Some mathematical models with a relatively bounded operator and additive "white noise" in spaces of sequences

Let us consider an abstract mathematical model of the form

Lu = Mu + f,

which is the prototype of the Barenblatt - Zheltov - Kochina model [1] and Hoff model [2]. Operators L = L (Л) and M = M (Л) are the polynomials with real coefficients; the operator Л u = ( A 1 u 1 , A ₂ u2,... ) acts in the spaces of sequences [3,4]

m lq

u

∞ mq

= ^{uk^{} :} X ^Ak^{2 |u}k I

<∞ ,

m G R , q G [1 , + to ) ,

k =1

which are the analogues of Sobolev spaces Wqm ; the sequence {Ak} C R₊ is a monotonically increasing sequence, such that lim Ak = + to. For the equation (1) we will consider the k→∞

Showalter - Sidorov condition

P ( u (0) - u o ) = 0 ,

where P is a relatively spectral projector. A study of positive solutions is started in [5], where the sufficient conditions for the existence of a single positive solution of the Showalter - Sidorov problem (1), (2) in Sobolev spaces of sequences that can be interpreted as the space of Fourier coefficients of solutions initial-boundary value problems are given [6].

K.V. Vasyuchkova, N.A. Manakova, G.A. Sviridyuk

This article (as, indeed, [5]) is inspired by the fundamental work [7], where it is suggested to use degenerate holomorphic resolving groups of operators whose construction is based on the theory of degenerate groups of operators to find solutions to the problem (1), (2). We apply developed in [5] methods to the search of the stochastic solutions of the ecpiation

Ln = Mn + N © , (3) ecpiipped with a weak (in the sense of S.G. Kreign) condition of Showalter - Sidorov

tH m+ P ⁽ n ⁽ t ) - n o) = ⁰ .

Here, operators L,M and P are the same as above, the operator N will be defined later. П = n(t) is a stochastic process we are looking for, and 0 are the given stochastic process; o by the symbol П the Nelson - Gliklikh derivative of the stochastic process n = П(t) is denoted.

The paper is organized as follows. The first part is preliminary nature. It presents our approach to studying stochastic ^ -processes. The foundations of this approach are laid in [8], then developed in [9-11]. Here also by analogy with [12] the spaces of sequences of differentiable "noises" are introduced into consideration. In the second section the abstract problem (3), (4), where the operator M is ( L,p )-bounded, p E { 0 } U N, is considered. We notice that our approach is based on the Nelson - Gliklikh derivative [13], the theory of (semi)groups of operators [7], which distinguishes it from the classical Ito - Stratonovich -Skorokhod approach (see, for example, [15]) and recently the emerged Melnikova - Filinkov approach [16].

1. Stochastic ^-Processes

Let us consider the complete probability space G = (G , A, P) and the set of real numbers R, endowed with a Borel a -algebra. Following [8], we call a measurable mapping £ : G ^ R _a random, variable. A set of random variables, which mathematical expectations are equal to zero, forms a Hilbert space L2 with scalar product ( £ 1 , £ 2) = E £ 1 £2, where E is the mathematical expectation of a random variable. We denote by A ₀- a the subalgebra of the a -algebra A and construct the space L0 of random variables that are measurable with respect to A ₀ . Obviously, L0 is a subspace of the space L2 . Let £ E L2, then П £, where П : L2 ^ L2 is an orthoprojector, is called a conditional expectation of a random variable £ and is de noted by E( £|A o) .

Let us consider two mappings: the first mapping f : I ^ L2, which puts to each t E I , I C R the random variable £ E L2 in the correspondence, and the second mapping g : L2 x G ^ R. which puts to each pair ( £,ш ) the iroint £ ( ш ) E R in the correspondence.

The mapping n : I x G ^ R. having the form n = n ( t,w ) = g ( f ( t ) , ш ). is called the (one-dimensional) random process. Following [8], the random process n I^s called continuous, if a.s. all its trajectories are continuous. Let us denote by the CL2 the set of the continuous random processes, which forms a Banach space. Continuous random process, which (independent) random variables are Gaussian (i.e. have the normal (Gauss) distribution), is called Gaussian. A one-dimensional Wiener process в = в ( t ) is ^an example of the continuous process, and it has the following properties:

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

(W2) the mathematical expectation E ( в ( t )) = 0 and autocorrelation function E (( в ( t ) — в ( s ))2) = It — s| foi' fol s, t C R₊:

(W3) trajectories в ( t ) are not differentiaible in each point t C R₊ and they have an unbounded variation on the each arbitrarily small interval.

Remark 1. Note, that the properties (Wl) and (W2) played the important role for the definition of the Wiener process, and the property (W3) is followed from the first two.

Theorem 1. Then there exists the unique random process в, satisfying properties (Wl) - (W2) with probability one, and it can be present in the form oo

^{в (} t ) = ^ ^zk sin|⁽2 к + 1) t,                           (5)

k =0

where Zk awe. independent Gaussian variables. E Zk = 0 , D Zk = [ П (2 k + 1)] ² , where, by the. E , D the mathematical expectation and the dispertion of the random variable are denoted.

Gow fix n C CL2 arid t C I(= ( e,t ) C R) and I>y the Nt let us denote ст -algebra, generating by the random variable n ( t ) • Denote E ) = E ( fN ) .

Definition 1. Let be n C CL2, the random variable n n (t + △t, •) — П(t, •)

Dn ( t, • ) = lim E

• △ t^ o+ ^t           At

n n ⁽ t, • ) ^— n ⁽ t ^{— At}, • ) (^D*^{n (} t • ⁾= △ t m ^Et (-------At------))

is called the mean derivative on the right Dn ( t, • ) (on thic left D*n ( t, • V of the random process n at the point t C ( e,t ), if there exists the limit in the sense of the uniform metrics on R. The rand, от process n is called, mean differentiable, oil the. right (on the. left) oil ( e,t ). in the there exists the mean derivative on the right (on the left) in each point t C ( e,t ).

Let the random process n C CL2 be mean differentiable on the right and on the left o on (e,t). Define the symmcti'leal mean derivative n = DSn = 2 (D + D*) n- and further we o (l)

will call it by the Nelson - Gliklikh derivative. By the n , l C N , let us denote the / -th derivative of Nelson - Gliklikh of the random process n.

Theorem 2. (Yu.E. Gliklikh, [13]) Let в = в ( t ) be the Wiener process, then в ( t ) = 2 t for all t C R₊.

Remark 2. (i) Note, that the Nelson - Gliklikh derivative of the Wiener process has the property of linearity:

o   oo

^{( a} 1 ^в 1 ⁽ t ) ^{+ a} 2 ^в 2 ⁽ t ))= ^a 1 ^в 1 ⁽ t ) ^{+ a} 2 ^в 2 ⁽ t ) •

• (ii)    Note, that if the trajectories of the random process n ^{are a}'^s' continuously differentiable in the "ordinary case" on ( e,t ), then its Nelson - Gliklikh derivative coincides with the "ordinary" derivative. Therefore, consistently applying the formula of

K.V. Vasyuchkova, N.A. Manakova, G.A. Sviridyuk differentiation of the quotient, we obtain, that Z-th Nelson - Gliklikh derivative of Wiener process satisfies

o(l)                — в (t) = (-1)-1 • П(2i - 1) •      • Z e N• Z > 2.

Let us introduce space ClL2• Z e N• of the random processes from CL2• which trajectories are a.s. differentialble in the sense of Nelson - Gliklikh on the set I until the order Z. If I C R+, then it follows from the Theorem 2, that there exists the derivative o ве C1L2, which is called (a one-dimensial) "white noise". Futlier the spaces ClL2 we will call the spaces of differentiable "noises ".

Next we choose a monotonically decreasing numerical sequence K = {^k}, such that ∞ lim цк = 0 and the numerical series ^f

∞

0K (t) = 52 VMkПк (t) Фк •(6)

к =1

which provides that the series (6) converges uniformly on any compact set from I. We introduce the Nelson - Gliklikh derivatives of random K-process o (l )„.   ^ o-o (l )__

0K (t) = X V^ nk (t) Фк к=1

by the condition that on the right-hand side (7) there are derivatives up to the order Z inclusive and all series converge uniformly on any compact set in I. Futher consider the space C_KL₂ of the random K-processes, which trajectories are a.s. continuous, and the spaces CKL₂ of random K-processes, which trajectories are a.s. continuously differentianle in the sense of Nelson - Gliklikh up to the order Z e N inclusive.

We define the Wiener K-process on the R₊ by the formula,

∞

Wk (t) = ^ VhVвк (t) Фк •                            ( 8 )

к=1

where в_к(t) has the properties (Wl) и (W2).

Theorem 3. Let K = {р_к} be a monotone numerical sequence, such that lim ц_к = 0 and k→∞

∞

52 № < + to. Then with probability one. tltere exist the. unique Wiener K-process. which к=1

satisfies the conditions (Wl) - (W3), and it can be rewritten in the form (7).

Let us consider the spaces of sequences lmL₂ of random variables w = (w 1 • w₂•...) with the norm

∞1

IMim = (E<^xmD^шк)2)'• q e [1 • to)• m e R.

к=1

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Obviously, the imbeddings lmmL2 ^ l^L2 are dense and continuous for all m > n and q E [1, to), and the operator Л : lm+2L2 ^ lmL2 is linear, continuous and continuously invertible for all m E R arid q E [1, to). Futlrer let us consider the spaces C^l_KlmL2(= CK lm L2(е,т ), ( е,т ) C R) of raridorn ^-processes n = (П 1 ,П 2, •••), nk = nk (t) ,t E ( е,т ), k E N, which Nelson - Gliklikh derivatives up to order l E {0}UN inclusive are a.s. continuous on (e, т) • Consider the Wiener W-process

Wk = ( VH в 1 ,VH2 в 2 ,•••), where вк = вк(t), t E R+, are Brownian motions of the form (5). Note, that if the series

∞ q mq µk2 λk2

k=1

converges, then Wiener К-process W_K E CKlmL₂. By the thadition [8], the Nelson -Gliklikh derivative W_K (t) = (2t)-¹W_K(t) of the Wiener К-process is called "white noise".

• 2.    Stochastic Sobolev Type Equations with Relativelyp-Bounded Operators

Let us move on to the finding the sufficient conditions of the existence and uniqueness of the solution of Showaler - Sidorov problem (3), (4). The foundation of our research are the theory of degenerate groups of operators and the phase space method for a degenerate Sobolev type equation (1), described in [7] in the deterministic case, which were successfully transfered to the stochastic case [8-11]. We present the necessary information on the theory of degenerate groups operators [7], and consider Banach spaces U and F, linear and bounded operators L,M E L(U; F). Following the classical work [7], let us conside L-resoh'ent set p^L (M) = {ц E C :(p,L — M)-¹EL (F; U)} aiid L-spectrum a^L (M) = C \ p^L (M) of the cjperator M. If the L-specturm a^L (M) of the cjperator M is bounded, then the operator M is cailed (L, a)-bounded. If the operator M is (L, a)-bounded, then the operators

P =

Д [ RL (M) du E L (U), 2 ni

γ

Q = Д [ LL (M) du E L (F)

2 ni ^γ

are projectors. Here RL(M) = (uL — M)-¹L is a right. and LL(M) = L(uL — M)-¹ is a left L- resolvent of the operator M, y C C is the contour bounding the domain containing a^L(M)• Let U0 (U¹) = kerP (imP). F⁰ (F¹) = kerQ (imQ) and I>y the Lk (Mk) denote the restriction of the operator L (M) оn U^k, k = 0, 1.

Theorem 4. (Splitting theorem, [7]) Let the operator M be (L, a)-bounded. Then

• (i)    the action of the operators Lk(Mk) E L(U^k; F^k), k = 0, 1 is observed;

• (ii)    there exist operators M0¹E L(F⁰; U⁰) a/iid L^-1E L(F¹; U¹).

Then by Theorem 4 in the case of (L, a)-boundness of the operator M we can construct the operators H = M-¹L₀E L(U⁰). S = L-¹M1 E L(U¹)•

K.V. Vasyuchkova, N.A. Manakova, G.A. Sviridyuk

Definition 2. Operator M is cailed (L,p)-bounded, p E {0} U N, if to is a point of the removable singularity (i.e. H = O, p = 0/ or the pole of order p E N (i. e. H^p = O, H^p+1= O) of the L-resolrent (pL — M)-¹of opcerator M.

Let us consider Showalter - Sidorov problem (3), (4), where the operators L = L(Л) and M = M(Л) are the polynomials with real coefficients, and their degrees satisfy the relation degL > degM,                             (10)

acting in the Banach space of sequences. Futher let be U = lm^+2degLL₂, F = lmL₂, m E R, q E R₊. It was shown In [5]. that the operators L,M E L(U; F). Let K = {pk} be a monotonically decreasing numerical sequence, such that lim pk = 0 and the numerical k→∞

∞ series ^2 Pk < + to.

k=1

Lemma 1. [5] Let the condition (10) be satisfied and the polynomials L = L(s) and M = M(s) hare only real roots and have no common roots. Then the operator M is (L, 0)L)ounded.

Definition 3. Let us call a random K-process n E CKlmL₂(0, т) a classical solution of equation (3), if a.s. all its trajectories satisfy equation (3) for all t E (0, т). Moreover, if the solution n = n(t) °/ equation (3) satisfies condition (4), we will call it a classical solution of problem (3), (4).

In the work [12] the sufficient conditions of the existence of unique classical solution of abstract problem (3), (4) were found. Let write the initial random variable nо in the form

∞

По = 52 nо k sp ek.                         UU k=1

Here the random variables nоk E L₂are uniformly bounded, i.e. there exists such number Cо > 0. tlrat Dnоk < Cо, ^k E N. Then In the case of arbitrarily operators L and M the Theorem is hold.

Theorem 5. [12] Let the. operator M bc (L,p)-bowtdcd. p E {0} U N. Then for each N E L (U; F), each random K-process 0 = 0( t) such, that (I — Q) N 0 E C p+1l m L2 an id. QN 0 E CK lm l2 is satisfied, and for each random variable nо E lmL2, not depending on 0 for every fixed t, there exists the unique solution n E CKlmL2 of problem (3), (4), which has the form n (t) = Utn о +

t

^∫U^t

о

sL^-1QN0(s)ds — 52 HqM-¹(I — Q)N q=о

0⁽q) Z X

0    (t).

Here

Utn о = 52 e^Vktn о k VPke ek. k: Vk EaL (M)

The proof of Theorem 5 is based on the methods of the theory of degenerate groups of solving operators [7] as in the deterministic case. Now we turn to problem (3), (4), where as an external impact we consider "white noise" 0 = W_K (t).

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

Theorem 6. Let condition (10) be satisfied and the polynomials L = L (s) a nd M = M (s) have only real roots and have no common roots. Then for each N E L(U; F), every sequence K such, that tlte conditions (I — Q) NW_K (t) E CKlm L2 an id QNW_K (t) E C к lm L2 arc satisfied, and for each random variable nо E lmL2, not depending on WK for every fixed t, there exists the unique solution n E CKlm L2 of problem (3), (4), which have the form

П (t) = Utn о + L i¹

QNWk (t) — Mi

t

I U^t-sL-¹QNWk(s)ds о

o

— M-¹(I — Q) NWk (t) .

Here

∞

Utn 0 = E' e^Vktn о k Vikek, k =1

L1

∞

= E'

k=1

ξk ek

LX)

∞

M12 = E' M (Xk) (k ek,  Mo~ ^

k=1

E

k:L(Xk )=0

ξkek

MX) ’

2 = {(k}, vk = MLX) are ^l,e Poi/n;ts of L-spectrum aL (M) of op orator M: vectors ek = (0,..., 0,1, 0,...). where the. unit stands oil the. k-th place: the. sign means the. absence, of k-th members of a. series of such, that the. polynomial L(Xk) is equal to zero.

Remark 3. The proof of Theorem 6 is based on the results of Theorem 5 and the limiting transition proposed in the paper [8]. Since "white noise" 0(t) = (2t)-¹W_K(t) is not differentiable in the case t = 0, by the analogue in [8] and by the definition of the Nelson - Gliklikh derivative for all e E (0, t), t E R₊, integrating by parts the second term in right side (12) we get:

to

J U^t-sL^-1QN Wк (s)ds = L^-1QNWk(t) — U^t-eL^-1QNWk (e) —

-

t

SP J U^t-sL^-1QNWk(s)ds.

ε

Then passing to the limit in (13) for e ^ 0, we obtain the reiquired, i.e.

t

t

I U^t-sL^-1QN Wk (s)ds = L^-1QNWk (t) — SP j U^t-sL^-1QNWk(s)ds.

о

о

Acknowledgements. The work was supported by Act 211 Government of the Russian Federation, contract No. 02.A03.21.0011.