Gauss-Markov processes on Hilbert spaces
GGAUSS-MARKOV PROCESSES ON HILBERT SPACES
BEN GOLDYS, SZYMON PESZAT, AND JERZY ZABCZYK
Abstract.
K. Itˆo characterised in [13] zero-mean stationary Gauss Markov-processesevolving on a class of infinite-dimensional spaces. In this work we extend the work ofItˆo in the case of Hilbert spaces: Gauss-Markov families that are time-homogenous areidentified as solutions to linear stochastic differential equations with singular coefficients.Choosing an appropriate locally convex topology on the space of weakly sequentiallycontinuous functions we also characterize the transition semigroup, the generator and itscore thus providing an infinite-dimensional extension of the classical result of Courr`ege [3]in the case of Gauss-Markov semigroups.
Contents
1. Introduction 12. Main results 43. Proof of Theorem 2.2 63.1. Conditional Gaussian measures 63.2. Proof of Theorem 2.2 74. Proof of Theorem 2.3 135. Proof of Theorem 2.4 156. Example: boundary noise 16References 171.
Introduction
The aim of the paper is to derive several characterizations of Gauss-Markov processeson infinite dimensional Hilbert spaces.
Mathematics Subject Classification.
Key words and phrases.
Gauss–Markov process, Ornstein–Uhlenbeck process, Gaussian measure, bw-topology, strict topology, generator.The work of Ben Goldys was partially supported by the ARC Discovery Grant DP120101886. Partof this work was prepared during his visit to the Institute of Mathematics of the Polish Academy ofSciences. Ben Goldys gratefully acknowledges excellent woking conditions and stimulating atmosphere ofthe Institute.The work of Szymon Peszat and Jerzy Zabczyk was supported by Polish Ministry of Science and HigherEducation Grant N N201 419039 “Stochastic Equations in Infinite Dimensional Spaces”.
Let H be a real separable Hilbert space and let { µ ( t, x ) : x ∈ H, t > } be a family of Gaussian transition kernels on H , that is µ ( t, x ) = N ( m ( t, x ) , Q ( t, x )) , (1.1)and for every Borel set B ⊂ H , µ ( s + t, x )( B ) = (cid:90) H µ ( s, x )(d y ) µ ( t, y )( B ) , s, t (cid:62) , x ∈ H. (1.2)It is well known, see for example Theorem 7.4 in [14], that there exist a measurable space(Ω , F ) endowed with a filtration ( F t ), an H -valued and ( F t )-adapted stochastic process Z defined on H , and a family of measures { P x : x ∈ H } on Ω such that for every x ∈ H the process Z is Markov on the probability space (Ω , F , ( F t ) , P x ), its transition kernel is µ ( t, x ) and P x ( Z (0) = x ) = 1. Let us recall that the system (Ω , F , ( F t ) , Z, { P x : x ∈ H } )is called a Markov family .Our aim in this paper is to derive three explicit characterizations of such processes. OurTheorem 2.2 describes the structure of the transition semigroup, Theorem 2.3 provides aconstruction of a stochastic equation which the process satisfies and Theorem 2.4 charac-terizes the generator of the transitions semigroup. In particular, we will prove that thereexist • a C -semigroup ( L ( t )) (with the generator A ) acting on H , • a vector b H = b H ( λ ) ∈ H defined for a certain λ > • a selfadjoint operator Q (cid:62) H ,such that for any h ∈ dom ( A (cid:63) ) the mean value and the covariance operator of Z ( t, x )satisfy for certain λ > (cid:104) m ( t, x ) , h (cid:105) = (cid:104) L ( t ) x, h (cid:105) + (cid:90) t (cid:104) L ( s ) b H , ( λ − A (cid:63) ) h (cid:105) d s, t (cid:62) , x ∈ H, and (cid:104) Q ( t, x ) h, h (cid:105) = (cid:90) t (cid:12)(cid:12) Q / L (cid:63) ( s ) h (cid:12)(cid:12) d s, t (cid:62) , x ∈ H. Given this representation of m and Q we derive the second and the third characteriza-tions. The second one is an extension of the work of K. Itˆo who obtained in [13] a similarrepresentation for a stationary Gauss-Markov process with zero mean. Since we do not as-sume stationarity and we work with a Gauss-Markov family of processes instead of a singleprocess, the approach of [13] can not be adopted and a different argument is required toderive representations for m ( t, x ) and Q ( t, x ). Given these representations, we prove thatthere exists an ( F t )-adapted standard cylindrical Wiener process on H such that (cid:26) d Z = ( AZ + b V ) d t + Q / d W,Z (0) = x ∈ H, (1.3)where b V = ( λ − A ) b H is uniquely defined in a larger Hilbert space V ⊃ H to which thesemigroup ( L ( t )) can be extended. Our third, purely analytic result is in the spirit of a theorem of Ph. Courr`ege [3]. Namely,Courr`ege obtained a representation of generator for an arbitrary Markov semigroup, thatis strongly continuous in C (cid:0) R d (cid:1) and such that C ∞ c (cid:0) R d (cid:1) is a core for its generator. It iswell known, see for example [5] that the space C normb ( H ) of bounded continuous functionson H endowed with the supremum norm is not appropriate for the analysis of Markoviansemigroups on Hilbert spaces. In order to obtain a counterpart of the Courr`ege theorem weneed to introduce two locally convex topologies. The first one is the bounded weak topologyon H as defined in [10]. It was demonstrated in [17] and recently in [2] that, in fact, it is anatural concept for Markovian transition semigroups and particularly useful for proving theexistence of an invariant measure. We will use the notation H bw for the space H endowedwith the bounded weak topology, C b ( H bw ) for the space of bounded continuous functionson H bw , and C normb ( H bw ) for the space C b ( H bw ) endowed with the norm topology. It wasshown in [17] that C b ( H bw ) is precisely the space of bounded functions that are weaklysequentially continuous. Another locally convex topology, known as the strict topology,should be introduced on the space C b ( H bw ). The strict topology can be defined as thestrongest topology on C b ( H bw ) which on norm bounded sets of C normb ( H bw ) coincides withthe topology of uniform convergence on compacts of H bw . For alternative definitions of thistopology and its applications to the theory of Markov semigroups, see [12] and referencestherein.Using the representations for m ( t, x ) and Q ( t, x ) we show that the transition semigroup P t φ ( x ) = E φ ( Z ( t, x )) of the process Z is strongly continuous in C strictb ( H bw ). Moreover, wewill identify the core of the generator L in C strictb ( H bw ) and will derive an explicit formulafor the generator acting on functions from the core. In the case when A has boundedinverse this formula takes a simple form Lφ ( x ) = 12 Tr (cid:0) QD φ ( x ) (cid:1) + (cid:104) x + b H , A (cid:63) Dφ ( x ) (cid:105) , x ∈ H. For a general version of this formula see Section 2. These results provide a version of theCourr`ege theorem for a Gauss-Markov semigroup in a Hilbert space.Although the extension concerns a rather narrow class of transition semigroups we believethat it does suggest how a general infinite dimensional version of the Courr`ege theoremshould look like.If the state space H is one-dimensional then results of our paper are part of the mathe-matical folklore. More generally, it is not very difficult to obtain our result if dim( H ) < ∞ .Let us note here that in finite dimensional spaces a similar problem is considered in theframework of affine processes, see for example [9]. In the theory of affine models a muchwider class of processes is considered but the linearity of the function x (cid:55)→ m ( t, x ) and thefact that x (cid:55)→ Q ( t, x ) is constant are assumed from the very beginning while we do derivethese properties as a result of a careful analysis. We use standard notations C (cid:0) R d (cid:1) for the space of continuous functions vanishing at infinity, and C ∞ c (cid:0) R d (cid:1) for the space of infinitely differentiable functions with compact supports. BEN GOLDYS, SZYMON PESZAT, AND JERZY ZABCZYK
In infinite dimensions the only result in this directions was obtained by Itˆo in [13]. Let usalso recall a related problem to characterise a family of measures that satisfy the so-calledskew convolution equation [19]. In that paper it is assumed again from the very beginningthat the expectation m ( t, x ) is linear in x and the covariance Q ( t, x ) constant in x .Main results are formulated in Section 2. The proofs are presented in the followingsections. In the final Section 6 we discuss an application of our results to models withboundary noise. 2. Main results
We give now precise formulation of our main theorems. The proofs will be postponed tothe following sections.By definition we have m ( t, x ) = E x Z ( t ) , x ∈ H, t (cid:62) , and for any h, k ∈ H (cid:104) Q ( t, x ) h, k (cid:105) = E x (cid:104) Z ( t ) − m ( t, x ) , h (cid:105) (cid:104) Z ( t ) − m ( t, x ) , k (cid:105) , x ∈ H, t (cid:62) . Let φ : H → R be continuous and such that | φ ( x ) | (cid:54) C (1 + | x | ). Then for any s, t (cid:62) E y ( φ ( Z ( t + s )) | Z s = x ) = E x φ ( Z ( t )) P y − a.s., x, y ∈ H. The following hypothesis is a standing assumption of the rest of the paper.
Hypothesis 2.1. (1) For every x ∈ H and h, k ∈ H the functions t (cid:55)→ (cid:104) m ( t, x ) , h (cid:63) (cid:105) and t (cid:55)→ (cid:104) Q ( t, x ) h (cid:63) , k (cid:63) (cid:105) are continuous.(2) For any t (cid:62) and h, k ∈ H the functions x → (cid:104) m ( t, x ) , h (cid:105) and x → (cid:104) Q ( t, x ) h, k (cid:105) are continuous on H .(3) For any t > and any x ∈ H , Im Q ( t, x ) = H. We note that for every x ∈ H , Hypothesis 2.1 yieldslim t ↓ (cid:104) m ( t, x ) , h (cid:105) = (cid:104) x, h (cid:105) , h ∈ H, and lim t ↓ (cid:104) Q ( t, x ) h, k (cid:105) = 0 , h, k ∈ H. In order to formulate our results we need some preparations. Let L = ( L ( t )) be any C -semigroup on H and let A denote its generator. Then for λ > λ − A ) isboundedly invertible on H and a new norm on H can be defined by the formula | x | − = (cid:12)(cid:12) ( λ − A ) − x (cid:12)(cid:12) , x ∈ H. The space V is defined as a completion of H with respect to the norm | · | − . Clearly,the topological space V does not depend on λ >
0. The semigroup ( L ( t )) extends to a C -semigroup ( L V ( t )) on V with the generator A V whose domain is equal to H .We can formulate now our first main result. It gives a precise description of the functions m and Q and will also play the crucial role in the proofs of Theorems 2.3 and 2.4 below.The proof of Theorem 2.2 is postponed to Section 3. Theorem 2.2.
There exists a strongly continuous semigroup ( L ( t )) with the generator A on H , a vector b H ∈ H and a selfadjoint operator Q (cid:62) in H such that: (1) We have dom ( A (cid:63) ) ⊂ dom (cid:0) Q / (cid:1) . (2.1)(2) For any x ∈ H , m ( t, x ) = L ( t ) x + (cid:90) t L V ( s ) b V d s, t (cid:62) , (2.2) where b V = A V b H ∈ V . In particular, for every h ∈ dom ( A (cid:63) ) , (cid:104) m ( t, x ) , h (cid:105) = (cid:104) L ( t ) x, h (cid:105) + (cid:90) t (cid:104) L ( s ) b H , ( λ − A (cid:63) ) h (cid:105) d s, t (cid:62) , x ∈ H. (2.3)(3) The covariance operator Q ( t ) = Q ( t, x ) is independent of x ∈ H , Im ( L (cid:63) ( t )) ⊂ Dom (cid:0) Q / (cid:1) , t > , and Q ( t ) = (cid:90) t (cid:0) Q / L (cid:63) ( s ) (cid:1) (cid:63) (cid:0) Q / L (cid:63) ( s ) (cid:1) d s. (2.4) In particular, (cid:104) Q ( t ) h, h (cid:105) = (cid:90) t (cid:12)(cid:12) Q / L (cid:63) ( s ) h (cid:12)(cid:12) d s, t (cid:62) , x ∈ H, (2.5) and (cid:90) T (cid:13)(cid:13) Q / L (cid:63) ( t ) (cid:13)(cid:13) HS d t < ∞ . (4) For any x ∈ H , any h, k ∈ H and (cid:54) s (cid:54) t , (cid:104) Q ( s, t, x ) h, k (cid:105) = E x (cid:104) Z ( s ) − m ( s, x ) , h (cid:105)(cid:104) Z ( t ) − m ( t, x ) , k (cid:105) = (cid:104) L ( t − s ) Q ( s ) h, k (cid:105) , (2.6) hence the operator Q ( s, t ) = Q ( s, t, x ) is independent of x ∈ H . Now we can formulate our second main result. Its proof is postponed to Section 4.
Theorem 2.3.
Assume that Hypothesis 2.1 holds. Then there exist a strongly continuoussemigroup of operators ( L ( t )) on the space H with its associated space V , an element BEN GOLDYS, SZYMON PESZAT, AND JERZY ZABCZYK b V ∈ V , a selfadjoint and non-negative operator Q in H , and there exists a standardcylindrical Wiener process W on H such that for every x ∈ H we have Z ( t ) = L ( t ) x + (cid:90) t L V ( t − s ) b V d s + (cid:90) t L ( t − s ) Q / d W ( s ) , t (cid:62) , P x − a.s. (2.7)Our final result is an extension of the Courr`ege theorem. We introduce first certainconvenient locally convex topologies.In the terminology of [10] the bounded weak topology τ bw on H is the strongest locallyconvex topology on H that coincides with the weak topology on every ball. We will denoteby H bw the space H endowed with the topology τ bw . It has been proved in [17] that φ ∈ C b ( H bw ) if and only if φ is bounded and weakly sequentially continuous on H .Let τ n and τ c denote, respectively, the norm topology and the topology of the uniformconvergence on compacts on the space C b ( H bw ). The strict topology τ s is defined asthe strongest locally convex topology on C b ( H bw ) that coincides with the topology τ c on compacts. We will use a short notation C b for the space C b ( H bw ) endowed with thetopology τ s .We will say that φ ∈ F C ∞ b ( A (cid:63) ) if there exist n (cid:62) f ∈ C ∞ b ( R n ) and h , . . . , h n ∈ dom ( A (cid:63) ) such that φ ( x ) = f ( (cid:104) x, h (cid:105) , . . . , (cid:104) x, h n (cid:105) )and sup x ∈ H |(cid:104) x, A (cid:63) Dφ ( x ) (cid:105)| < ∞ . It can be easily shown that F C ∞ b ( A (cid:63) ) is dense in C strictb ( H bw ). The proof of the resultbelow is postponed to Section 5. Theorem 2.4.
The semigroup ( P t ) is strongly continuous in C strictb ( H bw ) . Let L denotethe generator of the semigroup ( P t ) in C strictb ( H bw ) . Then the space F C ∞ b ( A (cid:63) ) is a corefor L and for every φ ∈ F C ∞ b ( A (cid:63) ) Lφ ( x ) = 12 Tr (cid:0) QD φ ( x ) (cid:1) + (cid:104) x, A (cid:63) Dφ ( x ) (cid:105) + (cid:104) b H , ( λ − A (cid:63) ) Dφ ( x ) (cid:105) . (2.8) Remark . Let us note that b H depends on λ but the formula (2.8) uniquely defines Lφ since b V = ( λ − A ) b H does not depend on λ .3. Proof of Theorem 2.2
We recall first a basic fact about the conditional Gaussian measures.3.1.
Conditional Gaussian measures.
Let H and H be two real separable Hilbertspaces and let ( X, Y ) be a Gaussian vector defined on a probability space (Ω , F , P ) andtaking values in H × H . Let m X = E X and m Y = E Y. The covariance operator C X of X is determined by the equation E (cid:104) X − m X , h (cid:105) (cid:104) X − m X , k (cid:105) = (cid:104) C X h, k (cid:105) , h, k ∈ H , (3.1) and a similar condition determines the covariance C Y of Y . The covariance operator C XY : H → H is defined by the condition (cid:104) C XY h, k (cid:105) = E (cid:104) X − m X , h (cid:105) (cid:104) Y − m Y , k (cid:105) , h ∈ H , k ∈ H , and then C ∗ XY = C Y X . For a linear closable operator G on H the closure of G will bedenoted by G . For a symmetric and compact operator K : H → H we will denote by K − its pseudo inverse. For the convenience of the reader we present the following known result(see. e.g. [16]). Theorem 3.1.
The following holds.(a) We have
Im ( C Y X ) ⊂ Im (cid:16) C / X (cid:17) , (3.2) and the operator K := C − / X C Y X is of Hilbert–Schmidt type on H and K ∗ = C XY C − / X .(b) We have E ( Y | X ) = m Y + K ∗ C − / X ( X − m X ) , P − a.s. (c) The conditional distribution of Y given X is Gaussian N (cid:0) E ( Y | X ) , C Y | X (cid:1) , where C Y | X := C Y − K ∗ K. Moreover, the random variables K ∗ C − / X X and (cid:16) Y − K ∗ C − / X X (cid:17) are independent. Proof of Theorem 2.2.
Let Q ( s, t, y ) denote the covariance operator, (cid:104) Q ( s, t, y ) h, k (cid:105) = E y (cid:104) Z ( s ) − m ( s, y ) , h (cid:105) (cid:104) Z ( t ) − m ( t, y ) , k (cid:105) , h, k ∈ H. By (3.2) for any y ∈ H and any 0 < s (cid:54) t , K ( s, t, y ) := Q − / ( s, y ) Q (cid:63) ( s, t, y )is a well defined Hilbert–Schmidt operator on H . The proof is divided into a sequence oflemmas. Lemma 3.2. (1) For any s, t such that (cid:54) s (cid:54) t the operator-valued mappings x (cid:55)→ Q ( s, t, x ) and x (cid:55)→ K (cid:63) ( s, t, x ) K ( s, t, x ) are constant in x ∈ H and for Q ( t ) = Q ( t, t, and K ( s, t ) = K ( s, t, we have Q ( t − s ) = Q ( t ) − K (cid:63) ( s, t ) K ( s, t ) , (3.3) (2) For any s, t such that (cid:54) s (cid:54) t the operator L ( t, s ) := K (cid:63) ( s, t ) Q − / ( s ) with thedomain Im (cid:0) Q / ( s ) (cid:1) has a unique extension to a bounded linear operator L ( t, s ) : H → H and for any x, y ∈ H , m ( t − s, x ) = m ( t, y ) − L ( t, s ) m ( s, y ) + L ( t, s ) x. (3.4) Moreover, m ( t − s, m ( s, y )) = m ( t, y ) , (cid:54) s (cid:54) t, y ∈ H. (3.5) BEN GOLDYS, SZYMON PESZAT, AND JERZY ZABCZYK
Proof.
Invoking Theorem 3.1 we find that for any 0 < s < t and y ∈ H the followingequality holds P y -a.s. E y ( ( Z ( t ) − E y ( Z ( t ) | Z ( s ))) ⊗ ( Z ( t ) − E y ( Z ( t ) | Z ( s ))) | Z ( s ))= Q ( t, y ) − K (cid:63) ( s, t, y ) K ( s, t, y ) . (3.6)Applying the Markov property to (3.6) we obtain for 0 < s < t , Q ( t − s, x ) = Q ( t, y ) − K (cid:63) ( s, t, y ) K ( s, t, y ) for µ ( s, y ) − a.e. x. (3.7)For any h, k ∈ H , (cid:104) Q ( t − s, x ) h, k (cid:105) = (cid:104) Q ( t, y ) h, k (cid:105) − (cid:104) K (cid:63) ( s, t, y ) K ( s, t, y ) h, k (cid:105) for µ ( s, y ) − a.e. x. (3.8)Therefore the function x (cid:55)→ (cid:104) Q ( t − s, x ) h, k (cid:105) is constant on a dense set and by Hypothesis2.1 is continuous on H . It follows that equality (3.8) holds for every x ∈ H . Finally, forall h, k ∈ H the function y → (cid:104) K (cid:63) ( s, t, y ) K ( s, t, y ) h, k (cid:105) is constant in y ∈ H .Invoking Theorem 3.1 and the first part of the proof we obtain for 0 < s < t and y ∈ H , E y ( Z ( t ) | Z ( s )) = m ( t, y ) + K (cid:63) ( s, t, y ) Q − / ( s )( Z ( s ) − m ( s, y )) , P y − a.s. (3.9)Then the Markov property yields for 0 < s < t , m ( t − s, x ) = m ( t, y ) + K (cid:63) ( s, t, y ) Q − / ( s )( x − m ( s, y )) for µ ( s, y ) − a.e. x, (3.10)Putting x − m ( s, y ) = z ∈ H equation (3.10) takes the form m ( t − s, z + m ( s, y )) = m ( t, y ) + K (cid:63) ( s, t, y ) Q − / ( s ) z for ν s − a.e. z, where ν s = N (0 , Q ( s )). The measurable linear operator K (cid:63) ( s, t, y ) Q − / ( s ) is well definedlinear on a dense linear space Im (cid:0) Q / ( s ) (cid:1) . By Hypothesis 2.1 the operator K (cid:63) ( s, t, y ) Q − / ( s )has a unique extension to a bounded linear operator L ( t, s, y ) on H . Therefore, for any u, v ∈ H and x = Q / ( s ) u and z = Q / ( s ) v we have m ( t − s, x ) − m ( t − s, z ) = K (cid:63) ( s, t, y )( u − v ) . Hence K (cid:63) ( t, s, y ) is constant in y ∈ H and (3.4) easily follows. Putting in equation (3.4) x = m ( s, y ) we obtain (3.5). (cid:3) Lemma 3.3.
For (cid:54) s (cid:54) t we have L ( t, s ) = L ( t − s ) and ( L ( t )) is a strongly continuoussemigroup on H .Proof. For any x, z ∈ H (3.4) yields m ( t − s, x ) − m ( t − s, z ) = L ( t, s )( x − z ) . (3.11)Therefore, L ( t, s ) x = m ( t − s, x ) − m ( t − s,
0) (3.12)depends on t − s only and there exists a function, still denoted by L , such that L ( t, s ) = L ( t − s ) , (cid:54) s (cid:54) t. Invoking (3.11) we obtain L ( s + t )( x − y ) = m ( s + t, x ) − m ( s + t, y )= m ( s, m ( t, x )) − m ( s, m ( t, y )) = L ( s )( m ( t, x ) − m ( t, y ))= L ( s ) L ( t )( x − y ) . Finally, L ( s + t ) x = L ( s ) L ( t ) x, x ∈ H, and L ( t ) x = m ( t, x ) − m ( t, . (3.13)By Hypothesis 2.1, lim t ↓ (cid:104) L ( t ) x, h (cid:105) = (cid:104) x, h (cid:105) , h ∈ H. Since t (cid:55)→ m ( t, y ) is weakly continuous for every y ∈ H , we obtainsup t (cid:54) T | L ( t ) y | (cid:54) C T ( y ) . Therefore, the semigroup property of ( L ( t )) and a well known result (see e.g. [18]) implylim t ↓ L ( t ) x = x x ∈ H, which completes proof of the lemma. (cid:3) Lemma 3.4.
There exist λ , C > such that | m ( t, | (cid:54) C e λ t , t (cid:62) . Proof.
We have m ( t, x ) = m ( t, y ) − L ( t ) y + L ( t ) x, and therefore, the function g ( t ) := m ( t, y ) − L ( t ) y is independent of y ∈ H . Putting m ( t,
0) = g ( t ) it is easy to check that g ( t + s ) = L ( t ) g ( s ) + g ( t ) , s, t (cid:62) . (3.14)For any k (cid:62) g ( k ) = k − (cid:88) i =0 L ( i ) g (1) . Since ( L ( t )) is a C -semigroup in H there exist finite M and λ such that (cid:107) L ( t ) (cid:107) (cid:54) M e λ t , t (cid:62) . Therefore, | g ( k ) | (cid:54) (cid:32) k − (cid:88) i =0 (cid:107) L ( i ) (cid:107) (cid:33) | g (1) | (cid:54) (cid:32) k − (cid:88) i =0 M e λ i (cid:33) | g (1) | with c := M e λ − | g (1) | . Now, take any t = k + s where k (cid:62) s ∈ [0 , g ( t ) = g ( k + s ) = L ( k ) g ( s ) + g ( k ) , and hence | g ( t ) | (cid:54) M e λ k sup t (cid:54) | g ( s ) | + c e λ k (cid:54) C e λ t , and the lemma follows. (cid:3) Lemma 3.5.
There exists b H ∈ H such that for b V = ( λ − A ) b H ∈ V we have m ( t,
0) = (cid:90) t L V ( s ) b V d s. Proof.
Let g ( t ) = m ( t, λ > λ the function (cid:98) g ( λ ) = (cid:90) + ∞ e − λt g ( t )d t is well defined by Lemma 3.4. Taking the Laplace transform of both sides of equation(3.14) we obtain for λ > λ ,e λs (cid:90) ∞ e − λ ( t + s ) g ( t + s )d t − (cid:98) g ( λ ) = ( λ − A ) − g ( s ) , hence e λs (cid:18)(cid:98) g ( λ ) − (cid:90) s e − λu g ( u )d u (cid:19) − (cid:98) g ( λ ) = ( λ − A ) − g ( s ) . Therefore, e λs − s (cid:98) g ( λ ) − e λs s (cid:90) s e − λu g ( u )d u = ( λ − A ) − (cid:18) g ( s ) s (cid:19) . The left-had-side of the above equality converges to b H = λ (cid:98) g ( λ )for s ↓ g ( s ) s has a limit b V in V for s ↓
0. More precisely, b V = lim s ↓ g ( s ) s = ( λ − A ) ( λ (cid:98) g ( λ )) . Invoking again equation (3.14) we obtainlim s ↓ g ( s + t ) − g ( t ) s = lim s ↓ (cid:18) L V ( t ) g ( s ) s (cid:19) = L V ( t ) b V , where the convergence holds in V . Finally,d g d t ( t ) = L V ( t ) b V , and the lemma easily follows. (cid:3) Let us recall that L ( t − s ) = K (cid:63) ( s, t ) Q − / ( s ) . Hence Lemma 3.5 implies that (3.3) can be written in the form Q ( t + s ) = Q ( t ) + L ( t ) Q ( s ) L (cid:63) ( t ) , s, t (cid:62) . (3.15)It is easy to see that V (cid:48) = Dom ( A (cid:63) ) endowed with the graph norm and V (cid:48) ⊂ H ⊂ V. If R : H → H is a trace class operator, such that R = R (cid:63) (cid:62) R : V (cid:48) → V which is again of trace class, and ˜ R = ˜ R (cid:48) (cid:62)
0. In the sequel we willuse the same notation R for both operators. For any bounded R : H → H let L ( t ) R = L ( t ) RL (cid:63) ( t ) . It may be checked that ( L ( t )) is a C -semigroup on the space of trace-class operators on H . Let ( L V ( t )) denote an extension of ( L ( t )) to V . Then L (cid:48) V ( t ) can be identified withthe restriction of L (cid:63) ( t ) to V (cid:48) . Let E denote a separable Banach space of symmetric traceclass operators R : V (cid:48) → V endowed with the nuclear norm. The dual space E (cid:63) can beidentified as the space of bounded operators from V to V (cid:48) and E (cid:63) (cid:104) P, R (cid:105) E = Tr ( RP ) , see pp. 34 and 65 of [8] for details. It is easy to see that the adjoint semigroup acting on E (cid:63) has the form L (cid:63) ( t ) R = L (cid:48)− ( t ) RL V ( t ) , R : V → V (cid:48) . Lemma 3.6.
There exist λ > and C > such that (cid:107) Q ( t ) (cid:107) E (cid:54) C e λ t , t (cid:62) . Proof.
The proof is similar to the proof of Lemma 3.4 hence omitted. (cid:3)
Lemma 3.7.
There exists Q ∈ E such that Q ( t ) = (cid:90) t L V ( s ) QL (cid:48) V d s. Moreover, the restriction of Q to an unbounded operator acting in H (still denoted by Q )is selfadjoint in H , and for every t > , L ( t ) Q / has an extension to a Hilbert–Schmidtoperator on H and (cid:90) t (cid:13)(cid:13) L ( s ) Q / (cid:13)(cid:13) HS d s < ∞ . Proof.
Using the same arguments as in the proof of Lemma 3.4 we can show that thereexist λ , C > (cid:107) Q ( t ) (cid:107) E (cid:54) C e λ t , t (cid:62) . Therefore, for any λ > λ the operator (cid:98) Q ( λ ) = (cid:90) + ∞ e − λt Q ( t )d t, where the integral is the Bochner integral in E is a well defined element of E . Let A denote the generator of the semigroup L on E . In view of (3.15) we can apply the samearguments as in the proof of Lemma 3.5 to obtainlim s ↓ (cid:18) ( λ − A ) − s Q ( s ) (cid:19) = λ (cid:98) Q ( λ ) , (3.16)where the convergence holds in E . Let us recall that for h, k ∈ V (cid:48) a bounded operator h ⊗ k : V → V (cid:48) is defined by the formula( h ⊗ k ) x = (cid:104) h, x (cid:105) k, x ∈ V. For u, v ∈ Dom( A ) ⊂ V we have u ⊗ v : V (cid:48) → V . Moreover, u ⊗ u ∈ Dom( A ) and A ( u ⊗ u ) = u ⊗ ( Au ) + ( Au ) ⊗ u. (3.17)Similarly, if u ∈ V (cid:48) then u ⊗ u : V → V (cid:48) , u ⊗ u ∈ Dom ( A (cid:63) ) and A (cid:63) ( u ⊗ u ) = u ⊗ ( A (cid:63) u ) + ( A (cid:63) u ) ⊗ u. (3.18)Therefore, for h ∈ H ⊂ V and u ∈ V (cid:48) , E (cid:104) h ⊗ h, A (cid:63) ( u ⊗ u ) (cid:105) E (cid:63) = E (cid:104) h ⊗ h, u ⊗ ( A (cid:63) u ) + ( A (cid:63) u ) ⊗ u (cid:105) E (cid:63) = E (cid:104) h ⊗ h, u ⊗ ( A (cid:63) u ) (cid:105) E (cid:63) + E (cid:104) h ⊗ h, ( A (cid:63) u ) ⊗ u (cid:105) E (cid:63) = (cid:104) h, u (cid:105) (cid:104) h, A (cid:63) u (cid:105) + (cid:104) h, A (cid:63) u (cid:105) (cid:104) h, u (cid:105) = 2 (cid:104) h, u (cid:105) (cid:104) h, A (cid:63) u (cid:105) = 2 (cid:104) ( h ⊗ h ) u, A (cid:63) u (cid:105) , where the first step follows from (3.18). Therefore, by the properties of the nuclear normon E we find that for any R ∈ E and u ∈ V (cid:48) , E (cid:104) R, A (cid:63) ( u ⊗ u ) (cid:105) E (cid:63) = 2 (cid:104) Ru, A (cid:63) u (cid:105) . (3.19)Equation (3.16) yieldslim s ↓ E (cid:28) ( λ − A ) − s Q ( s ) , P (cid:29) E (cid:63) = E (cid:68) λ (cid:98) Q ( λ ) , P (cid:69) E (cid:63) for any P ∈ E (cid:63) . Taking P = ( λ − A (cid:63) ) ( u ⊗ u ) with u ∈ Dom ( A (cid:48) V ) ⊂ V (cid:48) and using (3.19)we obtain lim s ↓ (cid:28) s Q ( s ) u, u (cid:29) = E (cid:68) λ (cid:98) Q ( λ ) , ( λ − A (cid:63) ) ( u ⊗ u ) (cid:69) E (cid:63) = λ (cid:68) λ (cid:98) Q ( λ ) u, u (cid:69) − λ (cid:68) (cid:98) Q ( λ ) u, A (cid:63) u (cid:69) . The bilinear form B ( u, v ) = λ (cid:68) λ (cid:98) Q ( λ ) u, v (cid:69) − λ (cid:68) (cid:98) Q ( λ ) u, A (cid:63) v (cid:69) − λ (cid:68) (cid:98) Q ( λ ) v, A (cid:63) u (cid:69) , defined for u, v ∈ Dom ( A (cid:48) V ) is well defined and symmetric as a limit in s ↓ | B ( u, v ) | (cid:54) c | u | V (cid:48) | v | V (cid:48) . Therefore, there exists Q : V (cid:48) → V such that V (cid:28) lim s ↓ s Q ( s ) u, u (cid:29) V (cid:48) = V (cid:104) Qu, u (cid:105) V (cid:48) . Clearly, Q is non-negative and V (cid:104) Qu, v (cid:105) V (cid:48) = V (cid:104) Qv, u (cid:105) V (cid:48) , u, v ∈ V (cid:48) . Therefore, the bilinear form B H ( u, v ) = V (cid:104) Qu, v (cid:105) V (cid:48) , u, v ∈ V (cid:48) ⊂ H, defines a selfadjoint operator in H , still denoted by Q and such thatdom ( A (cid:63) ) ⊂ dom (cid:0) Q / (cid:1) , which proves (2.1). Invoking (3.15) again we readily obtaindd t (cid:104) Q ( t ) u, u (cid:105) = 2 (cid:10) Q / A (cid:63) u, Q / A (cid:63) u (cid:11) , hence (cid:104) Q ( t ) u, u (cid:105) = (cid:90) t (cid:10) Q / L (cid:63) ( s ) u, Q / L (cid:63) ( s ) u (cid:11) d s, u ∈ dom (cid:0) ( A (cid:63) ) (cid:1) . Finally, by polarisation and the density of dom (cid:0) ( A (cid:63) ) (cid:1) in H we obtain (2.4). Moreover, Q : V (cid:48) → V is a trace-class operator since (cid:98) Q ( λ ) is. (cid:3) Proof of Theorem 2.3
We define a V -valued and ( F t )-adapted process M ( t ) = Z ( t ) − Z (0) − tb V − (cid:90) t A V Z ( s ) d s. It is easy to check that M ( t ) = Z ( t ) − m ( t, x ) − (cid:90) t A V ( Z ( s ) − m ( s, x )) d s, (4.1)where we put Z (0) = x . Then P x -a.s. E x ( M ( t ) | F t ) = E x ( Z ( t ) − m ( t , x ) | Z ( t )) − (cid:90) t A V ( Z ( s ) − m ( s, x )) d s − (cid:90) t t A V E x ( Z ( s ) − m ( s, x ) | Z ( t ))= L ( t − t ) ( Z ( t ) − m ( t , x )) − (cid:90) t A V ( Z ( s ) − m ( s, x )) d s − (cid:90) t t A V L ( s − t ) ( Z ( t ) − m ( t , x )) d s = Z ( t ) − m ( t , x ) − (cid:90) t A V ( Z ( s ) − m ( s, x )) d s = M ( t ) . Let M h ( t ) = (cid:104) M ( t ) , h (cid:105) for h ∈ V (cid:48) = dom ( A (cid:63) ). Then M h is a continuous Gaussianmartingale with M h (0) = 0. Since the martingale is Gaussian, we have (cid:10) M h (cid:11) t = E (cid:12)(cid:12) M h ( t ) (cid:12)(cid:12) . It remains to compute E (cid:12)(cid:12) M h ( t ) (cid:12)(cid:12) . To this end we write (cid:12)(cid:12) M h ( t ) (cid:12)(cid:12) = (cid:104) Z ( t ) − m ( t, x ) , h (cid:105) + (cid:18)(cid:90) t (cid:104) Z ( s ) − m ( s, x ) , A (cid:63) h (cid:105) d s (cid:19) − (cid:90) t (cid:104) Z ( t ) − m ( t, x ) , h (cid:105) (cid:104) Z ( s ) − m ( s, x ) , A (cid:63) h (cid:105) d s. Then E (cid:12)(cid:12) M h ( t ) (cid:12)(cid:12) = (cid:104) Q ( t ) h, h (cid:105) + (cid:90) t (cid:90) t E (cid:104) Z ( s ) − m ( s, x ) , A (cid:63) h (cid:105) (cid:104) Z ( u ) − m ( u, x ) , A (cid:63) h (cid:105) d u d s − (cid:90) t (cid:104) L ( t − s ) Q ( s ) A (cid:63) h, h (cid:105) d s. Since (cid:104) Q ( t ) h, h (cid:105) can be written in the form (cid:104) Q ( t ) h, h (cid:105) = (cid:90) t (cid:10) Q / L (cid:63) ( t − s ) h, Q / L (cid:63) ( t − s ) h (cid:11) d s we obtain ddt E (cid:12)(cid:12) M h ( t ) (cid:12)(cid:12) = 2 (cid:104) Q ( t ) h, A (cid:63) h (cid:105) + (cid:12)(cid:12) Q / h (cid:12)(cid:12) + (cid:90) t E (cid:104) Z ( s ) − m ( s, x ) , A (cid:63) h (cid:105) (cid:104) Z ( t ) − m ( t, x ) , A (cid:63) h (cid:105) d s + (cid:90) t E (cid:104) Z ( t ) − m ( t, x ) , A (cid:63) h (cid:105) (cid:104) Z ( u ) − m ( u, x ) , A (cid:63) h (cid:105) d u − (cid:90) t (cid:104) L ( t − s ) Q ( s ) A (cid:63) h, A (cid:63) h (cid:105) d s − (cid:104) Q ( t ) h, A (cid:63) h (cid:105) = (cid:12)(cid:12) Q / h (cid:12)(cid:12) . Finally, by the martingale representation theorem, see e.g. [6], there exists a standardcylindrical Wiener process W on H such that M h ( t ) = (cid:10) W ( t ) , Q / h (cid:11) is adapted to ( F t )and for any h ∈ dom ( A (cid:63) ), (cid:104) Z ( t ) , h (cid:105) = (cid:104) Z (0) , h (cid:105) + (cid:90) t (cid:104) Z ( s ) , A (cid:63) h (cid:105) d s + (cid:90) t (cid:104) b V , h (cid:105) d s + (cid:10) W ( t ) , Q / h (cid:11) . Now, Theorem 2.3 follows from a result in [4]. Proof of Theorem 2.4
It was proved in [17] that P t C b ( H bw ) ⊂ C b ( H bw ). Let B r ⊂ H denote the centered closedball of radius r . Let us recall that bounded sets of H bw are precisely the bounded sets ofthe norm topology and the sets B r are compact in H bw . The proof is a simple modificationof the proof of Theorem 4.2 in [12]. Let us note here that by definition of the bw -topology,see [10] the space H bw is a k -space and therefore the space C strictb ( H bw ) is complete. Weprovide details only for the proof of strong continuity. To this end it is enough to showthat for every weakly sequentially continuous φ : H → R and every r > t → sup x ∈ B r | P t φ ( x ) − φ ( x ) | = 0 . Assume that this convergence does not hold. Then there exist ε >
0, a sequence t n → x n ) ⊂ B r such that | P t n φ ( x n ) − φ ( x n ) | > ε. Therefore (cid:90) H | φ ( m ( t n , x n ) + y ) − φ ( x n ) | µ ( t n , dy ) > ε, (5.1)where by Lemma 2.2 the measure µ ( t, dy ) = N (0 , Q ( t ))( dy ) is independent of x ∈ H .Hypothesis 2.1 yields (cid:104) m ( t n , , h (cid:105) → , h ∈ H, and clearly lim n →∞ (cid:104) x n , L (cid:63) ( t n ) h (cid:105) = (cid:104) x, h (cid:105) , h ∈ H. Note that (2.3) yields (cid:104) m ( t, , h (cid:105) = (cid:28) ( λ − A ) (cid:90) t L ( s ) b H d s, h (cid:29) and therefore (2.3) holds for all h ∈ H . Finally, (cid:104) m ( t n , x n ) , h (cid:105) = (cid:104) x n , L (cid:63) ( t n ) h (cid:105) + (cid:104) m ( t n , , h (cid:105) −→ (cid:104) x, h (cid:105) , h ∈ H. Now, using the dominated convergence, we can pass to the limit in (5.1) obtaining thedesired contradiction.We pass now to the proof of the second part of the theorem. It is easy to check that forany φ ∈ F C ∞ b ( A (cid:63) ) formula (5.1) holds and by the definition of the space F C ∞ b ( A (cid:63) ) thefunction Lφ is a well defined element of C strictb ( H bw ). The proof that F C ∞ b ( A (cid:63) ) is a corefor L is an easy modification of the proofs of Lemma 4.4 and Theorem 4.5 in [12] and thusomitted. Example: boundary noise
We will consider here a stochastic PDE with boundary noise introduced in [1], see also[11]: ∂u∂t ( t, ξ ) = ∂ u∂ξ ( t, ξ ) , t > , ξ > ,u ( t,
0) = ˙ W ( t ) , t > ,u (0 , ξ ) = x ( ξ ) , ξ (cid:62) . (6.1)In [1] solution to (6.1) is defined for all t > ξ (cid:62) u ( t, ξ ) = (cid:90) + ∞ G ( t, ξ, η ) x ( η ) d η + (cid:90) t ∂G∂η ( t − s, ξ,
0) d W ( s ) , (6.2)where G ( t, ξ, η ) = 1 √ πt (cid:16) e −| ξ − η | / t − e −| ξ + η | / t (cid:17) , ξ, η (cid:62) . Let ρ ( ξ ) = min (1 , ξ α ) with α ∈ (0 ,
1) and H = L ([0 , + ∞ ) , ρ ( ξ )d ξ ). Finally, let A denote the Dirichlet Laplacian in L ([0 , + ∞ )). Proposition 6.1. (1) The operator A has an extension to the generator (still denoted by A ) of an analytic C -semigroup (cid:0) e tA (cid:1) on H .(2) There exists a selfadjoint unbounded operator Q in H , such that u ( t ) = e tA x + (cid:90) t e ( t − s ) A Q / d W ( s ) . Proof.
Part 1 of the proposition has been proved in [15]. The results in [1] imply that forany
T >
0, sup t (cid:54) T E | u ( t ) | H < ∞ , (6.3)and obviously the process is Gaussian. Clearly, m ( t, x ) = e tA x, t (cid:62) , x ∈ H, and Q ( t ) x ( ξ ) = (cid:90) ∞ q ( t, ξ, η ) ρ ( η ) x ( η ) d η, x ∈ H, ξ ∈ [0 , + ∞ ) , where q ( t, ξ, η ) = (cid:90) t ∂G∂η ( s, ξ, ∂G∂η ( s, η,
0) d s. It is not very difficult to show that E | u ( t ) | H < ∞ , and that u is a mean-square continuous,Gaussian and Markov process in H . For a fixed λ > D the Dirichlet map,e.a. for a > Da as a solution to the equation ( λ − A ) φ = 0, that is (cid:18) λ − ∂ ∂ξ (cid:19) φ = 0 , φ (0) = a. It is easy to see that Da ( ξ ) = a e − ξ √ λ , ξ (cid:62) . Let H − s (0 , + ∞ ) denote the negative Sobolev space. Using arguments similar to those in [7]one can show that a unique mild solution to equation (6.1) exists in H − s (0 , + ∞ ) provided s > . Moreover, u ( t ) = e tA x + ( λ − A ) (cid:90) t e ( t − s ) A D d W ( s ) . (cid:3) References [1] Al`os E. and Bonaccorsi S.:
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