# Linear Sobolev Type Equations with Relativelyp-Sectorial Operators in Space of “Noises”

Authors: A. Favini; G. A. Sviridyuk; N. A. Manakova
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

The concept of “white noise,” initially established in finite-dimensional spaces, is transferred to infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical interest. To reach this goal the Nelson-Gliklikh derivative is introduced and the spaces of “noises” are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable “noises.” The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition is considered as an application.


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## Body

## 1. Introduction

LetU and F be Banach spaces, the operator L∈L(U;F) (linear and continuous), and the operator M∈clU;F (linear, closed, and densely defined). Consider the equation(1)Lu˙=Mu,ker⁡L≠0.Equations of the form (1) were firstly studied in the works of A. Poincare. Then they appeared in the works of S. V. Oseen, J. V. Boussinesq, S. G. Rossby, and other researchers that were dedicated to the investigation of some hydrodynamics problems. Their systematical study started in the middle of the XX century with the works of S. L. Sobolev. The first monograph [1] devoted to the study of equations of the form (1) appeared in 1999. Nowadays the number of works devoted to such equations is increasing extensively [1–3]. Sometimes such equations are called “equations that are not of Cauchy-Kovalevskaya type,” “pseudoparabolic equations,” “degenerate equations,” or “equations unsolved with respect to the higher derivative.” We call equations of the form (1) the Sobolev type equations. This term was firstly proposed in the works of Carroll and Showalter [4]. The Sobolev type equations constitute the vast area in nonclassical equations of mathematical physics [5]. The theory of degenerate semigroups of operators is a suitable mathematical tool for the study of such problems [2].The right part of (1) can be subjected to random perturbations, such as white noise. Abstract stochastic equations are of great interest nowadays due to the large amount of applications. Linear stochastic differential equation in the simplest case can be represented in the form(2)dη=Sη+ψdt+Adw,where S and A are some linear operators; ψ=ψ(t) is a deterministic external influence and w=w(t) is a stochastic external influence; η=η(t) is unknown random process. Firstly dw was understood in the sense of differential of the Wiener process w=W(t) and was traditionally treated as white noise. K. Ito was the first to study ordinary differential equations of the form (2); then R. L. Stratonovich and A. V. Skorokhod developed research. The Ito-Stratonovich-Skorokhod approach in the finite-dimensional case remains popular to this day [6, 7]. Moreover, it was successfully distributed to infinite-dimensional situation [8, 9], and even it was applied to studies of the Sobolev type equations [10, 11]. Another approach was presented in [12], where (2) was considered in the Schwartz spaces and the distributional derivative of the Wiener process makes sense.A new approach to studying (2), where the noise is defined by the Nelson-Gliklikh derivative of the Wiener process, appeared recently and is actively developing [13, 14]. At first white noise was used in the theory of optimal measurements [15], where a special noises space was constructed [16]. In [17] the concept of white noise was also extended to the infinite-dimensional space CKL2 ofK-random processes with a.s. continuous trajectories and the space CKlL2 ofK-random processes, whose trajectories are a.s. continuously differentiable in the Nelson-Gliklikh sense up to order l∈N. The solvability of Showalter-Sidorov problem for linear stochastic Sobolev type equations with relatively bounded operators was studied in [17]. Our purpose is to study the solvability of weakened (in sense of S. G. Krein) Showalter-Sidorov problem for linear stochastic Sobolev type equation with relatively sectorial operator. The purpose of such extention is the development of the theory of stochastic Sobolev type equations and application of this theory to nonclassical models of mathematical physics of practical value.The paper is organized as follows. In the second section we introduce the definition of a strongly relativelyp-sectorial operator and construct semigroups of the resolving operators. In the third section the Nelson-Gliklikh derivative ofK-random process with values in real separable Hilbert spaces is considered. In particular theK-Wiener process is studied. Then the space of such processes, containing theK-Wiener process and its Nelson-Gliklikh derivative (i.e., white noise), is constructed. In the fourth section the theory of stochastic Sobolev type equations with relatively p-sectorial operators is developed; namely, the stochastic Sobolev type equation(3)Lη∘=Mη+Nwis considered. Here η=η(t) is the unknown random process, η∘ is its Nelson-Gliklikh derivative, w=w(t) is a random process, responsible for external influence; the operators L,M,N∈LU;F; moreover the operator M is (L,p)-sectorial, p∈{0}∪N. Add to (3) with a weakened Showalter-Sidorov condition(4)limt→0+⁡RαLMp+1ηt-ξ0=0,where RαL(M)=(αL-M)-1L, α∈ρL(M). Condition (4) is a natural generalization of condition(5)Lu0-u0=0which is in its turn the generalization of the Cauchy condition(6)u0=u0.Note that condition (4) is more natural for the Leontieff type system and for the Sobolev type equations [5] than the traditional Cauchy condition (6). Problem (5) for the deterministic Sobolev type equation was firstly studied in [1]. This investigation formed the basis of the study of problem (5) for linear stochastic Sobolev type equation (3) [11]. The existence and the uniqueness of classical solution for problem (3), (4) are proved in the fourth section of our paper. In the fifth section we apply the abstract scheme to the investigation of the Dzektser model [18], describing free surface evolution of filtered liquid.


## 2. Holomorphic Degenerate Semigroups of Operators

LetU and F be Banach spaces, and let the operators L∈L(U;F), M∈cl⁡(U;F). Consider the L-resolvent set of M,  ρLM=μ∈C:μL-M-1∈LF;U, theL-spectrum σL(M)=C∖ρL(M) of the operator M, and the right and the leftL-resolvents of the operator M, RμL(M)=μL-M-1L, LμL(M)=LμL-M-1, respectively. Let μk∈ρLM,k=0,1,…,p. The operator-functions(7)Rμ,pL=∏k=0pRμkLM,Lμ,pLM=∏k=0pLμkLMare calledthe right andthe left(L,p)-resolvents of the operator M.Definition 1.
OperatorM is said to bep-sectorial, p∈{0}∪N with respect to the operator L (or shortly (L,p)-sectorial), if(i)
∃a∈R, θ∈(π/2,π) such that the sector(8)Sa,θL=μ∈C:arg⁡μ-a<θ,μ≠a⊂ρLM;(ii)
∃K>0 such that(9)max⁡Rμ,pLMLU,Lμ,pLMLF≤K∏q=0pμq-a,for all μq∈Sa,θL, q=0,1,…,p.Remark 2.
Without loss of generality we can puta=0 in Definition 1. Indeed, if we find a resolving semigroup of (1) Vt∣t∈R+ for a=0, then the semigroup eatVt∣t∈R+ will be resolving when a≠0.Letα∈ρL(M). Consider two equivalent forms of the linear homogeneous Sobolev type equation (1)(10)RαLMu˙=αL-M-1Mu,(11)LαLMf˙=MαL-M-1fas concrete interpretations of the equation(12)Av˙=Bv,defined on a Banach space V, where the operators A,B∈L(V). Operator (αL-M)-1M=-I+α(αL-M)-1L is linear bounded on a dense set in U and it can be uniquely continued to a bounded operator -I+α(αL-M)-1L defined on U. For (10) the space V=U and for (11) V=F.Definition 3.
The vector functionv∈C∞(R+;V) satisfying (12) on R+≡(0,+∞) is calleda solution of (12).Definition 4.
The mappingV•∈C∞(R+;L(V)) is calleda semigroup of the resolving operators(a resolving semigroup) of (12), if(i)
VsVtv=Vs+tv for all s,t>0 and any v∈V;(ii)
v(t)=Vtv is a solution of (12) for any v from a dense set in V.The semigroup is calleduniformly bounded, if(13)∃C>0VtLV≤C∀t∈R+.The semigroup is calledanalytic, if it can be extended to some sector containing the ray R+ with fulfillment of properties (i), (ii) in Definition 4.Theorem 5 (see [2, p. 60]).
Let the operatorM be (L,p)-sectorial, p∈{0}∪N. Then there exists a uniformly bounded and analytic resolving semigroup of (10) and (11) and it is represented by(14)Ut=12πi∫ΓRμLMeμtdμ,Ft=12πi∫ΓLμLMeμtdμ,where t∈R+ and contour Γ⊂ρL(M) is such that arg⁡μ→θ for μ→∞, μ∈Γ.For example, contourΓ=μ∈C:arg⁡μ=θ, where θ∈(π/2,π) is taken from Definition 1.LetU1(F1) be the closure of im⁡R(μ,p)L(M) (im⁡L(μ,p)L(M)) in the norm of the space U(F). The set(15)ker⁡V•=v∈V:Vtv=0∃t∈R+is calleda kernel [2, p. 61] of the semigrop V• and the set(16)im⁡V•=v∈V:limt→0+⁡Vtv=vis calledan image [2, p. 61] of the semigrop V•.Theorem 6 (see [2, p. 62]).
Let the operatorM be (L,p)-sectorial. Then im⁡U•=U1, im⁡F•=F1.Further we assume that the operatorM is (L,p)-sectorial. Set U0=ker⁡U•, F0=ker⁡F•. By L0(M0) denote the restriction of the operator L(M) on U0(U0∩dom⁡M).Theorem 7 (see [2, pp. 63, 64]).
Let the operatorM be (L,p)-sectorial. Then(i)
the operatorL0∈L(U0;F0), and the operator M0:U0∩dom⁡M→F0;(ii)
there exists the operatorM0-1∈L(F0;U0);(iii)
the operatorH=M0-1L0∈L(U0) is nilpotent with degree less or equal to p.ByL1(M1) denote the restriction of the operator L(M) on U1(U1∩dom⁡M).Consider the following conditions:A1U0⊕U1=U,F0⊕F1=F,and there exists the operatorA2L1-1∈LF1;U1.Remark 8.
ConditionA1 holds, for example, in the case where M is strongly (L,p)-sectorial on the right (left) or when the space U(F) is reflexive [2, page 69]. Condition A2 holds in the case when the operator M is strongly (L,p)-sectorial or when it is (L,p)-sectorial, condition A1 is fulfilled and U1=im⁡L1.ConditionA1 is equivalent to the existence of the projector P(Q) along U0(F0) on U1(F1).Theorem 9 (see [2, pp. 69, 71, 73]).
Let the operatorM be (L,p)-sectorial and let conditions A1, A2 be fulfilled. Then(i)
the projectorP (Q) can be represented as(17)P=U0=s-limt→0+⁡Ut,Q=F0=s-limt→0+⁡Ft;(ii)
the operatorM1∈cl(U1;F1) and the operator M0∈cl(U0;F0);(iii)
the operatorS=L1-1M1∈cl(U1) is sectorial.The solution to (12) is calleda solution to a Cauchy problem if it also satisfies the condition(18)limt→0+⁡vt=v0.Definition 10.
The setP⊂U is calleda phase space of (12), if(i)
any solutionv=v(t) of (12) lies in P; that is, vt∈P for all t∈R+;(ii)
for anyv0∈P there exists a unique solution of problem (12), (18).Theorem 11 (see [2, p. 67]).
Let the operatorM be (L,p)-sectorial. Then phase space of (10) and (11) coincides with the image of semigroup im⁡U•(im⁡F•).


## 3. The Spaces of “Noises”

LetΩ≡(Ω,A,P) be a complete probability space and let R be the set of real numbers endowed with Boreal σ-algebra. The measurable mapping ξ:Ω→R is calleda random variable. The set of random variables with zero mean and finite variances forms a Hilbert space with the scalar product (ξ1,ξ2)=Eξ1ξ2, where E denotes the mathematical expectation. This Hilbert space will be denoted by L2. The random variables ξ∈L2, with normal (Gaussian) distribution, will be very important later on; they are called Gaussian random variables. Let A0 be a σ-subalgebra of σ-algebraA. Construct the space L20 of random variables, measurable with respect to A0. Obviously, L20 is a subset of L2; denote by Π:L2→L20 the orthoprojector. Let ξ∈L2, then Πξ is calledconditional expectation of the random variable ξ and is denoted by E(ξ∣A0). It is easy to see that E(ξ∣A0)=Eξ, if A0=⌀,Ω; and E(ξ∣A0)=ξ, if A0=A. Finally, the minimal σ-subalgebra A0⊂A, regarding which random variable ξ is measurable, is calledtheσ-algebra generated byξ.LetI⊂R be some interval. Consider two mappings: the first one f:I→L2, which maps each t∈I to a random variable ξ∈L2, and the second one g:L2×Ω→R, which maps every pair (ξ,ω) to the point ξ(ω)∈R. The composition η:I×Ω→R, η=η(t,ω)=g(f(t),ω), is calleda (one-dimensional) random process. Thus, for every fixed t∈I the random process η=η(t,·) is a random variable; that is, η(t,·)∈L2, and for every fixed ω∈Ω the random process η=η(·,ω) is calledthe (sample) trajectory. The random process η is calledcontinuousif almost surely (a.s.) all its trajectories are continuous; that is, for almost every (a.e.) ω∈Ω the trajectories η(·,ω) are continuous. The set of continuous random processes form a Banach space, which will be denoted by CL2. The continuous random process, whose random variables are Gaussian, is calledGaussian.The (one-dimensional) Wiener processβ=β(t), modeling Brownian motion on the line in Einstein-Smolukhovsky theory, is one of the most important examples of the continuous Gaussian random processes. It has the following properties:(W1)
a.s.β(0)=0; a.s. all its trajectories β(t) are continuous, and for all t∈R¯+(={0}∪R+) the random variable β(t) is Gaussian;(W2)
the mathematical expectationEβt=0 and autocorrelation function Eβt-βs2=t-s for all s,t∈R¯+;(W3)
the trajectoriesβ(t) are nondifferentiable at any point t∈R¯+ and have unbounded variation on any small interval.Example 12.
There exists a random processβ, satisfying properties (W1), (W2); moreover, it can be represented in the form(19)βt=∑k=0∞ξksin⁡π22k+1t,where ξk are independent Gaussian variables, Eξk=0, and Dξk=[π/2(2k+1)]-2, where D denotes the dispersion.The random processβ, satisfying properties (W1)-(W2), will be calledBrownian motion.Now fixη∈CL2 and t∈I(=(ε,τ)⊂R) and by Ntη denote the σ-algebra, generated by the random variable η(t). For the sake of brevity, we introduce the notation Etη=E(·∣Ntη).Definition 13.
Letη∈CL2, and the random variable(20)Dηt,·=limΔt→0+⁡Etηηt+Δt,·-ηt,·Δt,D∗ηt,·=limΔt→0+⁡Etηηt,·-ηt-Δt,·Δtis calleda forwardDη(t,·) (a backwardD∗η(t,·))mean derivative of the random processηat the pointt∈(ε,τ) if the limit exists in the sense of uniform metric on R. The random process η is calledforward(backward) mean differentiable on(ε,τ), if for every point t∈(ε,τ) there exists the forward (backward) mean derivative.Now let the random processη∈CL2 be forward (backward) mean differentiable on (ε,τ). Its forward (backward) mean derivative is also a random process; we denote it by Dη (D∗η). If the random process η∈CL2 is forward (backward) mean differentiable on (ε,τ), thenthe symmetric mean derivative(21)DSη=12D+D∗ηcan be defined. Since the mean derivatives were introduced by Nelson [19], and the theory of these derivatives was developed by Gliklikh [7], the symmetric mean derivative DS or the random process η will henceforth be calledthe Nelson-Gliklikh derivative for brevity and will be denoted by η∘; that is, DSη≡η∘. By η∘(l), l∈N denote the lth Nelson-Gliklikh derivative of the random process η. Note that if the trajectories of the random process η are a.s. continuously differentiable in a “common sense” on (ε,τ), then the Nelson-Gliklikh derivative of η coincides with the “regular” derivative.Theorem 14 (see [14]).
Letβ∘(l)(t)=-1l+1∏i=1l-1(2i-1)2t-lβ(t) for all t∈R+ and l∈N.Now letV≡(V,·,·) be a real separable Hilbert space; consider the operator K∈L(V) with spectrum σ(K) whose elements are nonnegative, discrete, with finite multiplicity tending only to zero. By {λj} denote the sequence of eigenvalues of operator K, numbered in decreasing order according to their multiplicity. Note that the linear span of related orthonormal eigenfunctions {φj} of operator K is dense in V. Suppose that the operator K is nuclear (i.e., its trace Tr⁡K=∑j=1∞λj<+∞).Take the sequence of independent random processes{ηj} and define the K-random process(22)ΘKt=∑j=1∞λjηjtφjprovided that the series (22) converges uniformly on any compact subset of I. Note that if {ηj}⊂CL2 and the K-random process ΘK exists, then a.s. its trajectories are continuous. Denote the space of such processes by the symbol CK≡CK(I×Ω;V). Consider in CK the subspace CKL2 of random processes, whose random variables belong to L2(Ω;V)=ξ:∫Ωξ(ω)2dP(ω)<+∞; that is, η∈CKL2, if η(t,·)∈L2(Ω;V) for each t∈I. Note that the space CKL2 contains, in particular, those K-random processes for which almost surely all trajectories are continuous, and all (independent) random variables are Gaussian.We now introduce the Nelson-Gliklikh derivatives of aK-random process(23)Θ∘Klt=∑j=1∞λjη∘jltφjprovided that the derivatives up to degree of l in the right hand side of (23) exist and the series uniformly converges on any compact subset of I.Similarly, introduce the spaceCKlL2 of K-random processes with a.s. continuous Nelson-Gliklikh derivatives up to order l∈N, whose random variables belong to L2(Ω;V).As an example consider theK-Wiener process(24)WKt=∑j=1∞λjβjtφj,which is defined on R¯+.Corollary 15.
LetW∘Klt=-1l+1∏i=1l-1(2i-1)2t-lWK(t) for all t∈R+, l∈N, and nuclear operator K∈L(V).Moreover, theK-Wiener process (24) satisfies conditions (W1) a.s. WK(0)=0, a.s. all its trajectories β(t) are continuous, and for all t∈R¯+ the random variable WK(t,·) is Gaussian; (W2) the mathematical expectation EWKt=0 and autocorrelation function(25)Eβt-βs2=t-sK∀s,t∈R¯+.


## 4. The Stochastic Sobolev Type Equation with Relativelyp-Sectorial Operator

LetU, F, V be real separable Hilbert spaces. Let the operator M be (L,p)-sectorial, let p∈{0}∪N and conditions A1, A2 be fulfilled, and let the operator N∈L(V;F). Let I=[0,τ). Let the operator K∈L(V) be nuclear with eigenvalues {λj}⊂R+. Consider the linear stochastic Sobolev type equation (3) with condition (4).Remark 16.
Due to Theorem6 condition (4) is equivalent to the following condition:(26)limt→0+⁡Pηt-ξ0=0.Definition 17.
TheK-random process η∈CK1L2 is calleda (classical)solution of(3), if a.s. all its trajectories satisfy (3) with some K-random process w∈CKL2 for all t∈(0,τ). The solution of (3) is calleda solution of weakened Showalter-Sidorov problem (3), (4), if it also satisfies condition (4).Suppose that theK-random process w=w(t),t∈[0,τ), satisfies condition(27)I-QNw∈CKp+1L2,QNw∈CKL2.Theorem 18.
Let the operatorM be (L,p)-sectorial, let p∈{0}∪N and conditions A1, A2 be fulfilled, and let operator N∈L(V;F). For any K-random process w=w(t) satisfying (27) and for any U-valued random variable ξ0∈L2, independent of w, there exists a unique solution η∈CK1L2 to problem (3), (4), given by(28)ηt=Utξ0+∫0tUt-sL1-1QNwsds-∑q=0pHqM0-1I-QNw∘qt.Proof.
Proof of the theorem is analogous to the deterministic case [2]. Acting on (3) and condition (4) by projectors Q and I-Q and using Theorems 7 and 9, reduce it to the equivalent system of two independent problems(29)Hη∘0=η0+M0-1I-QNw,(30)η∘1=Sη1+L1-1QNw,limt→0+⁡η1t=ξ01,where η0=(I-P)η,  η1=Pη,  ξ01=Pξ0. Since the operator H is nilpotent, it follows from (29) that necessarily(31)η0t=-∑q=0pHqM0-1I-QNw∘qt.Since the operator S∈cl⁡(U1) the solution of problem (30) exists and can be represented in the form(32)η1t=Utξ01+∫0tUt-sL1-1QNwsds.Consider theweakened Showalter-Sidorov problem (4) for equation(33)Lη∘=Mη+NW∘K,here the right hand side includes the Nelson-Gliklikh derivative of theK-Wiener process WK(t). The white noise w(t)=(2t)-1WK(t) does not satisfy condition (27). One of feasible approaches to overcome this difficulty was proposed in [10, 11]. The advantage of this approach comes from transformation of the second summand in the right hand side of (28) as follows:(34)∫εtUt-sL1-1QNW∘Ksds=L1-1QNWKt-Ut-εL1-1QNWKε-SP∫εtUt-sL1-1QNWKsds.By virtue of the definition of Nelson-Gliklikh derivative for all ε∈(0,t), t∈R+, we can make integration by parts. Letting ε→0 in (34) we get(35)∫0tUt-sL1-1QNW∘Ksds=L1-1QNWKt-SP∫0tUt-sL1-1QNWKsds.Theorem 19.
Let the operator M be(L,p)-sectorial and let p∈{0}∪N and conditions A1, A2 be fulfilled. For any N∈L(V;F) and for any U-valued random variable ξ0∈L2, independent of WK, there exists a unique solution η=η(t) of problem (4), (33), given by(36)ηt=Utξ0+L1-1QNWKt-M1∫0tUt-sL1-1QNWKsds-∑q=0pHqM0-1I-QNW∘Kq+1t.


## 5. Dzektser Stochastic Model

LetD⊂Rn be a bounded domain with a boundary ∂D of class C∞. Consider a boundary value (37)Δηx,t=ηx,t=0,x,t∈∂D×0,Tand initial value(38)limt→0+⁡λ-Δηt-ξ0=0problems for the stochastic equation(39)λ-Δη∘=βΔ-αΔ2η+w.Here the parameters λ,β∈R, α∈R+. This model describes evolution of free surface of filtered liquid.Define the spaceU=W22(D)∩W∘21(D) and the space F=L2(D) with the scalar product(40)u,v=∫Duvdx.Denote by {λj} the sequence of eigenvalues of the homogeneous Direchlet problem for the operator Δ, numbered in nonincreasing order with regard to multiplicities and tending to -∞. By {φj} denote the orthonormal (in the sense of L2(D)) family of corresponding eigenfunctions φj∈C∞(D), j∈N. Introduce theF-valued K-random process. Define the operator Λ=(-1)m-1Δm with the domain(41)dom⁡Λ=W22mD:Δkux=0,x∈∂D,k∈0,1,…,m-1,m∈N.It is rather easy to find such a number m according to fixed number n (which is the dimension of the domain D) that the mentioned series converges. For example, m can be equal to n. Note that the operator Λ has the same eigenfunctions {φj}, as the Laplace operator, but its spectrum consists of eigenvalues λjm. Since their asymptotic λjm~j2m/n→∞, j→∞, we take such number m∈N that the series ∑j=1∞λj-m converges. Then the operator Λ is continuously invertable on dom⁡Λ, whereas the inverse operator (i.e., the Green operator) has the spectrum consisting of eigenvalues νj=λj-m. We take this operator as the nuclear operator K forF-valued K-random process.Fixλ,β∈R, α∈R+, and define the operators L=λI-Δ,M=βΔ-αΔ2. The operator L∈L(U;F), and the operator M∈cl⁡(U;F) with(42)dom⁡M=u∈W24D:Δux=ux=0,x∈∂D.Lemma 20 (see [2, p. 198]).
For anyλ,β∈R, α∈R+ the operator M is (L,0)-sectorial.TheL-spectrum of the operator M consists of all points of the form(43)μk∈C:μk=βλk-αλk2λ-λk,ifk:λk≠λ.By Theorem 5 there exists a holomorphic resolving semigroup for (39) in the form(44)
U
t
=
∑
k
=
1
∞
e
μ
k
t
·
,
φ
k
φ
k
,
if
λ
k
≠
λ
∀
k
∈
N
;
∑
k
∈
N
:
k
≠
l
e
μ
k
t
·
,
φ
k
φ
k
,
if
∃
l
∈
N
:
λ
l
=
λ
.Lemma 21.
For anyλ,β∈R,  α∈R+ conditions A1, A2 are fulfilled.Construct subsets(45)U0=0,ifλk≠λ∀k∈N;u∈U:u,φk=0,k∈N∖l:λl=λ;U1=U,ifλk≠λ∀k∈N;u∈U:u,φk=0,λk=λ.Obviously U0⊕U1=U. Thus the projector P has the form(46)P=s-limt→0+⁡Ut=I,ifλk≠λ∀k∈N;∑k∈N:k≠l·,φkφk,if∃l∈N:λl=λ.The projector Q is constructed analogously.Moreover there exists the operator(47)
L
1
-
1
=
∑
k
=
1
∞
λ
-
λ
k
-
1
·
,
φ
k
φ
k
,
if
λ
k
≠
λ
∀
k
∈
N
;
∑
k
∈
N
:
k
≠
l
λ
-
λ
k
-
1
·
,
φ
k
φ
k
,
if
∃
l
∈
N
:
λ
l
=
λ
.Conditions (3) and (38) take the form(48)limt→0+⁡ηt-ξ0=0,ifλk≠λ∀k∈N;limt→0+⁡∑k≠lηt-ξ0,φkφk=0,if∃l∈N:λl=λ.Thus, we have reduced problem (37)–(39) to problem (3), (4). From Theorem 18 we have the following assertion.Theorem 22.
For anyλ,β∈R, τ,α∈R+ and for any K-random process w=w(t) satisfying (27) and for any U-valued random variable ξ0∈L2, independent of w, there exists a unique solution η∈CK1L2 for problem (37)–(39), given by(49)ηt=-M0-1ξ0t+Utξ0+∫0τUt-sL1-1Qwsds.Here(50)M0-1=O,ifλk≠λ∀k∈N;∑k∈N:λk=λβλk-αλk2-1·,φkφk.Consider the initial-boundary value problem (37), (38) for equation(51)λ-Δη∘=βΔ-αΔ2η+W∘K,where the right part includes the Nelson-Gliklikh derivative of theK-Wiener process WK(t). From Theorem 19 we have the following assertion.Theorem 23.
For anyλ,β∈R, τ,α∈R+, and U-valued random variable ξ0∈L2, independent of WK there exists a unique solution η=η(t) of problem (37), (38), (51) given by(52)ηt=Utξ0+L1-1QWKt-M1∫0tUt-sL1-1QWKsds-∑q=0pHqM0-1I-QW∘Kq+1t.Here(53)
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*Source: 697410-2015-11-23.xml*