Stochastic Volterra equations driven by 7 cylindrical Wiener process 0 0 2 Anna Karczewska and Carlos Lizama n a J Department of Mathematics, University of Zielona G´ora 0 1 ul. Szafrana 4a, 65-246 Zielona G´ora, Poland, e-mail: [email protected] ] R Universidad de Santiago de Chile, Departamento de Matem´atica, Facultad de Ciencias P . Casilla 307-Correo 2, Santiago, Chile, e-mail: [email protected] h t a m Abstract [ In this paper, stochastic Volterra equations driven by cylindrical Wiener pro- 2 v cess in Hilbert space are investigated. Sufficient conditions for existence of strong 1 solutions are given. The key role is played by convergence of α-times resolvent 4 2 families. 0 1 6 0 1 Introduction / h t a Let H be a separable Hilbert space with a norm |·| and A be a closed linear unbounded m H operator with dense domain D(A) ⊂ H equipped with the graph norm | · | . The : D(A) v purpose of this paper is to study the existence of strong solutions for a class of stochastic i X Volterra equations of the form r a t t X(t) = X + a(t−τ)AX(τ)dτ + Ψ(τ)dW(τ), t ≥ 0, (1) 0 Z0 Z0 tα−1 wherea(t) = , α > 0,andW,Ψareappropriatestochasticprocesses. Itiswellknown Γ(α) that there are several situations that can be modeled by stochastic Volterra equations (see e.g. [7, Section 3.4 ] and references therein). We note that stochastic Volterra equations driven by white noise have been studied in [3] among other authors. A similar equation and very related to our case appears first studied in [2]. Here we are interested in the study of strong solutions when equation (1) is driven by a cylindrical Wiener process W. 2000 Mathematics Subject Classification: primary: 60H20; secondary: 60H05, 45D05. Keywords and phrases: stochasticVolterraequation,α-timesresolventfamily,strongsolution,stochastic convolution, convergence of resolvent families. 1 Whena(t)isacompletely positivefunction, sufficient conditionsforexistence ofstrong solutions for (1) were obtained in [9]. This was done using a method which involves the use of a resolvent family associated to the deterministic version of equation (1): t u(t) = a(t−τ)Au(τ)dτ +f(t), t ≥ 0, (2) Z0 where f is an H-valued function. However, there are two kinds of problems that arise when we study (1). On the one hand, the kernels tα−1 are α-regular and απ -sectorial but not completely positive Γ(α) 2 functions for α > 1, so e.g. the results in [9] cannot be used directly for α > 1. On the other hand, for α ∈ (0,1), we have a singularity of the kernel in t = 0. This fact strongly suggests the use of α-times resolvent families associated to equation (2). This new tool appears carefully studied in [1] as well as their relationship with fractional derivatives. For convenience of the reader, we provide below the main results on α-times resolvent families to be used in this paper. Our second main ingredient to obtain strong solutions of (1) relies on approximation of α-times resolvent families. This kind of result was very recently formulated by Li and Zheng [10]. It enables us to prove a key result on convergence of α-times resolvent families (see Theorem 2 below). Then we canfollowthe methods employed in [9]to obtain existence of strong solution for the stochastic equation (1) (see Theorem 4). Our plan for the paper is the following. In section 2 we formulate the deterministic results which will play the key role for the paper. Section 3 is devoted to weak and mild solutions while in section 4 we provide strong solution to (1). More precisely, we give sufficient condition for a stochastic convolution to be a strong solution to (1). 2 Convergence of α-times resolvent families In this section we formulate the main deterministic results on convergence of resolvents. We denote tα−1 g (t) = , α > 0, t > 0, α Γ(α) where Γ is the gamma function. By S (t), t ≥ 0, we denote the family of α-times resolvent families corresponding to α the Volterra equation (2), if it exists, and defined as follows. Definition 1 (see [1]) A family (S (t)) of bounded linear operators in a Banach space B is called α-times α t≥0 resolvent family for (2) if the following conditions are satisfied: 1. S (t) is strongly continuous on R and S (0) = I; α + α 2 2. S (t) commutes with the operator A, that is, S (t)(D(A)) ⊂ D(A) and AS (t)x = α α α S (t)Ax for all x ∈ D(A) and t ≥ 0; α 3. the following resolvent equation holds t S (t)x = x+ g (t−τ)AS (τ)xdτ (3) α α α Z0 for all x ∈ D(A), t ≥ 0. Necessary and sufficient conditions for existence of the α-times resolvent family have been studied in [1]. Observe that the α-times resolvent family corresponds to a C - 0 semigroup in case α = 1 and a cosine family in case α = 2. In consequence, when 1 < α < 2 such resolvent families interpolate C -semigroups and cosine functions. In 0 particular, for A = ∆, the integrodifferential equation corresponding to such resolvent family interpolates the heat equation and the wave equation (see [6]). Definition 2 An α-times resolvent family (S (t)) is called exponentially bounded α t≥0 if there are constants M ≥ 1 and ω ≥ 0 such that kS (t)k ≤ Meωt, t ≥ 0. (4) α If there is the α-times resolvent family (S (t)) for A and satisfying (4), we write α t≥0 A ∈ Cα(M,ω). Also, set Cα(ω) := ∪ Cα(M,ω) and Cα := ∪ Cα(ω). M≥1 ω≥0 Remark 1 It was proved by Bazhlekova [1, Theorem 2.6] that if A ∈ Cα for some α > 2, then A is bounded. The following subordination principle is very important in the theory of α-times re- solvent families (see [1, Theorem 3.1]). Theorem 1 Let 0 < α < β ≤ 2,γ = α/β,ω ≥ 0. If A ∈ Cβ(ω) then A ∈ Cα(ω1/γ) and the following representation holds ∞ S (t)x = ϕ (s)S (s)xds, t > 0, (5) α t,γ β Z0 where ϕ (s) := t−γΦ (st−γ) and Φ (z) is the Wright function defined as t,γ γ γ ∞ (−z)n Φ (z) := , 0 < γ < 1. (6) γ n!Γ(−γn+1−γ) n=0 X 3 Remark 2 (i)WerecallthattheLaplacetransformoftheWrightfunctioncorrespondsto E (−z) where E denotes theMittag-Leffler function. Inparticular, Φ (z) isa probability γ γ γ density function. (ii) Also we recall from [1, (2.9)] that the continuity in t ≥ 0 of the Mittag-Leffler function together with the asymptotic behavior of it, imply that for ω ≥ 0 there exists a constant C > 0 such that E (ωtα) ≤ Ceω1/αt, t ≥ 0, α ∈ (0,2). (7) α As we have already written, in this paper the results concerning convergence of α- times resolvent families in a Banach space B will play the key role. Using a very recent result due to Li and Zheng [10] we are able to prove the following theorem. Theorem 2 Let A be the generator of a C -semigroup (T(t)) in a Banach space B 0 t≥0 such that kT(t)k ≤ Meωt, t ≥ 0. (8) Then, for each 0 < α < 1 we have A ∈ Cα(M,ω1/α). Moreover, there exist bounded opera- tors A and α-times resolvent families S (t) for A satisfying ||S (t)|| ≤ MCe(2ω)1/αt, n α,n n α,n for all t ≥ 0, n ∈ N, and S (t)x → S (t)x as n → +∞ (9) α,n α for all x ∈ B, t ≥ 0. Moreover, the convergence is uniform in t on every compact subset of R . + Proof Since A is the generator of a C semigroup satisfying (8), we have A ∈ C1(ω). 0 Hence, the first assertion follows directly from Theorem 1, that is, for each 0 < α < 1 there is an α-times resolvent family (S (t)) for A given by α t≥0 ∞ S (t)x = ϕ (s)T(s)xds, t > 0. (10) α t,α Z0 Since A generates a C -semigroup, the resolvent set ρ(A) of A contains the ray [w,∞) 0 and M ||R(λ,A)k|| ≤ for λ > w, k ∈ N. (λ−w)k Define A := nAR(n,A) = n2R(n,A)−nI, n > w, (11) n the Yosida approximation of A. Then ∞ n2ktk ||etAn|| = e−nt||en2R(n,A)t|| ≤ e−nt ||R(n,A)k|| k! k=0 X 2 ≤ Me(−n+nn−w)t = Menn−wwt . 4 Hence, for n > 2w we obtain ||eAnt|| ≤ Me2wt. (12) Next, since each A is bounded, it follows also from Theorem 1 that for each 0 < α < 1 n there exists an α-times resolvent family (S (t)) for A given as α,n t≥0 n ∞ S (t) = ϕ (s)esAnds, t > 0. (13) α,n t,α Z0 By (12) and Remark 2(i) it follows that ∞ kS (t)k ≤ ϕ (s)kesAnkds α,n t,α Z0 ∞ ∞ ≤ M ϕ (s)e2ωsds = M Φ (τ)e2ωtατdτ = ME (2ωtα), t ≥ 0. t,α α α Z0 Z0 This together with Remark 2(ii), gives kS (t)k ≤ MCe(2ω)1/αt, t ≥ 0. (14) α,n Now, we recall the fact that R(λ,A )x → R(λ,A)x as n → ∞ for all λ sufficiently n large (see e.g. [11, Lemma 7.3]), so we can conclude from [10, Theorem 4.2] that S (t)x → S (t)x as n → +∞ (15) α,n α for all x ∈ B, uniformly for t on every compact subset of R . + An analogous result can be proved in the case when A is the generator of a strongly continuous cosine family. Theorem 3 Let A be the generator of a C -cosine family (T(t)) in a Banach space B. 0 t≥0 Then, for each 0 < α < 2 we have A ∈ Cα(M,ω2/α). Moreover, there exist bounded opera- tors A and α-times resolvent families S (t) for A satisfying ||S (t)|| ≤ MCe(2ω)1/αt, n α,n n α,n for all t ≥ 0, n ∈ N, and S (t)x → S (t)x as n → +∞ α,n α for all x ∈ B, t ≥ 0. Moreover, the convergence is uniform in t on every compact subset of R . + In the following, we denote by Σ (ω) the open sector with vertex ω ∈ R and opening θ angle 2θ in the complex plane which is symmetric with respect to the real positive axis, i.e. Σ (ω) := {λ ∈ C : |arg(λ−ω)| < θ}. θ 5 We recall from [1, Definition 2.13] that an α-times resolvent family S (t) is called α analytic if S (t) admits an analytic extension to a sector Σ for some θ ∈ (0,π/2]. An α θ0 0 α-times analytic resolvent family is said to be of analyticity type (θ ,ω ) if for each 0 0 θ < θ and ω > ω there is M = M(θ,ω) such that 0 0 kS (t)k ≤ MeωRet, t ∈ Σ . α θ The set of all operators A ∈ Cα generating α-times analytic resolvent families S (t) of α type (θ ,ω ) is denoted by Aα(θ ,ω ). In addition, denote Aα(θ ) := {Aα(θ ,ω );ω ∈ 0 0 0 0 0 0 0 0 R }, Aα := {Aα(θ );θ ∈ (0,π/2]}. For α = 1 we obtain the set of all generators of + 0 0 S analytic semigroups. S Remark 3 We note that the spatial regularity condition R(S (t)) ⊂ D(A) for all t > 0 α is satisfied by α-times resolvent families whose generator A belongs to the set Aα(θ ,ω ) 0 0 where 0 < α < 2 (see [1, Proposition 2.15]). In particular, setting ω = 0 we have that 0 A ∈ Aα(θ ,0) if and only if −A is a positive operator with spectral angle less or equal to 0 π −α(π/2+θ). Note that such condition is also equivalent to the following Σ ⊂ ρ(A) and kλ(λI −A)−1k ≤ M, λ ∈ Σ . (16) α(π/2+θ) α(π/2+θ) The above considerations give us the following remarkable corollary. Corollary 1 Suppose A generates an analytic semigroup of angle π/2 and α ∈ (0,1). Then A generates an α-times analytic resolvent family. Proof Since A generates an analytic semigroup of angle π/2 we have kλ(λI −A)−1k ≤ M, λ ∈ Σ . π−ǫ Thenthecondition(16)(seealso[1,Corollary2.16])impliesA ∈ Aα(min{2−απ, 1π},0), 2α 2 α ∈ (0,2), that is A generates an α-times analytic resolvent family. In the sequel we will use the following assumptions concerning Volterra equations: (A1) A is the generator of C -semigroup and α ∈ (0,1); or 0 (A2) A is the generator of a strongly continuous cosine family and α ∈ (0,2). Observe that (A2) implies (A1) but not vice versa. 6 3 Weak vs. mild solutions Assume that H and U are separable Hilbert spaces. Let the cylindrical Wiener process W be defined on a stochastic basis (Ω,F,(F) ,P), with the positive symmetric covariance t≥0 operator Q ∈ L(U). This is known that the process W takes values in some superspace of U. (For more details concerning cylindrical Wiener process we refer to [4] or [8].) We define the subspace U := Q1/2(U) of the space U, endowed with the inner product 0 hu,vi := hQ−1/2u,Q−1/2vi . The set L0 := L (U ,H) of all Hilbert-Schmidt operators U0 U 2 2 0 fromU intoH,equippedwiththenorm|C| := ( +∞|Cf |2 )1/2,where{f } ⊂ U 0 L2(U0,H) k=1 k H k 0 is an orthonormal basis of U , is a separable Hilbert space. We assume that Ψ belongs to 0 P the class of measurable L0-valued processes. 2 By N2(0,T;L0) we denote a Hilbert space of all L0-predictable processes Ψ such that 2 2 ||Ψ|| < +∞, where T 1 T 2 ||Ψ|| := E |Ψ(τ)|2 dτ T L02 (cid:26) (cid:18)Z0 (cid:19)(cid:27) 1 T 2 = E Tr(Ψ(τ)Q21)(Ψ(τ)Q12)∗ dτ . (cid:26) Z0 h i (cid:27) We shall use the following Probability Assumptions (abbr. (PA)): 1. X is an H-valued, F -measurable random variable; 0 0 2. Ψ ∈ N2(0,T;L0) and the interval [0,T] is fixed. 2 Definition 3 Assume that (PA) hold. An H-valued predictable process X(t), t ∈ [0,T], is said to be a strong solution to (1), if X takes values in D(A), P-a.s., t for any t ∈ [0,T], |g (t−τ)AX(τ)| dτ < +∞, P −a.s., α > 0, (17) α H Z0 and for any t ∈ [0,T] the equation (1) holds P-a.s. Let A∗ denote the adjoint of A with a dense domain D(A∗) ⊂ H and the graph norm |·|D(A∗). Definition 4 Let (PA) hold. An H-valued predictable process X(t), t ∈ [0,T], is said to t be a weak solution to (1), if P( |g (t−τ)X(τ)| dτ < +∞) = 1, α > 0, and if for all 0 α H ξ ∈ D(A∗) and all t ∈ [0,T] the following equation holds R t t hX(t),ξi = hX ,ξi +h g (t−τ)X(τ)dτ,A∗ξi +h Ψ(τ)dW(τ),ξi , P−a.s. H 0 H α H H Z0 Z0 7 Definition 5 Assume that X is F -measurable random variable. An H-valued pre- 0 0 dictable process X(t), t ∈ [0,T], is said to be a mild solution to the stochastic Volterra equation (1), if E( t|S (t−τ)Ψ(τ)|2 dτ) < +∞, α > 0, for t ≤ T and, for arbitrary 0 α L02 t ∈ [0,T], R t X(t) = S (t)X + S (t−τ)Ψ(τ)dW(τ), P −a.s. (18) α 0 α Z0 where S (t) is the α-times resolvent family. α We will use the following result. Proposition 1 (see, e.g. [4, Proposition 4.15]) Assume that A is a closed linear unbounded operator with the dense domain D(A) ⊂ H. Let Φ(t),t ∈ [0,T], be an L (U ,H)-predictable process. If Φ(t) ∈ D(A), P − a.s. for 2 0 all t ∈ [0,T] and T T P |Φ(s)|2 ds < ∞ = 1, P |AΦ(s)|2 ds < ∞ = 1, L02 L02 (cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19) T then P Φ(s)dW(s) ∈ D(A) = 1 and (cid:18)Z0 (cid:19) T T A Φ(s)dW(s) = AΦ(s)dW(s), P −a.s. Z0 Z0 We define the stochastic convolution t WΨ(t) := S (t−τ)Ψ(τ)dW(τ), (19) α α Z0 where Ψ ∈ N2(0,T;L0). Because α-times resolvent families S (t), t ≥ 0, are bounded, 2 α then S (t−·)Ψ(·) ∈ N2(0,T;L0), too. α 2 Analogously like in [8], we can formulate the following result. Proposition 2 Assume that S (t),t ≥ 0, are the resolvent operators to (2). Then, for α any process Ψ ∈ N2(0,T;L0), the convolution WΨ(t), t ≥ 0, α > 0, given by (19) has a 2 α predictable version. Additionally, the process WΨ(t), t ≥ 0, α > 0, has square integrable α trajectories. Under some conditions a mild solution to Volterra equations is a weak solution and vice versa, see [8, Propositions 4 and 5]. Now, we can prove that a mild solution to the equation (1) is a weak solution to (1). 8 Proposition 3 If Ψ ∈ N2(0,T;L0) and Ψ(·,·)(U ) ⊂ D(A), P-a.s., then the stochastic 2 0 convolution WΨ(t), t ≥ 0, α > 0, given by (19), fulfills the equation α t t hWΨ(t),ξi = hg (t−τ)WΨ(τ),A∗ξi + hξ,Ψ(τ)dW(τ)i , α ∈ (0,2), (20) α H α α H H Z0 Z0 for any t ∈ [0,T] and ξ ∈ D(A∗). Proof Let us notice that the process WΨ has integrable trajectories. For any ξ ∈ D(A∗) α we have t hg (t−τ)WΨ(τ),A∗ξi dτ ≡ (from (19)) α α H Z0 t τ ≡ hg (t−τ) S (τ −σ)Ψ(σ)dW(σ),A∗ξi dτ = α α H Z0 Z0 (from Dirichlet’s formula and stochastic Fubini’s theorem) t t = h g (t−τ)S (τ −σ)dτ Ψ(σ)dW(σ),A∗ξi α α H Z0 (cid:20)Zσ (cid:21) t t−σ = h g (t−σ −z)S (z)dz Ψ(σ)dW(σ),A∗ξi α α H Z0 (cid:20)Z0 (cid:21) (where z := τ −σ and from definition of convolution) t = h A[(g ⋆S )(t−σ)]Ψ(σ)dW(σ),ξi = α α H Z0 (from the resolvent equation (3) because A(g ⋆S )(t−σ)x = (S (t−σ)−I)x, α α α where x ∈ D(A)) t = h [S (t−σ)−I]Ψ(σ)dW(σ),ξi = α H Z0 t t = h S (t−σ)Ψ(σ)dW(σ),ξi − h Ψ(σ)dW(σ),ξi . α H H Z0 Z0 Hence, we obtained the following equation t t hWΨ(t),ξi = hg (t−τ)WΨ(τ),A∗ξi dτ + hξ,Ψ(τ)dW(τ)i α H α α H H Z0 Z0 for any ξ ∈ D(A∗). Immediately from the equation (20) we deduce the following result. Corollary 2 If A is a bounded operator and Ψ ∈ N2(0,T;L0), then the following equality 2 holds t t WΨ(t) = g (t−τ)AWΨ(τ)dτ + Ψ(τ)dW(τ), (21) α α α Z0 Z0 for t ∈ [0,T], α > 0. Remark 4 The formula (21) says that the convolution WΨ(t), t ≥ 0, α > 0, is a strong α solution to (1) with X ≡ 0 if the operator A is bounded. 0 9 4 Strong solutions In this section we provide sufficient conditions under which the stochastic convolution WΨ(t), t ≥ 0, α > 0, defined by (19) is a strong solution to the equation (1). α Lemma 1 Let A be a closed linear unbounded operator with dense domain D(A) equipped with the graph norm |·| . Assume that (A1) or (A2) holds. If Ψ and AΨ belong to D(A) N2(0,T;L0) and in addition Ψ(·,·)(U ) ⊂ D(A), P-a.s., then (21) holds. 2 0 Proof Because formula (21) holds for any bounded operator, then it holds for the Yosida approximation A of the operator A, too, that is n t t WΨ (t) = g (t−τ)A WΨ (τ)dτ + Ψ(τ)dW(τ), α,n α n α,n Z0 Z0 where t WΨ (t) := S (t−τ)Ψ(τ)dW(τ). α,n α,n Z0 By Proposition 1, we have t A WΨ (t) = A S (t−τ)Ψ(τ)dW(τ). n α,n n α,n Z0 By assumption Ψ ∈ N2(0,T;L0). Because the operators S (t) are deterministic and 2 α,n bounded for any t ∈ [0,T], α > 0, n ∈ N, then the operators S (t −·)Ψ(·) belong to α,n N2(0,T;L0), too. In consequence, the difference 2 Φ (t−·) := S (t−·)Ψ(·)−S (t−·)Ψ(·) (22) α,n α,n α belongs to N2(0,T;L0) for any t ∈ [0,T], α > 0 and n ∈ N. This means that 2 t E |Φ (t−τ)|2 dτ < +∞ (23) α,n L02 (cid:18)Z0 (cid:19) for any t ∈ [0,T]. Let us recall that the cylindrical Wiener process W(t), t ≥ 0, can be written in the form +∞ W(t) = f β (t), (24) j j j=1 X where {f } is an orthonormal basis of U and β (t) are independent real Wiener processes. j 0 j From (24) we have t +∞ t Φ (t−τ)dW(τ) = Φ (t−τ)f dβ (τ). (25) α,n α,n j j Z0 j=1 Z0 X 10