
Lemma 3. For all A G S and for all e > 0 there exist n, k G N : A(A A y hi +
i=i
Qi x [0; 1] x [0; 1] x •••) < e, where Qi G B([0; 1]k), and hi + Qi f| hj + Qj = 0, when i = j .
Lemma 4. B(l) C S11.
Proof. It is enough to prove that B(RC S .A space R™ is a separable Frechet space. Let us define linear continuous mappings Pi : R™ ^ R, where i G N in the following way: Pi(xi,x2,...) = xi. Mappings Pi separate points of a space R™. That is why, according to the paragraph A.3.7. from [5], it is enough to prove that for all B G B(R),n G N : Rn x B x R™ G S. So, according to lemma 2, we have that
B(l™) C S\l™ . □
™
Lemma 5. A set {Ih+Qx[0;i]x[0;i]x---\h G R™,Q G U B([0;1]i)} is a total set in
i=1
a space L2(l™, B(l™),A\l™).
Lemma 6. Bs(l2) = B(R™)p| l2 , where Bs(l2) is a Borel a -algebra corresponding to the standard topology on a space l2 .
Hereafter, we consider A as a measure on a space (l™, B(l™)).
2. Operator semigroups of shifts
Let us define operators Ah : L2(l™, B(l™),A) ^ L2(l™, B(l™),A), where t > 0, h G l™, in the following way:
Afhu(x) = u(x + th).
For each h G lœ, operators Ah form a semigroup, besides for each t > 0, h G lœ : \\AhW = 1. Let us explore conditions, under which operator semigroups Ah should satisfy in order to be strongly continuous.
Lemma 7. Let for all n G N : tn > 0, hn > 0; tn ^ 0; (h\,h<2,...) G li ; Q G B([0;1]k); b G hœ . Then
lim SUp \\A(" 2 /b+Qx[0;l]x[0;l]x- - ^b+Qx [0;1] x [0;1] x-\\l2 = 0
Proof. It is enough to prove the statement in the case b = 0. We have that for all n G N :
sup \\At(X, x2 )Ib+Qx[0;1]x[0;1]x^ - h+Q x [0;1] x [0;1] x...\\l2 =
\xi\<hi,i£N V 2 '
, 1/2
2,
= SUP / lIQx[0;1]x[0;1]x^(I+tn(l1,l2,--.))-IQx[0;1]x[0;1]x-(I)|2^A(in =
\xH\<hi,ieN yj J
= sup (\(tn(-X1, -X2,...) + Q x [0; 1] x [0; 1] x-^A Q x [0; 1] x [0; 1] x .. .)1/2 <
\xi\<hi,ieN
< 2 sup (A(tn(x1,X2,...) + Q x [0; 1] x [0; 1] x • ••X Q x [0; 1] x [0; 1] x ••• )1/2 <
\xi\<hi,ieN
< 2 sup Ak (tn(x1,X2,...Xk ) + Q \ Q)+
\xi\<hi,ieN
+C sup (tn|Xk+1 | + (1-tn|Xk + 1|)tn|Xk+2| + (1-tn|Xk+1|)(1-tn |Xk+2|)tn |Xk+3| + ^ • • ),
\xi\<hi,ieN
where Xk is the Lebesgue measure on a space Rk and C = 2Ak(Q). We know that (h1, h2,...) G l1. That is why
sup (tn|Xk + 1| + (1-tn|Xk+1|)tn|Xk+2| + (1-tn |Xk+1|)(1-tn|Xk+2|)tn|Xk+3|H----) ^ 0.
\xi\<hi,i£N
As a result, we have sup
1 \xH\<hi,ieN
lim sup \\ A(x! ,x2,...)Ib+Q x [0;1] x [0;1] x — - ^b+Qx [0;1] x [0;1] x-\\.L2
Theorem 1. An operator semigroup Ah, t G R+ is strongly continuous if and only if h G li ■
Proof. If h G l\, then, according to lemma 7, a semigroup Ah is a strongly contin-
oo
uous operator semigroup, because a set {/h+QX[0;i]x[0;i]x^ |h G lo, Q G |J B([0; 1]*)}
i=i
is total in a space L2(l0, B(lo), A).
Suppose that h = (hi, h2,...) G li. Let us calculate a value of the following measure:
A(-th + [0; 1] x [0; 1] x • • • f| [0; 1] x [0; 1] x • • •),
where t > 0. We have that
n
A(-th + [0; 1] x [0; 1] x^f| [0; 1] x [0; 1] x^ )= lim [I (1 - th |) =
n—0 k=i
( n \ ( n \
= lim exp > ln(1 — tlhk |) < lim exp — > tlhk | =0,
n—>0 -i I n—>o \ -k-i I
when t is close enough to 0. Hence,
Jim IIAih1,h2,...)I[0;1]x[0;1]x... - I[0;1]x[0;1]x...yL2 = 21/2 ■ That is why an operator semigroup Ah is not strongly continuous when h £ ¡1. □
3. Gaussian measures concentrated on a space
Our goal in this paragraph is to construct a Gaussian measure on a space ¡2, which is concentrated on a space ¡1.
Let us take a sequence of numbers an, which satisfies the following properties: for all ne N : an e (0,1),
tt
0 < J} an < 1.
n=1
Let us take h = (h1, h2,...) e ¡1, where Vn e N : h1 > 0. The next step is to choose a sequence an, where for all n e N : an > 0 and for all n e N :
hn
1 f ( X2\ d
exp ( dx = an.
V2nan J V 2al
We can rewrite the equation in the following way:
hn/v-n
exp ( — —2 ) dx = _ I exp ( —— )dx = an.
y/2nan J V 2al) J \ 2
—hn -hn/vn
hn
As a result, we obtain that av = —=-. So, let us denote a Gaussian measure
' _ n V2erf-1(an) ' _ _
on R, corresponding to an, as Yn and construct a measure f in the following way:
tt
f = Yn. According to the paragraph 2.2 from [5], a measure f is well defined on
n=1
B(Rtt), and f\l2 is a centered Gaussian measure on a space ¡2. A measure f depends on a sequence an and a vector h. Let us denote a class of measures which can be constructed by the suggested scheme as M.
Lemma 8. For all f e M : f(l1) = 1.
Proof. According to the definition of a sequence an, there exists n e N for all e > 0:
tt
an > 1 — e.
k=n+1
For this reason, we can conclude that f(Rn x [—hn+1; hn+1\ x [—hn+2; hn+2\ x • • •) > 1 — e . Hence, we obtain that
00
f ( U Rn x —hn+1; hn+1] x —hn+2; hn+2] X •••) = 1.
n=1
As a result, we have that f(l1) = 1. □
Let us define measures ft, where t > 0, by the formula:
VAe Bs(h) : ft(A) =
1
where f e M. Measures ft are Gaussian on a space ¡2, and we have that for all
t, s > 0 : ft+s = ft * fs.
1
4. Averaging of operator semigroups
Let us define an operator set Ht : L2(l^, B(lTO),A) ^ L2(l^, B(lTO),A), where t > 0, in the following way:
Htu(x)=!u(x+^h) Mh)-
The vector Htu should be understood as the Pettis integral:
V v € B(lx),A): (Ht u; v)l2 = J u(x + Vth)v(x) dA(xd^(h),
( o
where ^ € M. Let us note that
/ (/u<x + dA<x>) dMh) = f (j u(x + vih)v(x)dA(x^ d,(h),
(^ (^ (^ SUpp(v)
J \u(x + Vth)v(x)\ dA(x) < \\u\\L2\\v\\l2 .
supp(v)
That is why, according to Fubini's theorem, a function
h ^J u(x + Vth)v{x) dA(x)
^ I u(x
is defined everywhere and measurable. Hence, the definition on an operator set Ht is correct. Moreover, for all t > 0 : \\Ht \\ < 1.
Theorem 2. For an arbitrary measure ^ € M, a set Ht, t € R+ is an operator semigroup.
Proof. Let us fix elements u,v € L2(l^, B(lTO),A) and introduce a function f : l^o —R as:
Let us fix t, s > 0. Then, we have that
f (h) = j u(x + h)v(x) dA(x).
(Ht+Su; v)l2 = J ^ j u(x + Vt + sh)v(x)dA(xd^(h) =
( ©o ( ©o
f (Vt + Sh) d^(h) = J f (Vt + Sh) d^(h) = J f (h) d^t+s(h).
(^ (2 According to paragraph 3, we obtain that
(Ht+S u; v)l2 = J f (x + y) d(fa ® Vs)(x,y) =
(2^(2
= j ^ j f (x + y)d^t(x)^ d^s(y) = j ^ j f (x + y)d^s(y^j d^t(x).
(2 (2 (2 (2
On the other hand,
(HtHsu; v)l2 = J ^ J (Hsu)(x + Vthi)v(x) dX(xd^(hi) =
loo l oo
u(x + \fth\ + ^/sh2)v(x) dX(x) d^(h2)
l oo lo
dp(hi) =
f (x + y) d^s(y)\ d^t(x).
l2 I2
That is why Ht+s = HtHs and, as a result, Ht is an operator semigroup.
□
Theorem 3. For an arbitrary measure ^ G M, a set Ht, t G R+ is an operator continuous operator semigroup.
Proof. Let us fix arbitrary functions u,v G B(lx),X). Let us consider
a sequence tn > 0 and let lim tn =0. Then
n—
\(Htnu - u; v)l2 \ <
<J \(u(x + y/t^h) — u(x))v(x)\dX(x)J d^(h) <
100
\u(x + Vt^h) — u(x)\2 dX(x) d^(h)
= IML
i\
\u(x + \ftñ,h) — u(x)\2 dX(x) d^(h).
According to Lebesgue's dominated convergence theorem, we obtain that
j j \u(x + \Jtñh) — u(x)\2 dX(x) d^(h) ^ 0. ii \
So, we have that sup \(Htnu — u)L2 \ ^ 0. That is why Htnu — u ^ 0. Therefore,
\\v\\L2 =1
an operator semigroup Ht is strongly continuous.
□
Conclusions
In this paper, we proved the criterion of strong continuity for shift operator semigroups, along a constant vector field in a space l, on a space of complex square integrable by shift-invariant measure on a space lfunctions. In addition, it was shown that averaging of shift-invariant operator semigroups by Gaussian measures concentrated on a space li is a strongly continuous contraction operator semigroup.
Acknowledgements. The work was performed according to the Russian Government Program of Competitive Growth of Moscow Institute of Physics and Technology (project 5-100).
< v l
2
References
1. Baker R. Lebesgue measure on Rx. Proc. Am. Math. Soc., 1991, vol. 113, no. 4, pp. 10231029. doi: 10.2307/2048779.
2. Zavadsky D.V. Shift-invariant measures on sequence spaces. Tr. Fiz.-Tekh. Inst., 2017, vol. 9, no. 4, pp. 142-148. (In Russian)
3. Sakbaev V.Zh. Averaging of random walks and shift-invariant measures on a Hilbert space. Theor. Math. Phys., 2017, vol. 191, no. 3, pp. 886-909. doi: 10.1134/S0040577917060083.
4. Sakbaev V.Z. Random walks and measures on Hilbert space that are invariant with respect to shifts and rotations. Itogi Nauki Tekh, Ser.: Sovrem. Mat. Prilozh. Temat. Obz., 2017, vol. 140, pp. 88-118. (In Russian)
5. Bogachev V.I. Gaussovskie mery [Gaussian Measures]. Moscow, Fizmatlit, 1997. 352 p. (In Russian).
Recieved October 17, 2017
Sakbaev Vsevolod Zhanovich, Doctor of Physics and Mathematics, Professor of the Higher Mathematics Department
Moscow Institute of Physics and Technology
Institutskiy per., 9, Dolgoprudny, Moscow Region, 141701 Russia E-mail: [email protected]
Zavadsky Dmitrii Viktorovich, Student of the Department of Control and Applied Mathematics
Moscow Institute of Physics and Technology
Institutskiy per., 9, Dolgoprudny, Moscow Region, 141701 Russia E-mail: [email protected]
УДК 517.98+519.2
Трансляционно-инвариантные меры на бесконечномерных пространствах, интегрируемые функции и случайные блуждания
В.Ж. Сакбаев, Д.В. Завадский Московский физико-технический институт, г. Долгопрудный, 141701, Россия
Аннотация
В работе изучается усреднение случайных операторов сдвига аргумента в пространстве квадратично интегрируемых по трансляционно-инвариантной мере комплекснознач-ных функций на линейных топологических пространствах. В качестве примера рассмотрен случай пространства . Трансляционно-инвариантная мера на пространстве , построенная при помощи схемы Каратеодори, обладает свойством счетной аддитивности, но не обладает свойством а-конечности. Также рассматриваются различные приближения измеримых множеств. Рассматриваются однопараметрические группы сдвигов вдоль постоянного векторного поля в пространстве и полугруппы сдвигов на случайный вектор, распределение которого задается семейством гауссовских мер. Получен критерий сильной непрерывности группы сдвигов вдоль постоянного векторного поля. Установлены условия на семейство гауссовских мер, достаточные для сохранения полугруппового свойства усредненного однопараметрического семейства линейных операторов и его сильной непрерывности.
Ключевые слова: сильно непрерывные полугруппы, усреднение операторных полугрупп, трансляционно-инвариантные меры, квадратично интегрируемые функции
Поступила в редакцию 17.10.17
Сакбаев Всеволод ^Канович, доктор физико-математических наук, профессор кафедры высшей математики
Московский физико-технический институт
Институтский пер., д. 9, г. Долгопрудный, Московская обл., 141701, Россия E-mail: [email protected]
Завадский Дмитрий Викторович, студент факультета управления и прикладной математики
Московский физико-технический институт
Институтский пер., д. 9, г. Долгопрудный, Московская обл., 141701, Россия E-mail: [email protected]
I For citation: Sakbaev V.Zh., Zavadsky D.V. Shift-invariant measures on infinite-di-( mensional spaces: Integrable functions and random walks. Uchenye Zapiski Kazanskogo \ Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, pp. 384-391.
/ Для цитирования: Sakbaev V.Zh., Zavadsky D.V. Shift-invariant measures on infinite-( dimensional spaces: Integrable functions and random walks // Учен. зап. Казан. ун-та. \ Сер. Физ.-матем. науки. - 2018. - Т. 160, кн. 2. - С. 384-391.
