Lévy Laplacians and annihilation process Текст научной статьи по специальности «Математика»

2018, Т. 160, кн. 2 С. 399-409

УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА. СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)

UDK 517.9

LEVY LAPLACIANS AND ANNIHILATION PROCESS

B.O. Volkov

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia Bauman Moscow State Technical University, Moscow, 105005 Russia

Abstract

The Levy Laplacians are infinite-dimensional Laplace operators defined as the Cesaro mean of the second-order directional derivatives. In the theory of Sobolev-Schwarz distributions over a Gaussian measure on an infinite-dimensional space (the Hida calculus), we can consider two canonical Levy Laplacians. The first Laplacian, the so-called classical Levy Laplacian, has been well studied. The interest in the second Laplacian is due to its connection with the Malliavin calculus (the theory of Sobolev spaces over the Wiener measure) and the Yang-Mills gauge theory. The representation in the form of the quadratic function of the annihilation process for the classical Levy-Laplacian is known. This representation can be obtained using the S-transform (the Segal-Bargmann transform). In the paper, we show, by analogy, that the representation in the form of the quadratic function of the derivative of the annihilation process exists for the second Levy-Laplacian. The obtained representation can be used for studying the gauge fields and the Levy Laplacian in the Malliavin calculus.

Keywords: Levy Laplacian, Hida calculus, quantum probability, annihilation process

Introduction

In the present paper, we study some relationships between infinite-dimensional Laplacians and quantum stochastic processes.

Let us recall the definition of the Levy Laplacian. Let E be a real locally convex space continuously embedded into a separable Hilbert space H. Let the image of E under the embedding be dense in H. The value of the Levy Laplacian on a function f on E is determined by the formula

1 n

ALf (x) = lim - J^if"(x)ek, ek), (1)

k=1

where {en} is an orthonormal basis in H such that all its elements belong to E. This definition depends on a choice of the orthonormal basis {en}. For so-called weakly uniformly dense bases in H = L2([0,1],R), this definition coincides with the definition of the Levy Laplacian as an integral functional determined by the special form of the second derivative (see [1, 2]). If we replace in (1) the Cesaro mean by the sum of the series, we obtain the definition of the Volterra-Gross Laplacian.

Now let

E = Wd'2([0,1], R) := {7 e AC([0,1],R): 7(0) =0,7 e L2([0,1],R)}.

This space is canonically isomorphic to L2Q0,1], R) (the isomorphism is determined by the differentiation). We can define the Levy Laplacian on the functions on E by the following two ways. On the one hand, we can choose some good orthonormal basis

in Wq ([0,1], R). Then we obtain the Levy Laplacian AL of order 1. On the other hand, Wq1,2([0,1], R) is embedded into H = L2([0,1],R). We can choose some good orthonormal basis in H and obtain the Levy Laplacian A^ 1) of order ( —1) .1

The theory of the Sobolev-Schwartz distributions over the abstract Wiener measure is called the Hida calculus or the white noise theory. It is known that the Volterra-

Gross Laplacian avg can be represented by the formula avg = j o?t dt, where a is

R

the annihilation process (see, e.g., [7, 8]). An analogue of the Levy Laplacian A^ in

the Hida calculus we will call the classical Levy Laplacian. The classical Levy Laplacian

1

AL can be in^ed as / a^) (see [8] and also [9], where this formn,a appear,

0

with the reference to Kuo). In this paper, we will show that the analogue of the Levy

1

Laplacian A^ 1) can be interpreted as ~ J a2 (dt2). The connections between the

0

Levy Laplacians and quantum stochastic processes were also discussed in [5, 6, 10, 11]. In the latter paper, a rigorous meaning for the formulas

Al = lim / as at ds dt

E^0 J

\s-t\<£

and 1

aL 1) = —2 lim asat ds dt

n2 e^0

\s-t\<E

was given Note t

its connection to the gauge fields (see [4, 12-16]).

Note that one of the main reasons for the interest in the Levy Laplacian A^ 1) is

1. Levy Laplacians

{p1,---,pd} is an orthonormal basis in Rd everywhere below. If E is a locally convex space (LCS), its dual space E* is equipped with strong topology. If E and V are LCSs, the space Lb(E, V) is the space of all continuous linear operators from E to V. We assume that Lb(E, V) is equipped with the topology of the uniform convergence on bounded sets.

Let EC = S(R, Cd) be the Schwartz space of Cd-valued rapidly decreasing functions and EC*C = S*(R, Cd) be the space of generalized functions of slow growth. Let EM = {£ =(t\---,td) € Ec : f = 0, if n = v }. Then Ec = Ex ,®Ed. Let T2 = T2(Rd, C) and T2sym be the space of all C-valued tensors of type (0, 2) and the space of all symmetric C-valued tensors of type (0, 2) on Rd, respectively. Let L2ym(R2, T2) = {g € L2 (R2,T2): (t,s) = gv^(s,t)}. Let C^ (EC, C) be the space of all two times Frechet complex differentiable C-valued functions on EC = S(R, Cd) satisfying the following condition:

the second derivative of f € C^(EC, C) has the form (f"(0C,v) =jj KVv(£; s,t)z»(t)vv(s) dtds+JK^v(£;t)z^(t)vu(t) dt, z, n € Ec, (2)

R R R

1One can consider the family of so-called exotic Levy Laplacian A® , where l > 0 (see [3]). The Lap-

lacian A^ 1) belongs to the extension of this family for negative l < 0 (see [4] and also [5, 6]).

where KV(£, ■, •) e Ls2vm{R2,T2), KL(£, ■) e LTO(R,T2svm) for any £ e Ec. (KV is the Volterra kernel and KL is the Levy kernel of the second derivative).

If f e CL(EC,C), it is possible to extend f"(£) as a bilinear jointly continuous functional on L2(R, Cd) x L2 (R, Cd). We will denote this extension by the same symbol.

Let {en} be an orthonormal basis in L2([0,1], R). We identify any element h e L2 ([0,1], r) with h e L2(R, R) defined by

h(t) = {h(t) if 4 e [01]' (3)

0, otherwise.

We do not require {en} to have its elements from

Definition 1. The Levy Laplacian of order s e {—1,1} is a linear mapping

from Dom ALen}s to the space of all C-valued functions on EC defined by:

nd

ALen}'Sf (£) = Jim kl-S(f "(£)P^eu ,p,ek), (4)

fc = l ,= 1

where Dom ALe"}'s is the space of all f unctions f e CL(EC, C), for which the right side of (4) exists for any £ e EC.

The following definition is from [1].

Definition 2. An orthonormal basis {en} in L2([0,1], R) is weakly uniformly dense,

if

l

1n

/1 '

h(t)(- y^ efc(t) - l) dt = 0

\n z—' )

k=i

for any h e LTO([0,1],R).

Let ln(t) = %/2sin(nnt) and hn(t) = %/2cos(nnt) for n e N and h0(t) = 1. The orthonormal bases {hn}'^=1 and {ln}^=0 in L2([0,1],R) are weakly uniformly dense.

Proposition 1. Let {en} be a weakly uniformly dense basis in L2Q0, 1], R). Let f e CL (EC, C). Then

(0 = £ / KL.&t) dt. (5)

This is a well-known fact for d =1 (see, e.g., [1, 2, 7]). If d > 1 formula (5) can be proved by analogy (see [4]).

Due to the kernel theorem (see, e.g., [17]) for any f e C2(EC, C) the second partial derivative f^ E (£) belongs to 5*(R2, C). Let C2L — 1)(EC, C) be the space of all f e CL (Ec, C) such that for any j e {1,..., d} and for any £ e EC the second mixed generalized derivative of f e/j_e^(£) e 5*(R2, C) has the form

o2 f

(dtd~s fE (£)U) = J ^(s,t)v^(dsdt), 4 e ^(R2, c), (6)

R2

where vf.,, is a a -additive a-finite C-valued measure on R2 .

"" 2

Let I = {(s, t) e R2 : s = t, 0 < s < 1}. Let 1/ be the indicator of this set.

Proposition 2. If f e CL ( _1)(EC, C), then

d , i

A{Lhn}'- 1f (0 = £ -^i 11 (s,t)viß(dsdt). (7)

M=1

Proof. Let

exKt^r)' if ltl < 1

-(t) = r ^ - IP (8)

otherwise.

Let Ti(t) = r (t)/ j t (t) dt and re(t) = T1(t/e)/e .Let len = ln *re .Then len e S(R, R),

—^

the support of in belongs to [—e, 1 + e] and len converges to ln as e ^ 0 uniformly on any compact set. Let hn = hn * t£. Then hn converges to hn in L2(R, R) as e ^ 0. Due to l'n = nnhn, we have (ln)' = nnhEn . Hence, we obtain

n^ifEE» (O'hn ® hn) = lim nWif^ (Oh ® hn =

= lim|ln(s)len(t)v^(dtds) = j ln(s)ln(t)v^(dtds). (9)

R2 R2

The last equality is due to Lebesgue's dominated convergence theorem. For any (s,t) e R2 we have

1n

lim - Vln(s)ln(t) = 1i(s,t) (10)

n—n —

k=1

and

1

sup

nEN

~y/ln(s)ln(t) < 2. (11)

n z—'

k=1

Thus Lebesgue's dominated convergence theorem and (9) together imply (7). □

Remark 1. One of the approaches to define the Levy Laplacian is to define it as the integral functional determined by the special form of the second derivative (see [1, 2, 7]). Particularly, the operators A^1-1 and A^ 1) can be defined by formulas (5), and (7), respectively. It can be shown that form (6) is generalization of the form of the second derivative from [12].

2. Levy Laplacians in Hida calculus

The operator

2 d2 A = 1+ t2 - dP,

is a self-adjoint operator on HC = L2 (R, Cd). For any p > 0 let Ep be the domain of Ap equipped with the Hilbert norm |£|p = \ ApC\hc . For any p < 0 let Ep be the completion of HC with respect with the Hilbert norm |£|p = |Ap£|Hc . Then S(R, Cd) coincides

with the projective limit projlim Ep and S*(R, Cd) coincides with the inductive limit

p—^

indlim E-p . We have the real and complex Gelfand triplets:

p—

, Rd) = Er c L2(R, Rd) = HR c S*(R, Rd) = ER.

and

Ec C He C EC

The Fock space over the Hilbert space Ep is defined as

to

r(Ep) = {4 = (f)TO=o; fn e Efn, U\\p = £m\fn\p < ■

Let E = projlimr(Ep). Then £* = indlimr(E-p). Let the symbol ((■, ■)) denote

p^ + TO p^ + TO

the duality form on E* x E.

Let hi be the Gaussian pseudomeasure on S(R, Rd) with the Fourier transform HI(£) = exp( — (£, £)Hr/2). The Minlos-Sazonov theorem implies that hi is o-additive measure on S(R, Rd). The unitary Wiener-Ito-Segal isomorphism between r(HC) and L2 (ER*, hi, C) is determined by the values of this isomorphism on the coherent states:

^ = (U, Y'■■■) ^ ^ = , Z e Ec.

Below we will not distinguish between the spaces r(Hc) and L2(ER,hi, C). The Gelfand triplet E C L2(ER*, hi, C) C E* is called the Hida-Kubo-Takenaka space. E is the space of white noise test functionals (Hida test functionals), and E* is the space of white noise generalized functionals (Hida generalized functionals).

The S-transform (the Bargman-Segal transform) of generalized white noise functional $ e E* is the function S$: EC ^ C defined by the formula S$(£) = (($,)), Z e Ec . It is known that a complex function G on Ec is a S-transform of some generalized white noise functional if and only if G satisfies the following two conditions (see, e.g., [7, 18]):

A. for any e Ec the function (z) = G(zn + Z) is entire on C;

B. there exist constants Ci, C2 > 0 and p > 0 that for each £ e EC hold:

\G(Z)\ < Ci exp(C2\Z\2)■

If a complex-valued function on the space Ec satisfies the conditions above, it is called U-functional. Let the symbol Fu denote the space of all U-functionals.

Definition 3. The domain of the Levy Laplacian Aof order s e {—1,1} is the space Dom A[en}'s = {$ e E* : S$ e DomA^"1'8, A^6"}'sS$ e Fu}. The Levy Laplacian ALn}'s is a linear mapping from Dom ALn}'s to E* defined by

A Len}'s$ = S-1ALen}'s(S$)■ (12)

Remark 2. It is known that if $ e L2(ER, hI, C), then S$ e CL(EC, C) and the Levy kernel of S$" is equal to zero (see, e.g., theorem 6.42 from [18]). Hence, if $ e L2(ER, hi, C), then AL^'1 $ = 0, where {en} is a weakly uniformly dense basis.

The elements from E can be realized as entire functions on ER* (see, e.g., [7]). For any Z e ERR let a(Z) be the operator of differentiation in the direction:

a(Z)№ = lim^e + tZ) — m)/t, Z e ER*,4 eE■

If Z € ERR, then a(Z) € Lb(E, E). If Z € Er, then a(Z) can be extended to the operator a(C) € Lb(E*, E*). For any $ € E* and Z € ER the following holds:

S (a(Z )$)(£) = (snt),Z ).

A continuous mapping from R to Lb(E, E*) is a quantum stochastic process in the sense of the Hida calculus (see [19]). Note that there is the canonical embedding id of E* into L(E, E*) defined by the following way:

(id$)(0) = $0, $ ge*, 0 ge.

The mapping R 3 t ^ af = a(pfSt) € Lb(E, E*) is a quantum stochastic process, which is called the annihilation process. The mapping R 3 t ^ (af)* = a(pfSt)* € Lb(E, E*) is a quantum stochastic process, which is called the creation process. It is possible to show that these mappings are smooth and the mapping R 3 t ^ af € Lb(E, E*) is also quantum stochastic process (see [20]). The sum (af + (af)*) is the white noise process, which is the derivative of the Brownian motion.

Using Obata's result on the integral kernel operators (see [17, 20]), it is possi-

1 1

ble to give a rigorous sense for the integral // ^(t, *)«dtds, where k € E®2 .

0 0

If 0, y € E, then

WM) = (<>,t)) = (((afaV0,y))) € E®2.

If k € (E®2)*, then there exists the unique S0j2(k) € Lb(E, E) such that ((So,2(k)0, y)) = (K,ri4,^). If k € Ec®2, then So,2(k) can be extended to the operator S0j2(k) € Lb(E*, E*). For any $ € E* and k € Ec®2 the following holds:

1 1

s(// Kfv (t,s)af aV dtds $)(£) = S (S o,2(k)$)(0 = (S$"(£),k). (13)

00

Let {en} be a weakly uniformly dense basis. If $ € DomAL^'1 and S$ € CL(EC, C), proposition 1 implies

d 1 d

s(ALen}'1$)(e) = £ J KLf(£,t)dt = £(s$>^(o,h). (14) f=10 f=1

So, the right side of (13) has sense if Kfv = Sfv 1/, where Sfv is the Kronecker symbol.

1

Thus, formula (14) gives a rigorous sense for al^'1 = a'2(dt2) (see [8]). (The formula

0

1 1

n}'1 =

00

A{Lhn}'1 = / 1/(s,t)asat (dsdt) (15)

is probab, more accurate. However, there ,s a conjees«, the form„,a / ^

can be included into the quantum Ito table (see [9]).)

1

lEvy laplacians and annihilation process

405

If k e E®2, it is possible to give a rigorous sense for the integral

1 1

Vv(^ s)at as

(t, s)à»àV dt ds

0 0

dimes'

as s0,2( ^TTT"K) £ Lb(E*, E*). Let = 0 if ^ = v. Let S$ £ C2L ,(-1) (Ec, C). Then

i 1

S

(t, s)àfàv dtds $)(£) = S( s0,2

e2

dtds'

$ (0 = <s$"(e),

d2

ete&

00

= £ (0, ddrskw) = ^ / K^(s,t)v^(dsdt). (16)

M=1 M=10

The following theorem is a direct corollary of proposition 2.

Theorem 1. If $ G Dom À

{hn},-1

and S$ £ CL,(-1)(Ec, C). Then

S(À{Lhn}'-1$)(C) = £ ^J 11 (sM,(dsdt).

n

0

(17)

Due to (16), formula (17) gives a rigorous sense for al^' 1 = ~ J ^(dt2).

o

(Similarly to (15), the formula

1 1

À{Lhn}' 1 = I I 1i(s,t)àsàt(dsdt)

00

is probably more accurate.)

3. Yang—Mills equations

Let A(x) = AM(x)dxM be a smooth u(N)-valued 1-form on Rd. This form determines a connection in the trivial vector bundle with base Rd, fiber CN, and structure group U(N). The covariant derivative of C 1(Rd,u(N)) is defined by V ¡4 = + [AM, 4]. The curvature F(x) = ^^ F^v(x)dxM A dxv is determined by

¡1<v

F^v = dA v dv A^ + [A ¡, AvJ ■ The Yang-Mills equations on a connection A have the form

»v = 0.

(18)

In paper [12] by Accardi, Gibilisco, and Volovich, the following was proved. The parallel transport associated with the connection A is a solution to the Laplace equation for the Levy Laplacian if and only if A satisfies to the Yang-Mills equations. In [12], the parallel transport was considered as an operator-valued functional on the space of

k =

1

1

1

C1 -smooth curves in Rd and the Levy Laplacian was defined as an integral functional determined by the special form of the second derivative. In [14, 15], it was shown that this Laplacian can be defined as the Cesaro mean of the second directional derivatives (this Laplacian coincides with n2ALhn}' 1).

A stochastic parallel transport and its connection to the Levy Laplacian can be considered. Let {bt}t£[o,1] be a standard d-dimensional Brownian motion and (Q, F, P) be the probability space associated with this process. The stochastic parallel transport UA(b,t) is a solution to the stochastic equation:

t

U A(b, t) = IN -J A^ (ba)U A(b,s) o db%, o

where odb is the Stratonovich differential.

In paper [13] by Leandre and Volovich, the Levy Laplacian on the Sobolev space over the Wiener measure P was introduced. This Laplacian was defined as the integral functional. It was shown that the stochastic parallel transport UA(b, 1) is a solution to the Laplace equation for a such Levy Laplacian if and only if A satisfies to the Yang-Mills equations. In [16], the Levy Laplacian AL defined as the Cesaro mean of the second directional derivatives on the Sobolev space over the Wiener measure P was introduced. In [16], it was proven that A satisfies to the Yang-Mills equations if and only if UA(b, 1) satisfies

1

AlU A(b, 1) = U A(b, 1) j U A(b,t)-1F^v (bt)F^v (bt)U A(b,t) dt. o

Thus, in contrast to the deterministic case, the Levy Laplacian as the integral functional and the Levy Laplacian as the Cesaro mean are two different operators on the Sobolev space over P.

In [4], it was proven that the Levy Laplacian AL coincides with ^al^^' 1 under the canonical embedding J of the Sobolev space over the Wiener measure into MN (C) £ * (the space of MN (C)-valued Hida functionals). Moreover, it is possible to show that S-transform of JUA(b, 1) belongs to CL(EC,MN(C)) (this space is defined by analogy with C'(EC, C)). Thus,

1

J (AlU a (b, !))=(/ ¿?(dt2)) JU A(b, 1). o

It would be of interest to investigate whether the definition of the Levy Laplacian from [13] could be reformulated in terms of the quantum stochastic processes. It is still unknown how the Levy Laplacians from [13] and [16] are connected.

Conclusions

Further research is needed to see whether the Levy Laplacian A^ 1) could be included in the Ito quantum table and used in some areas related to quantum probability (see, e.g., [21-23] and, especially, [9]). In addition, it would be of relevance to study the approach based on the Levy Laplacian in some areas connected to the theory of gauge fields (see, e.g., [24-27]).

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Recieved

December 19, 2017

Volkov Boris Olegovich, Candidate of Physical and Mathematical Sciences, Senior Research Associate; Associate Professor

Steklov Mathematical Institute of Russian Academy of Sciences

ul. Gubkina, 8, Moscow, 119991 Russia Bauman Moscow State Technical University

ul. Vtoraya Baumanskaya, 5, str. 1, Moscow, 105005 Russia E-mail: [email protected]

УДК 517.9

Лапласиан Леви и процесс уничтожения Б.О. Волков

Математический институт им. В.А. Стеклова Российской академии наук,

г. Москва, 119991, Россия Московский государственным технический университет им. Н.Э. Баумана, г. Москва, 105005, Россия

Аннотация

Лапласианы Леви представляют собой бесконечномерные операторы Лапласа, определенные как среднее Чезаро вторых производных по направлению. В теории распределений Соболева - Шварца над гауссовской мерой на бесконечномерном пространстве (исчислении Хиды) можно рассмотреть два канонических лапласиана Леви. Первый из них, так называемый классический лапласиан Леви, хорошо изучен. Интерес ко второму лапласиану обусловлен его связью с исчислением Маллявэна (теорией пространств Соболева над мерой Винера) и калибровочной теорией Янга-Миллса. Для классического лапласиана Леви известно представление в виде квадратичной функции от процесса уничтожения. Это представление может быть получено с помощью Я-преобразования (преобразования

Сигала-Баргмана). В настоящей статье по аналогии показано, что для второго лапласиана Леви существует представление в виде квадратичной функции от производной процесса уничтожения. Полученное представление может оказаться полезным для изучения калибровочных полей и лапласиана Леви в исчислении Маллявэна.

Ключевые слова: Лапласиан Леви, исчисление Хиды, квантовая вероятность, процесс уничтожения

Поступила в редакцию 19.12.17

Волков Борис Олегович, кандидат физико-математических наук, старший научный сотрудник; доцент

Математический институт им. В.А. Стеклова Российской академии наук

ул. Губкина, д. 8, г. Москва, 119991, Россия Московский государственный технический университет им. Н.Э. Баумана

ул. 2-я Бауманская, д. 5, стр. 1, г. Москва, 105005, Россия E-mail: [email protected]

/ For citation: Volkov B.O. Levy laplacians and annihilation process. Uchenye Zapiski ( Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 2, \ pp. 399-409.

Для цитирования: Volkov B.O. Levy laplacians and annihilation process // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. - 2018. - Т. 160, кн. 2. - С. 399-409.