Cylindrical Fractional Brownian Motion in Banach Spaces
aa r X i v : . [ m a t h . P R ] J u l CYLINDRICAL FRACTIONAL BROWNIAN MOTIONIN BANACH SPACES
ELENA ISSOGLIO AND MARKUS RIEDLE
Abstract.
In this article we introduce cylindrical fractional Brownianmotions in Banach spaces and develop the related stochastic integrationtheory. Here a cylindrical fractional Brownian motion is understood inthe classical framework of cylindrical random variables and cylindricalmeasures. The developed stochastic integral for deterministic operatorvalued integrands is based on a series representation of the cylindricalfractional Brownian motion, which is analogous to the Karhunen-Lo`eveexpansion for genuine stochastic processes. In the last part we applyour results to study the abstract stochastic Cauchy problem in a Banachspace driven by cylindrical fractional Brownian motion. Introduction
In the past decades, a wide variety of infinite dimensional stochasticequations have been studied, due to their broad range of applications inphysics, biology, neuroscience and in numerous other areas. A compre-hensive study of stochastic evolution equations in Hilbert spaces driven bycylindrical Wiener processes, based on a semigroup approach, can be foundin the monograph of Da Prato and Zabczyk [8]. Various extensions andmodifications have been studied, such as different types of noises as well asgeneralisations to Banach spaces. For the latter see for example Brze´zniak[6] and van Neerven et al. [29, 30].Fractional Brownian motion (fBm) has become very popular in recentyears as driving noise in stochastic equations, in particular as an alternativeto the classical Wiener noise. This is mainly due to properties of fBms,such as long-term dependence, which leads to a memory effect, and self-similarity , features which show great potential for applications, for examplein hydrology, telecommunication traffic, queueing theory and mathematicalfinance. Since fBms are not semi-martingales, Itˆo-type calculus cannot beapplied. Several different stochastic integrals with respect to real valuedfBm have been introduced in the literature, e.g. Wiener integrals for deter-ministic integrands, Skorohod integrals using Malliavin calculus techniques,pathwise integrals using generalised Stieltjes integrals or integrals based onrough path theory. For more details see e.g. [5, 18, 19] and referencestherein.
Mathematics Subject Classification.
Primary 60G22; Secondary 60H05, 60H15,28C20.
Key words and phrases.
Cylindrical fractional Brownian motion, stochastic integrationin Banach spaces, stochastic partial differential equations, fractional Ornstein-Uhlenbeckprocess, γ -radonifying, cylindrical measures.The second named author acknowledges the EPSRC grant EP/I036990/1. The purpose of this paper is to begin a systematic study of cylindrical frac-tional Brownian motion in Banach spaces and, starting from this, to buildup a related stochastic calculus in Banach spaces with respect to cylindricalfBm. Our approach is based on cylindrical measures and cylindrical ran-dom variables which enables us to develop a theory that does not requirea Hilbert space structure of the underlying space because the cylindricalfBm is defined through finite dimensional projections. We can characterisethe cylindrical fBm by a series representation, which can be considered asthe analogue of the Karhunen-Lo`eve expansion in the classical situation ofgenuine stochastic processes. This representation is exploited to define thestochastic integral of deterministic, operator valued integrands with respectto a cylindrical fBm. The stochastic integral is defined as a stochastic ver-sion of a
Pettis integral, as accomplished in van Neerven and Weis [30] forWiener processes and in Riedle and van Gaans [24] for L´evy processes. Asthe integrand is deterministic, the integral process is Gaussian and thereforeit is characterised by its covariance operator.We apply our theory to a class of parabolic stochastic equations in Banachspaces of the form d Y ( t ) = AY ( t ) d t + C d B ( t ) , where B is a cylindrical fBm in a separable Banach space U , A is a generatorof a strongly continuous semigroup in a separable Banach space V and C is alinear and continuous operator from U to V . We give necessary and sufficientconditions for the existence and uniqueness of a weak solution, which is agenuine stochastic process in the Banach space U . For comparison, we applyour methods to an example often considered in the literature and typicallyformulated in a Hilbert space setting.The systematic approach adopted in this paper goes back to Kallianpurand Xiong [15] and to Metivier and Pellaumail [17], who treated the cylin-drical Wiener case and the cylindrical martingale case, respectively. In thispaper we consider an extension beyond the martingale case, since fractionalBrownian motion is not a semi-martingale. Our methodology, based oncylindrical measures and cylindrical random variables, has the advantagethat it is intrinsic in the sense that it does not require the construction ofa larger space in which the cylindrical noise exists as a genuine stochasticprocess. Due to the connection between cylindrical measures and the the-ory of geometry of Banach spaces, our methodology relates the study of fBmand stochastic differential equations driven by fBm to other areas of mathe-matics, such as operator theory, functional analysis and harmonic analysis,therefore providing a wider range of tools and techniques.Our long-term aim is to study general stochastic equations in Banachspaces driven by cylindrical fBms, which involves stochastic integration forrandom integrands. We are inspired by the paper of van Neerven et al.[29] in which they deal with the Wiener case. Here, the approach is basedon a two-sided decoupling inequality which enables the authors to definethe stochastic integral for random integrands by means of the integral fordeterministic integrands. The latter is introduced in van Neerven and Weis[30], and we hope that our present work will play an analogous role forequations driven by fractional Brownian motions. YLINDRICAL FRACTIONAL BROWNIAN MOTION 3
Only a few works deal with fBm in Banach spaces and related stochasticintegration theory. Brze´zniak et al. [7] consider abstract Cauchy problemsin Banach spaces driven by cylindrical Liouville fBm. It is shown that for
H < / H > / C , and therefore we keep theirregular character of the cylindrical noise in the space where the equationis considered. Note however, that in some special cases the authors in [7] getaround this restriction by means of interpolation techniques. Furthermore,our approach enables us to guarantee the existence of a solution for H > / anisotropic fBm, i.e. spatially non-symmetric noise, and we giveconditions under which such cylindrical noises are genuine fBms in the un-derlying space. Section 5 is dedicated to the construction and the study ofthe stochastic integral in a Banach space. In Section 6 we use this integralto construct the fractional Ornstein-Uhlenbeck process as the mild and weaksolution of a abstract stochastic Cauchy problem in a Banach space. Finally,in Section 7 we consider the special case of the stochastic heat equation withfractional noise in a Hilbert space and compare our results with the existingliterature. E. ISSOGLIO AND M. RIEDLE Preliminaries
Throughout this paper, U indicates a separable Banach space over R withnorm k · k U . The topological dual of U is denoted by U ∗ and the algebraicone by U ′ . For u ∗ ∈ U ∗ we indicate the dual pairing by h u, u ∗ i . If U is aHilbert space we identify the dual space U ∗ with U . The Borel σ -algebraon a Banach space U is denoted by B ( U ). If V is another Banach spacethen L ( U, V ) denotes the space of bounded, linear operators from U to V equipped with the operator norm topology.For a measure space ( S, S , µ ) we denote by L pµ ( S ; U ), p >
0, the space ofequivalence classes of measurable functions f : S → U with R k f ( s ) k pU µ (d s ) < ∞ . If S ∈ B ( R ) and µ is the Lebesgue measure we use the notation L p ( S ; U ).Next we recall some notions about cylindrical measures and cylindricalrandom variables as it can be found in Badrikian [3] or Schwartz [26]. Let Γbe a subset of U ∗ , n ∈ N , u ∗ , . . . , u ∗ n ∈ Γ and B ∈ B ( R n ). A set of the form Z ( u ∗ , . . . , u ∗ n ; B ) := { u ∈ U : ( h u, u ∗ i , . . . , h u, u ∗ n i ) ∈ B } , is called a cylindrical set . We denote by Z ( U, Γ) the set of all cylindricalsets in U for a given Γ. It turns out this is an algebra . Let C ( U, Γ) bethe generated σ -algebra . When Γ = U ∗ the notation is Z ( U ) and C ( U ),respectively. If U is separable then both the Borel σ -algebra B ( U ) and thecylindrical σ -algebra C ( U ) coincide.A function µ : Z ( U ) → [0 , ∞ ] is called a cylindrical measure on Z ( U ) iffor each finite subset Γ ⊆ U ∗ the restriction of µ to the σ -algebra C ( U, Γ)is a measure. It is called finite if µ ( U ) is finite and cylindrical probabilitymeasure if µ ( U ) = 1.For every function f : U → C which is measurable with respect to Z ( U, Γ)for a finite subset Γ ⊆ U ∗ the integral R f ( u ) µ (d u ) is well defined as acomplex valued Lebesgue integral if it exists. In particular, the characteristicfunction ϕ µ : U ∗ → C of a finite cylindrical measure µ is defined by ϕ µ ( u ∗ ) := Z U e ı h u,u ∗ i µ (d u ) for all u ∗ ∈ U ∗ . Let (Ω , A , P ) be a probability space. A cylindrical random variable Z in U is a linear and continuous map Z : U ∗ → L P (Ω; R ) , where L P (Ω; R ) is equipped with the topology of convergence in probability.The characteristic function of a cylindrical random variable Z is defined by ϕ Z : U ∗ → C , ϕ Z ( u ∗ ) = E [exp( ı Zu ∗ )] . A cylindrical process in U is a family ( Z ( t ) : t >
0) of cylindrical randomvariables in U .Let Z : U ∗ → L P (Ω; R ) be a cylindrical random variable in U . If Z = Z ( u ∗ , . . . , u ∗ n ; B ) is a cylindrical set for u ∗ , . . . , u ∗ n ∈ U ∗ and B ∈ B ( R n ), weobtain a cylindrical probability measure µ by the prescription µ ( Z ) := P (( Zu ∗ , . . . , Zu ∗ n ) ∈ B ) . We call µ the cylindrical distribution of Z and the characteristic functions ϕ µ and ϕ Z of µ and Z coincide. Conversely, for every cylindrical measure µ YLINDRICAL FRACTIONAL BROWNIAN MOTION 5 on Z ( U ) there exist a probability space (Ω , A , P ) and a cylindrical randomvariable Z : U ∗ → L P (Ω; R ) such that µ is the cylindrical distribution of Z .A cylindrical probability measure µ on Z ( U ) is called Gaussian if theimage measure µ ◦ ( u ∗ ) − is a Gaussian measure on B ( R ) for all u ∗ ∈ U ∗ .The characteristic function ϕ µ : U ∗ → C of a Gaussian cylindrical measure µ is of the form ϕ µ ( u ∗ ) = exp (cid:0) ı m ( u ∗ ) − s ( u ∗ ) (cid:1) for all u ∗ ∈ U ∗ , (2.1)where the mappings m : U ∗ → R and s : U ∗ → R + are given by m ( u ∗ ) = Z U h u, u ∗ i µ (d u ) , s ( u ∗ ) = Z U h u, u ∗ i µ (d u ) − m ( u ∗ ) . Conversely, if µ is a cylindrical measure with characteristic function of theform (2.1) for a linear functional m : U ∗ → R and a quadratic form s : U ∗ → R + , then µ is a Gaussian cylindrical measure.For a Gaussian cylindrical measure µ with characteristic function of theform (2.1) one defines the covariance operator Q : U ∗ → ( U ∗ ) ′ by( Qu ∗ ) v ∗ = Z U h u, u ∗ ih u, v ∗ i µ (d u ) − m ( u ∗ ) m ( v ∗ ) for all u ∗ , v ∗ ∈ U ∗ . On the contrary to Gaussian Radon measures, the covariance operator mighttake values only in the algebraic dual of U ∗ , that is the linear map Qu ∗ : U ∗ → R might be not continuous for some u ∗ ∈ U ∗ . However often weexclude this rather general situation by requiring at least that Qu ∗ is normcontinuous, that is Q : U ∗ → U ∗∗ . Note that in this situation the character-istic function ϕ µ of µ in (2.1) can be written as ϕ µ ( u ∗ ) = exp (cid:0) ı m ( u ∗ ) − h u ∗ , Qu ∗ i (cid:1) for all u ∗ ∈ U ∗ . A cylindrical random variable Z : U ∗ → L P (Ω; R ) is called Gaussian if itscylindrical distribution is Gaussian. Since we require from the cylindricalrandom variable Z to be continuous it follows that its characteristic function ϕ Z : U ∗ → C is continuous. The latter occurs if and only if the covarianceoperator Q maps to U ∗∗ .3. Wiener integrals for Hilbert space valued integrands
In the following we recall the construction of the Wiener integral withrespect to a real valued fractional Brownian motion for integrands whichare Hilbert space valued deterministic functions. For real valued integrandsthe construction is accomplished for example in [5] and for Hilbert spacevalued integrands in [9, 11, 21].We begin with recalling the definition of a fractional Brownian motion(fBm) and for later purpose, we introduce it in R n . A Gaussian process( b ( t ) : t >
0) in R n is a fractional Brownian motion with Hurst parameter H ∈ (0 ,
1) if there exists a matrix M ∈ R n × n such that E (cid:2) h α, b ( s ) i (cid:3) = 0 , E (cid:2) h α, b ( s ) ih β, b ( t ) i (cid:3) = h M α, β i R ( s, t )for all s, t > α, β ∈ R n , where R ( s, t ) := (cid:0) s H + t H − | s − t | H (cid:1) for all s, t > . E. ISSOGLIO AND M. RIEDLE
The matrix M = ( m i,j ) ni,j =1 is called the covariance matrix of the fBm (cid:0) ( b ( t ) , . . . , b n ( t )) : t >
0) in R n since it follows m i,j = E (cid:2) b i (1) b j (1) (cid:3) for all i, j = 1 , . . . , n .Thus, M is a positive and symmetric matrix. If M = Id then b is called standard fractional Brownian motion . It follows from Kolmogorov’s conti-nuity theorem by the Garsia-Rodemich-Rumsey inequality, that there existsa version of a fBm with H¨older continuous paths of any order smaller than H .We fix for the complete work the Hurst parameter and assume H ∈ (0 , \{ } . The covariance function has an integral representation given by R ( s, t ) = Z s ∧ t κ ( s, u ) κ ( t, u ) d u for all s, t > , (3.1)where the kernel κ has different expressions depending on the Hurst param-eter. If H > then κ ( t, u ) = b H u / − H Z tu ( r − u ) H − / r H − / d r for all 0 u < t, where b H = ( H (2 H − / ( β (2 − H, H − / − / and β denotes the Betafunction. If H < , we have κ ( t, u ) = b H (cid:16) (cid:0) tu (cid:1) H − / ( t − u ) H − / − (cid:0) H − (cid:1) u / − H Z tu ( r − u ) H − / r H − / d r (cid:17) for all 0 u < t, where b H = [2 H/ ((1 − H ) β (1 − H, H + 1 / / .Let X be a separable Hilbert space with scalar product [ · , · ]. A simple X -valued function f : [0 , T ] → X is of the form f ( t ) = n − X i =0 x i [ t i ,t i +1 ) ( t ) for all t ∈ [0 , T ] , (3.2)where x i ∈ X , 0 = t < t < · · · < t n = T and n ∈ N . The space of allsimple, X -valued functions is denoted by E and it is equipped with an innerproduct defined by * m − X i =0 x i [0 ,s i ) , n − X j =0 y j [0 ,t j ) + M := m − X i =0 n − X j =0 [ x i , y j ] R ( s i , t j ) . (3.3)Thus, E is a pre-Hilbert space. We denote the closure of E with respect to h· , ·i M by M .Let ( b ( t ) : t >
0) be a real valued fractional Brownian motion with Hurstparameter H . For a simple, X -valued function f : [0 , T ] → X of the form(3.2) we define the Wiener integral by Z T f d b := n − X i =0 x i (cid:0) b ( t i +1 ) − b ( t i ) (cid:1) . YLINDRICAL FRACTIONAL BROWNIAN MOTION 7
The integral R f d b is a random variable in L P (Ω; X ) and the map f R f d b defines an isometry between E and L P (Ω; X ), since (cid:13)(cid:13)(cid:13)(cid:13)Z T f d b (cid:13)(cid:13)(cid:13)(cid:13) L P = k f k M . (3.4)Consequently, we can extend the mapping f R f d b to the space M andthe extension still satisfies the isometry (3.4).There is an alternative prescription of the space M of possible integrands.For that purpose, we introduce the linear operator K ∗ : E → L ([0 , T ]; X ),which is defined for all t ∈ [0 , T ] in case H < by( K ∗ f )( t ) := f ( t ) κ ( T, t ) + Z Tt ( f ( s ) − f ( t )) ∂κ∂s ( s, t ) d s, and in case H > by( K ∗ f )( t ) := Z Tt f ( s ) ∂κ∂s ( s, t ) d s. The integrals appearing on the right-hand side are both Bochner integrals.Since the operator K ∗ satisfies h K ∗ f, K ∗ g i L = h f, g i M for all f, g ∈ E , (3.5)it can be extended to an isometry K ∗ between M and L ([0 , T ]; X ). To-gether with (3.4) we obtain (cid:13)(cid:13)(cid:13)(cid:13)Z T f d b (cid:13)(cid:13)(cid:13)(cid:13) L P = k K ∗ f k L = k f k M for all f ∈ M . (3.6)The operator K ∗ can be rewritten using the notion of fractional integralsand derivatives. For this purpose, define for α > fractional integraloperator I αT − : L ([0 , T ]; X ) → L ([0 , T ]; X ) by (cid:0) I αT − f (cid:1) ( t ) := 1Γ( α ) Z Tt ( s − t ) α − f ( s ) d s for all t ∈ [0 , T ] . Young’s inequality guarantees that I αT − f ∈ L ([0 , T ]; X ) and that the oper-ator I αT − is bounded on L ([0 , T ]; X ). We define the space H αT − ([0 , T ]; X ) := I αT − ( L ([0 , T ]; X ))and equip it with the norm (cid:13)(cid:13) I αT − f (cid:13)(cid:13) H αT − := k f k L for all f ∈ L ([0 , T ]; X ) . It follows that the space H αT − ([0 , T ]; X ) is a Hilbert space and it is continu-ously embedded in L ([0 , T ]; X ).For α ∈ (0 ,
1) the fractional differential operator D αT − : H αT − ([0 , T ]; X ) → L ([0 , T ]; X ) is defined by( D αT − f )( t ) := 1Γ(1 − α ) (cid:18) f ( t )( T − t ) α + α Z Tt f ( t ) − f ( s )( s − t ) α +1 d s (cid:19) E. ISSOGLIO AND M. RIEDLE for all t ∈ [0 , T ]. The fractional integral and differential operators obey theinversion formulas I αT − ( D αT − f ) = f for all f ∈ H αT − ([0 , T ]; X ) , and D αT − ( I αT − f ) = f for all f ∈ L ([0 , T ]; X ) . Let p H − / denote the function p H − / ( t ) = t H − / for all t ∈ [0 , T ]. Theoperator K ∗ can be rewritten in the case H > / K ∗ f )( t ) = b H Γ (cid:0) H − (cid:1) t / − H I H − / T − (cid:16) p H − / f (cid:17) ( t ) (3.7)for all t ∈ [0 , T ] and in the case of H < / K ∗ f )( t ) = b H Γ (cid:0) H + (cid:1) t / − H D / − HT − (cid:16) p H − / f (cid:17) ( t ) . (3.8)It can be seen from (3.7) that M contains distribution for H > . Thusit became standard to restrict the space M in this case, see for example[5, 11, 22]. It turns out that an appropriate choice is the function space |M| := (cid:26) f : [0 , T ] → X : Z T Z T k f ( s ) kk f ( t ) k| s − t | H − d s d t < ∞ (cid:27) , equipped with the norm k f k |M| := H (2 H − Z T Z T k f ( s ) kk f ( t ) k| s − t | H − d s d t. The space |M| is complete and it is continuously embedded in M . The proofof this fact is analogous to the real valued case, see e. g. [5, Pro.2.1.13]. If H > then the covariance function R is differentiable with ∂ R∂s∂t ( s, t ) = H (2 H − | s − t | H − for all s, t > , and we can rewrite (3.3) as h f, g i M = H (2 H − Z T Z T [ f ( s ) , g ( t )] | s − t | H − d s d t (3.9)for all simple functions f, g ∈ E . Since E is dense in |M| , equation (3.9) istrue for all f, g ∈ |M| , see [11, Eq.(2.14)].We summarise the two cases by defining c M := ( M if H ∈ (0 , / , |M| if H ∈ (1 / , . (3.10)Recall that c M is a Banach space and the operator K ∗ satisfies k K ∗ f k L c k f k c M for all f ∈ c M (3.11)for a constant c >
0. Inequality (3.11) follows from (3.6) and, if
H > ,from the continuous embedding |M| ֒ → M . If H < we can choose c = 1.In the sequel, we collect some properties of the spaces M and |M| . Recallthat the time interval [0 , T ] is fixed. In our first result the coincidence of YLINDRICAL FRACTIONAL BROWNIAN MOTION 9 the spaces are well known, whereas we are only aware that the equivalenceof the norms is stated in [7] but without a proof.
Proposition 3.1.
For
H < the spaces M and H / − HT − ([0 , T ]; X ) coincideand the norms are equivalent.Proof. The fact that the spaces coincide is shown in [1, Pro.6]. The proof ofthe equivalence of the norms is based on the following relation, which canbe found in the proof of [1, Pro.6]: K ∗ f = a (cid:16) D / − HT − f (cid:17) + Rf for all f ∈ M , (3.12)where a := b H Γ( H + ) and R : L ([0 , T ]; X ) → L ([0 , T ]; X ) is a linear andcontinuous operator. Since H / − HT − ([0 , T ]; X ) is continuously embedded in L ([0 , T ]; X ) there exists a constant c > f ∈ M we have k f k M = k K ∗ f k L a (cid:13)(cid:13)(cid:13) D / − HT − f (cid:13)(cid:13)(cid:13) L + k Rf k L ( a + c k R k ) k f k H / − HT − . On the other hand, the Hardy-Littlewood inequality in weighted spaces guar-antees that M is continuously embedded in L /H ([0 , T ]; X ). More specifi-cally, by choosing p = 2 , α = − H, m = 0 , q = H , µ = 2 α, ν = qα in [25,Th.5.4], we obtain for f ∈ Mk f k L q = (cid:18)Z T k f ( t ) k q d t (cid:19) /q = (cid:18)Z T t ν (cid:13)(cid:13) t − α f ( t ) (cid:13)(cid:13) q d t (cid:19) /q = (cid:18)Z T t ν (cid:13)(cid:13) ( I αT − D αT − p − α f )( t ) (cid:13)(cid:13) q d t (cid:19) /q c (cid:18)Z T t µ (cid:13)(cid:13) ( D αT − p − α f )( t ) (cid:13)(cid:13) p d t (cid:19) /p = c ( b H Γ( H + )) − /p k K ∗ f k L = c ( b H Γ( H + )) − /p k f k M , (3.13)for a constant c >
0. Consequently, together with the continuous embeddingof L /H ([0 , T ]; X ) in L ([0 , T ]; X ), it follows from (3.12) that each f ∈ M satisfies a k f k H / − HT − k K ∗ f k L + k Rf k L = (cid:16) c ( b H Γ( H + )) − / k R k (cid:17) k f k M , which completes the proof. (cid:3) Proposition 3.2.
For every t ∈ [0 , T ] there exists a constant c t > suchthat each f ∈ c M obeys: (a) [0 ,t ] f ∈ c M and (cid:13)(cid:13) [0 ,t ] f (cid:13)(cid:13) M c t k f k M . (b) [0 ,t ] f ( t − · ) ∈ c M and (cid:13)(cid:13) [0 ,t ] f ( t − · ) (cid:13)(cid:13) M = (cid:13)(cid:13) [0 ,t ] f (cid:13)(cid:13) M .Proof. If H > , both properties (a) and (b) follow from (3.9) with c t = 1 forall t ∈ [0 , T ]. If H < , note that it is known for f ∈ H / − HT − ([0 , T ]; X ) that [0 ,t ] f and [0 ,t ] f ( t − · ) are in H / − HT − ([0 , T ]; X ), see [25, Th.13.9, Th.13.10,Re.13.3] or [7, Le.2.1, Le.2.2]. Furthermore, there exists a constant a t > (cid:13)(cid:13) [0 ,t ] f (cid:13)(cid:13) H / − HT − a t k f k H / − HT − . Thus Proposition 3.1 implies part (a) and [0 ,t ] f ( t − · ) ∈ c M . To show thenorm equality in part (b), note the identity h g, h i = e H D D / − H − g, D / − H + h E L for all g, h ∈ M , (3.14)where e H denotes a constant depending only on H , see [20, page 286]. Here D α ± denote the right-sided/left-sided Weyl-Marchaud fractional derivativesdefined by D α ± g ( r ) := α Γ(1 − α ) Z ∞ g ( r ) − g ( r ∓ s ) s α d s for all r ∈ R . It follows from (3.14) that (cid:13)(cid:13) [0 ,t ] f ( t − · ) (cid:13)(cid:13) M = e H (cid:10)(cid:0) D / − H − [0 ,t ] f ( t − · ) (cid:1) ( · ) , (cid:0) D / − H + [0 ,t ] f ( t − · ) (cid:1) ( · ) (cid:11) = e H (cid:10)(cid:0) D / − H + [0 ,t ] f (cid:1) ( t − · ) , (cid:0) D / − H − [0 ,t ] f (cid:1) ( t − · ) (cid:11) = (cid:13)(cid:13) [0 ,t ] f (cid:13)(cid:13) M , which completes the proof. (cid:3) In the following we prove a technical result that links the real case, that is X = R in the above, with the Hilbert case. For this reason we will stress thedependence on the underlying space by writing either K ∗ R or K ∗ X . Analogousnotation will be adopted for the space c M . Proposition 3.3. (a)
Let f be in c M R and x ∈ X . Then F : [0 , T ] → X, F ( t ) = x f ( t ) , defines an element in c M X satisfying ( K ∗ X F )( · ) = x ( K ∗ R f )( · ) . (b) Let F be in c M X and x ∈ X . Then f : [0 , T ] → R f ( t ) = [ F ( t ) , x ] , defines an element in c M R satisfying h K ∗ X F ( · ) , x i = ( K ∗ R f )( · ) .Proof. We prove only part (a) as part (b) can be done analogously. If
H < then by Proposition 3.1 there exists ϕ f ∈ L ([0 , T ]; R ) such that f = I / − HT − ϕ f . Since xϕ f ∈ L ([0 , T ]; X ) and F = xI / − HT − ϕ f = I / − HT − xϕ f ,it follows that F ∈ c M X . If H > , the assumption f ∈ | M | R implies F ∈ | M | X . In both cases, the very definition of K ∗ X and K ∗ R shows K ∗ X F = xK ∗ R f . (cid:3) YLINDRICAL FRACTIONAL BROWNIAN MOTION 11 Cylindrical fractional Brownian motion
We define cylindrical fractional Brownian motions in a separable Banachspace U by following the classical approach of cylindrical processes. In thesame way, one can introduce cylindrical Wiener processes, see for instance[15, 17, 23], and recently, this approach has been accomplished in [2] to givethe first systematic treatment of cylindrical L´evy processes. Definition 4.1.
A cylindrical process ( B ( t ) : t > in U is a cylindricalfractional Brownian motion with Hurst parameter H ∈ (0 , if (a) for any u ∗ , . . . , u ∗ n ∈ U ∗ and n ∈ N , the stochastic process (cid:0) ( B ( t ) u ∗ , . . . , B ( t ) u ∗ n ) : t > (cid:1) is a fractional Brownian motion with Hurst parameter H in R n ; (b) the covariance operator Q : U ∗ → U ∗∗ of B (1) defined by h Qu ∗ , v ∗ i = E (cid:2)(cid:0) B (1) u ∗ (cid:1)(cid:0) B (1) v ∗ (cid:1)(cid:3) for all u ∗ , v ∗ ∈ U ∗ , is U -valued. By applying part (a) for n = 2 it follows that a cylindrical fBm ( B ( t ) : t >
0) with covariance operator Q obeys E (cid:2) ( B ( s ) u ∗ )( B ( t ) v ∗ ) (cid:3) = h Qu ∗ , v ∗ i R ( s, t )for all s, t > u ∗ , v ∗ ∈ U ∗ . Note that if H = then Definition 4.1covers the cylindrical Wiener process as defined in [15, 17, 23].Definition 4.1 involves all possible n -dimensional projections of the pro-cess, but since we are dealing with Gaussian processes the condition can besimplified using only two-dimensional projections. Lemma 4.2.
For a cylindrical process B := ( B ( t ) : t > in U the followingare equivalent: (a) B is a cylindrical fractional Brownian motion with Hurst parameter H ∈ (0 , ; (b) B satisfies: (i) for each u ∗ , v ∗ ∈ U ∗ the stochastic process (cid:0) ( B ( t ) u ∗ , B ( t ) v ∗ ) : t > (cid:1) is a two-dimensional fBm; (ii) the covariance operator of B (1) is U -valued.Proof. We have to prove only the implication (b) ⇒ (a). For u ∗ , . . . , u ∗ n ∈ U ∗ define the stochastic process Y = (cid:0) ( B ( t ) u ∗ , . . . , B ( t ) u ∗ n ) : t > (cid:1) . Itfollows that Y is Gaussian and satisfies E [ h α, Y ( t ) i ] = 0 for all t > α = ( α , . . . , α n ) ∈ R n since h α, Y ( t ) i = n X i =1 α i B ( t ) u ∗ i = B ( t ) n X i =1 α i u ∗ i ! . Let M = ( m i,j ) ni,j =1 be the n -dimensional matrix defined by m i,j = E (cid:2)(cid:0) B (1) u ∗ i (cid:1)(cid:0) B (1) u ∗ j (cid:1)(cid:3) , i, j = 1 , . . . , n. Since it follows from (b) that E (cid:2) ( B ( s ) u ∗ i )( B ( t ) u ∗ j ) (cid:3) = m i,j R ( s, t ) for all s, t > i, j = 1 , . . . , n we obtain E (cid:2) h α, Y ( s ) ih β, Y ( t ) i (cid:3) = E n X i =1 n X j =1 α i β j (cid:0) B ( s ) u ∗ i (cid:1)(cid:0) B ( t ) u ∗ j (cid:1) = n X i =1 n X j =1 α i β j m i,j R ( s, t )= h M α, β i R ( s, t )for each α = ( α , . . . , α n ) and β = ( β , . . . , β n ) in R n . (cid:3) The following result provides an analogue of the Karhunen-Lo`eve expan-sion for cylindrical Wiener processes.
Theorem 4.3.
For a cylindrical process B := ( B ( t ) : t > the followingare equivalent: (a) B is a cylindrical fractional Brownian motion with Hurst parameter H ∈ (0 , ; (b) there exist a Hilbert space X with an orthonormal basis ( e k ) k ∈ N , i ∈L ( X, U ) and a sequence ( b k ) k ∈ N of independent, real valued standardfBms with Hurst parameter H ∈ (0 , such that B ( t ) u ∗ = ∞ X k =1 h ie k , u ∗ i b k ( t ) (4.1) in L P (Ω; R ) for all u ∗ ∈ U ∗ and t > .In this situation the covariance operator of B (1) is given by Q = ii ∗ : U ∗ → U .Proof. The implication (a) ⇒ (b) can be proved as Theorem 4.8 in [2]. Forestablishing the implication (b) ⇒ (a), it is immediate that the right handside of (4.1) converges. Fix u ∗ , . . . , u ∗ n ∈ U ∗ and define the n -dimensionalstochastic process Y := ( Y ( t ) : t >
0) by Y ( t ) : = ( B ( t ) u ∗ , . . . , B ( t ) u ∗ n ) for all t > . It follows that Y is Gaussian and satisfies E [ h α, Y ( t ) i ] = 0 for all t > α = ( α , . . . , α n ) ∈ R n since h α, Y ( t ) i = n X i =1 α i B ( t ) u ∗ i = B ( t ) n X i =1 α i u ∗ i ! . Let M = ( m i,j ) ni,j =1 be the n × n -dimensional covariance matrix of therandom vector Y (1), that is m i,j := E (cid:2) ( B (1) u ∗ i )( B (1) u ∗ j ) (cid:3) . The definitionof Y yields m i,j = ∞ X k =1 ∞ X ℓ =1 h ie k , u ∗ i ih ie ℓ , u ∗ j i E [ b k (1) b ℓ (1)] = ∞ X k =1 h ie k , u ∗ i ih ie k , u ∗ j i . YLINDRICAL FRACTIONAL BROWNIAN MOTION 13
Let α = ( α , . . . , α n ) ∈ R n and β = ( β , . . . , β n ) ∈ R n . By using theindependence of b k and b ℓ for each k = ℓ we obtain for every s, t > E (cid:2) h α, Y ( s ) ih β, Y ( t ) i (cid:3) = E n X i =1 α i ∞ X k =1 h ie k , u ∗ i i b k ( s ) ! n X j =1 β j ∞ X ℓ =1 h ie ℓ , u ∗ j i b ℓ ( t ) = n X i =1 n X j =1 α i β j ∞ X k =1 ∞ X ℓ =1 h ie k , u ∗ i ih ie ℓ , u ∗ j i E [ b k ( s ) b ℓ ( t )]= n X i =1 n X j =1 α i β j ∞ X k =1 h ie k , u ∗ i ih ie k , u ∗ j i E [ b k ( s ) b k ( t )]= n X i =1 n X j =1 α i β j m i,j R ( s, t )= h M α, β i R ( s, t ) . It is left to prove that B (1) : U ∗ → L P (Ω; R ) is continuous and its covarianceoperator Q : U ∗ → U ∗′ is U -valued. By independence of b k and b ℓ for k = ℓ it follows for u ∗ ∈ U ∗ that ϕ B (1) ( u ∗ ) = ∞ Y k =1 E (cid:2) exp ( ı h ie k , u ∗ i b k (1)) (cid:3) = ∞ Y k =1 exp (cid:0) − h ie k , u ∗ i (cid:1) = exp (cid:0) − k i ∗ u ∗ k X (cid:1) . Thus, the characteristic function ϕ B (1) : U ∗ → C is continuous, which entailsthe continuity of B (1) by [28, Pro. IV.3.4]. Moreover, it follows that Q = ii ∗ ,that is the covariance operator Q is U -valued and of the claimed form. (cid:3) Example 4.4.
Let U be a Hilbert space with orthonormal basis ( e k ) k ∈ N ,identify the dual space U ∗ with U , and let ( q k ) k ∈ N ⊆ R be a sequencesatisfying sup k ∈ N | q k | < ∞ . It follows by Theorem 4.3 that for an arbitrarysequence ( b k ) k ∈ N of independent, real valued standard fBms, the series B ( t ) u := ∞ X k =1 q k h e k , u i b k ( t ) , u ∈ U, defines a cylindrical fBm ( B ( t ) : t >
0) in U . The covariance operator Q isgiven by Q = ii ∗ , where i : U → U is defined as iu = P ∞ k =1 q k h e k , u i e k . Example 4.5.
For a set D ∈ B ( R n ) let ( e k ) k ∈ N ⊆ L ( D ; R ) be an or-thonormal basis and let ( τ k ) k ∈ N be a sequence of functions τ k ∈ L ( D ; R )satisfying P ∞ k =1 k τ k k L < ∞ . Applying Cauchy-Schwarz inequality twiceshows that i : L ( D ; R ) → L ( D ; R ) , if = ∞ X k =1 h e k , f i τ k ( · ) e k ( · ) (4.2)defines a linear and continuous mapping. It follows from Theorem 4.3 thatfor an arbitrary sequence ( b k ) k ∈ N of independent, real valued standard fBm, the series B ( t ) f := ∞ X k =1 h ie k , f i b k ( t ) , f ∈ L ∞ ( D ; R ) , defines a cylindrical fBm ( B ( t ) : t >
0) in L ( D ; R ) with covariance operator Q = ii ∗ : L ∞ ( D ; R ) → L ( D ; R ). Example 4.6.
A special case of Example 4.5 is obtained by choosing thefunctions τ k ∈ L ( D ; R ) as τ k = q k A k for q k ∈ R and A k ∈ B ( D ) satisfying P ∞ k =1 q k Leb( A k ) < ∞ . Then the cylindrical fBm of Example 4.5 has theform B ( t ) f = ∞ X k =1 q k h A k e k , f i b k ( t ) . This process can be considered as an anisotropic cylindrical fractional Brow-nian sheet in L ( D ; R ) since its covariance structure might vary in differentdirections.In the final part of this section we consider the relation between cylindricaland genuine fractional Brownian motion in a separable Banach space U . Forthis purpose, we generalise the definition of a fractional Brownian motionin R n to Banach spaces. This definition is consistent with others in theliterature, in particular the one in [9] for Hilbert spaces. Definition 4.7. A U -valued Gaussian stochastic process ( Y ( t ) : t > iscalled a fractional Brownian motion in U with Hurst parameter H ∈ (0 , if there exists a mapping Q : U ∗ → U such that h Y ( t ) , u ∗ i = 0 , E (cid:2) h Y ( s ) , u ∗ ih Y ( t ) , v ∗ i (cid:3) = h Qu ∗ , v ∗ i R ( s, t ) for all s, t > and u ∗ , v ∗ ∈ U ∗ . By taking s = t = 1 it follows that h Qu ∗ , v ∗ i = E (cid:2) h Y (1) , u ∗ ih Y (1) , v ∗ i (cid:3) for all u ∗ , v ∗ ∈ U ∗ . Thus, Q is the covariance operator of the Gaussian measure P Y (1) and itmust be a symmetric and positive operator in L ( U ∗ , U ).Clearly every fBm in a Banach space U is a cylindrical fBm in U andthus, it obeys the representation (4.1). However, the operator i , or in otherwords the embedding of the reproducing kernel Hilbert space, must yield aRadon measure in U , which basically leads to the following result: Theorem 4.8.
For a U -valued stochastic process Y := ( Y ( t ) : t > thefollowing are equivalent: (a) Y is a fBm in U with Hurst parameter H ∈ (0 , ; (b) there exist a Hilbert space X with an orthonormal basis ( e k ) k ∈ N ,a γ -radonifying operator i ∈ L ( X, U ) and independent, real valuedstandard fBms ( b k ) k ∈ N such that Y ( t ) = ∞ X k =1 ie k b k ( t ) in L P (Ω; U ) for all t > . YLINDRICAL FRACTIONAL BROWNIAN MOTION 15
In this situation the covariance operator of Y (1) is given by Q = ii ∗ : U ∗ → U .Proof. The result can be proved as Theorem 23 in [23]. (cid:3)
In the literature a fractional Brownian motion in a Hilbert space is oftendefined by a series representation as in Theorem 4.8, in which case the spaceof γ -radonifying operators coincides with Hilbert-Schmidt operators.If ( B ( t ) : t >
0) is a cylindrical fBm which is induced by a U -valuedprocess ( Y ( t ) , t > B ( t ) u ∗ = h Y ( t ) , u ∗ i for all t > , u ∗ ∈ U ∗ , (4.3)then Y is a U -valued fBm. Vice versa, if Y is a U -valued fBm then B definedby (4.3) is a cylindrical fBm, and in both cases the covariance operatorscoincide. This can be seen by the fact that (4.3) determines uniquely thecharacteristic functions of (cid:0) B ( s ) , B ( t ) (cid:1) and (cid:0) Y ( s ) , Y ( t ) (cid:1) for all s, t > U -valued fBmif and only if the embedding i is γ -radonifying. This result can be establishedas in [23, Th.25]. Example 4.9.
If we assume in Example 4.5 that the functions τ k are in L ∞ ( D ; R ) and satisfy P k τ k k ∞ < ∞ then the mapping i , defined in (4.2),maps to L ( D ; R ). Moreover, i is a Hilbert-Schmidt operator, as ∞ X k =1 k ie k k L = ∞ X k =1 k τ k e k k L ∞ X k =1 k τ k k ∞ . Since γ -radonifying and Hilbert-Schmidt operators coincide in Hilbert spaces,Theorem 4.8 implies that the cylindrical fBm in Example 4.9 is induced bya genuine fractional Brownian motion in L ( D ; R ).5. Integration
In this section we introduce the stochastic integral R Ψ( s ) d B ( s ) as a V -valued random variable for deterministic, operator valued functions Ψ :[0 , T ] → L ( U, V ), where V is another separable Banach space. Our approachis based on the idea to introduce firstly a cylindrical random variable Z Ψ : V ∗ → L P (Ω; R ) as a cylindrical integral . Then we call a V -valued randomvariable I Ψ : Ω → V the stochastic integral of Ψ if it satisfies Z Ψ v ∗ = h I Ψ , v ∗ i for all v ∗ ∈ V ∗ . In this way, the stochastic integral I Ψ can be considered as a stochastic Pet-tis integral . This approach enables us to have a candidate of the stochasticintegral, i.e. the cylindrical random variable Z Ψ , under very mild condi-tions at hand because cylindrical random variables are more general objectsthan genuine random variables. The final requirement, that the cylindri-cal random variable Z Ψ is in fact a classical Radon random variable, canbe equivalently described in terms of the corresponding covariance operatorand thus, it solely depends on geometric properties of the underlying Banachspace V . For defining the cylindrical integral, recall the representation of a cylin-drical fBm ( B ( t ) : t >
0) with Hurst parameter H ∈ (0 ,
1) in the Banachspace U , according to Theorem 4.3: B ( t ) u ∗ = ∞ X k =1 h ie k , u ∗ i b k ( t ) for all u ∗ ∈ U ∗ , t > . (5.1)Here, X is a Hilbert space with an orthonormal basis ( e k ) k ∈ N , i : X → U isa linear, continuous mapping and ( b k ) k ∈ N is a sequence of independent, realvalued standard fBms. If we assume momentarily that we have already in-troduced a stochastic integral R T Ψ( t ) d B ( t ) as a V -valued random variable,then the representation (5.1) of B naturally results in ∞ X k =1 Z T h Ψ( t ) ie k , v ∗ i d b k ( t ) for all v ∗ ∈ V ∗ . (5.2)By swapping the terms in the dual pairing, the integrals can be consideredas the Fourier coefficients of the X -valued integral Z T i ∗ Ψ ∗ ( t ) v ∗ d b k ( t ) , which we introduce in Section 3. This results in the minimal requirementthat the function t i ∗ Ψ ∗ ( t ) v ∗ must be integrable with respect to the realvalued standard fBm b k for every v ∗ ∈ V ∗ and k ∈ N , that is the functionΨ must be in the linear space I := { Φ : [0 , T ] → L ( U, V ) : i ∗ Φ ∗ ( · ) v ∗ ∈ c M for all v ∗ ∈ V ∗ } . Here, c M = c M X denotes the Banach space of functions f : [0 , T ] → X introduced in Section 3. For this class of integrands we have the followingproperty. Proposition 5.1.
For each Ψ ∈ I the mapping L Ψ : V ∗ → c M , L Ψ v ∗ = i ∗ Ψ ∗ ( · ) v ∗ is linear and continuous.Proof. The operator L = L Ψ is linear and takes values in c M by definitionof I . We prove that L is continuous by the closed mapping theorem. Forthis purpose, let v ∗ n → v ∗ in V ∗ and Lv ∗ n → g ∈ c M . We consider the cases H < / H > / Case
H < / . From the Hardy-Littlewood inequality in weighted spaces,see (3.13), it follows that k f k L q c ( b H Γ( H + )) − / k f k M for all f ∈ M for a constant c > q = H . Consequently, the convergence Lv ∗ n → g in M implies that there exists a subsequence ( n k ) k ∈ N ⊆ N such that Lv ∗ n k ( t ) → g ( t ) as k → ∞ for Lebesgue almost all t ∈ [0 , T ]. On the other hand, we have i ∗ Ψ ∗ ( t ) v ∗ n k → i ∗ Ψ ∗ ( t ) v ∗ in X as k → ∞ for all t ∈ [0 , T ], because i ∗ and Ψ ∗ ( t )are continuous. Consequently, we arrive at g ( t ) = i ∗ Ψ ∗ ( t ) v ∗ for Lebesguealmost all t ∈ [0 , T ], and thus, g = Lv ∗ as functions in L ([0 , T ]; X ). YLINDRICAL FRACTIONAL BROWNIAN MOTION 17
Case
H > / . In this case c M = |M| . Let us remark, that if f ∈ |M| then f ∈ L ([0 , T ]; X ) and(2 T ) H − k f k L = (2 T ) H − Z T Z T k f ( s ) kk f ( t ) k d s d t Z T Z T k f ( s ) kk f ( t ) k| s − t | H − d s d t = 1 H (2 H − k f k |M| . Using this fact, the convergence Lv ∗ n → g in |M| implies that Lv ∗ n k ( t ) → g ( t )as k → ∞ for Lebesgue almost all t ∈ [0 , T ] for a subsequence ( n k ) k ∈ N ⊆ N .The continuity of the mapping v ∗ i ∗ Ψ ∗ ( t ) v ∗ for all t ∈ [0 , T ] shows that g ( t ) = i ∗ Ψ ∗ ( t ) v ∗ for Lebesgue almost all t ∈ [0 , T ] and thus, g = Lv ∗ in |M| . (cid:3) Before we establish the existence of the cylindrical integral as motivatedin (5.2), we introduce an operator which will turn out to be the factorisationof the covariance operator of the cylindrical integral.
Lemma 5.2.
For every Ψ ∈ I we define h Γ Ψ f, v ∗ i = Z T [ K ∗ ( i ∗ Ψ ∗ ( · ) v ∗ )( t ) , f ( t )] d t for all f ∈ L ([0 , T ]; X ) , v ∗ ∈ V ∗ . In this way, one obtains a linear, bounded operator Γ Ψ : L ([0 , T ]; X ) → V ∗∗ .Proof. Proposition 5.1, together with equation (3.11), implies |h Γ Ψ f, v ∗ i| = |h K ∗ ( i ∗ Ψ ∗ ( · ) v ∗ ) , f i L | c k i ∗ Ψ ∗ ( · ) v ∗ k c M k f k L c k v ∗ k V ∗ k f k L , for some constants c , c >
0, which shows boundedness of Γ Ψ . (cid:3) Proposition 5.3.
Let the fBm B be represented in the form (5.1) . Thenfor each Ψ ∈ I the mapping Z Ψ : V ∗ → L P (Ω; R ) , Z Ψ v ∗ := ∞ X k =1 Z T h Ψ( t ) ie k , v ∗ i d b k ( t ) (5.3) defines a Gaussian cylindrical random variable in V with covariance operator Q Ψ : V ∗ → V ∗∗ , factorised by Q Ψ = Γ Ψ Γ ∗ Ψ . Furthermore, the cylindricalrandom variable Z Ψ is independent of the representation (5.1) .Proof. Since h Ψ( · ) ie k , v ∗ i = [ e k , i ∗ Ψ ∗ ( · ) v ∗ ] and i ∗ Ψ ∗ ( · ) v ∗ ∈ c M for every v ∗ ∈ V ∗ , Proposition 3.3 guarantees that the one-dimensional integrals in (5.3) are well defined, and it implies that k Z Ψ v ∗ k L P = ∞ X k =1 E (cid:12)(cid:12)(cid:12)(cid:12)Z T h Ψ( t ) ie k , v ∗ i d b k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = ∞ X k =1 Z T (cid:12)(cid:12) K ∗ R (cid:0) h Ψ( · ) ie k , v ∗ i (cid:1) ( t ) (cid:12)(cid:12) d t = ∞ X k =1 Z T (cid:12)(cid:12) K ∗ R (cid:0) [ e k , i ∗ Ψ ∗ ( · ) v ∗ ] (cid:1) ( t ) (cid:12)(cid:12) d t = ∞ X k =1 Z T (cid:12)(cid:12) [ e k , K ∗ X (cid:0) i ∗ Ψ ∗ ( · ) v ∗ (cid:1) ( t )] (cid:12)(cid:12) d t = ∞ X k =1 Z T (cid:2) e k , (cid:0) Γ ∗ Ψ v ∗ (cid:1) ( t ) (cid:3) d t = k Γ ∗ Ψ v ∗ k L . Consequently, the sum in (5.3) converges in L P (Ω; R ) and the limit is a zeromean Gaussian random variable. The continuity of the operator Γ ∗ Ψ : V ∗ → L ([0 , T ]; X ) implies the continuity of Z Ψ : V ∗ → L P (Ω; R ). It follows forthe characteristic function of Z Ψ that ϕ Z Ψ ( v ∗ ) = exp (cid:16) − k Γ ∗ Ψ v ∗ k L (cid:17) for all v ∗ ∈ V ∗ . Since Lemma 5.2 implies that k Γ ∗ Ψ v ∗ k L = h Γ Ψ Γ ∗ Ψ v ∗ , v ∗ i for all v ∗ ∈ V ∗ , it follows that the covariance operator Q Ψ of Z Ψ obeys Q Ψ = Γ Ψ Γ ∗ Ψ .The independence of Z Ψ of the representation (5.1) can be established asin [23, Le.2]. (cid:3) For Ψ ∈ I we call the cylindrical random variable Z Ψ , defined in (5.3),the cylindrical integral of Ψ. Apart from the restriction of the space M ofall integrable distributions to c M , the condition for a mapping Ψ to be in I isthe minimal requirement to guarantee that the real valued integrals in (5.2)exist. Thus without any further condition the cylindrical integral Z Ψ existsin the Banach space U . However, in order to obtain that the cylindricalintegral Z Ψ extends to a genuine random variable in U , the integrand mustexhibit further properties. Definition 5.4.
A function Ψ ∈ I is called stochastically integrable if thereexists a random variable I Ψ : Ω → V such that Z Ψ v ∗ = h I Ψ , v ∗ i for all v ∗ ∈ V ∗ , where Z Ψ denotes the cylindrical integral of Ψ . We use the notation I Ψ := Z T Ψ( t ) d B ( t ) . In other words, a function Ψ ∈ I is stochastically integrable if and only ifthe cylindrical random variable Z Ψ is induced by a Radon random variable.This occurs if and only if the cylindrical distribution of Z Ψ extends to a YLINDRICAL FRACTIONAL BROWNIAN MOTION 19
Radon measure. In Sazonov spaces this is equivalent to the condition thatthe characteristic function of Z Ψ is Sazonov continuous. However, since thecylindrical distribution of Z Ψ is Gaussian, one can equivalently express thestochastic integrability in terms of the covariance operator. Theorem 5.5.
For Ψ ∈ I the following are equivalent: (a) Ψ is stochastically integrable; (b) the operator Γ Ψ is V -valued and γ -radonifying.Proof. (b) ⇒ (a). Let γ be the canonical Gaussian cylindrical measure on L ([0 , T ]; X ). It follows from Proposition 5.3 that the cylindrical distributionof Z Ψ is the image cylindrical measure γ ◦ Γ − . According to [28, Thm.IV.2.5,p.216], the cylindrical random variable Z Ψ is induced by a V -valued randomvariable if and only if its cylindrical distribution γ ◦ Γ − extends to a Radonmeasure on B ( V ), which is guaranteed by (b).(a) ⇒ (b). The proof follows closely some arguments in the proof ofTheorem 2.3 in [30]. Let Q : V ∗ → V be the covariance operator of theGaussian random variable R Ψ( t ) d B ( t ). Proposition 5.3 implies that Q =Γ Ψ Γ ∗ Ψ : V ∗ → V . Define the set S := { K ∗ ( i ∗ Ψ ∗ ( · ) v ∗ ) : v ∗ ∈ V ∗ } , whichis a subset of L ([0 , T ]; X ). By the very definition of Γ Ψ , a function f ∈ L ([0 , T ]; X ) is in ker Γ Ψ if and only if f ⊥ S , which yields L ([0 , T ]; X ) = ¯ S ⊕ ker Γ Ψ . (5.4)Since for all v ∗ , w ∗ ∈ V ∗ we have h Γ Ψ K ∗ ( i ∗ Ψ ∗ ( · ) v ∗ ) , w ∗ i = h Γ Ψ Γ ∗ Ψ v ∗ , w ∗ i = h Qv ∗ , w ∗ i , it follows that Γ Ψ K ∗ ( i ∗ Ψ ∗ ( · ) v ∗ ) = Qv ∗ for all v ∗ ∈ V ∗ . Consequently,Γ Ψ f ∈ V for all f ∈ S and the decomposition (5.4) implies that Γ Ψ f ∈ V for all f ∈ L ([0 , T ]; X ). Clearly, since Q is a Gaussian covariance operator,the operator Γ Ψ is γ -radonifying, which completes the proof. (cid:3) Corollary 5.6. If Φ and Ψ are mappings in I satisfying k i ∗ Φ ∗ ( · ) v ∗ k M c k i ∗ Ψ ∗ ( · ) v ∗ k M for all v ∗ ∈ V ∗ , for a constant c > and if Ψ is stochastically integrable then Φ is alsostochastically integrable.Proof. The proof follows some arguments in the proof of Theorem 2.3 in [30].Define the operator Q := Γ Φ Γ ∗ Φ : V ∗ → V ∗∗ . The isometry (3.6) implies forevery v ∗ ∈ V ∗ that h v ∗ , Qv ∗ i = h Γ ∗ Φ v ∗ , Γ ∗ Φ v ∗ i L = (cid:13)(cid:13) K ∗ (cid:0) i ∗ Φ ∗ ( · ) v ∗ (cid:1)(cid:13)(cid:13) L = k i ∗ Φ ∗ ( · ) v ∗ k M c k i ∗ Ψ ∗ ( · ) v ∗ k M = c h Γ Ψ Γ ∗ Ψ v ∗ , v ∗ i . Since Γ Ψ Γ ∗ Ψ and Q are positive, symmetric operators in L ( V ∗ , V ∗∗ ) and thefirst one is V -valued according to Theorem 5.5, it follows by an argumentbased on a result of the domination of Gaussian measures, see [30, Sec.1.1],that Q is also V -valued and a Gaussian covariance operator. As in the proofof the implication (a) ⇒ (b) in Theorem 5.5 we can conclude that Γ Φ is V -valued. (cid:3) If the mapping Ψ ∈ I is stochastically integrable, Proposition 3.2 impliesfor each t ∈ [0 , T ] that [0 ,t ] Ψ ∈ I and it satisfies (cid:13)(cid:13) [0 ,t ] i ∗ Ψ ∗ ( · ) v ∗ (cid:13)(cid:13) M c t k i ∗ Ψ ∗ ( · ) v ∗ k M for all v ∗ ∈ V ∗ , for a constant c t >
0. Corollary 5.6 enables us to conclude that [0 ,t ] Ψ isstochastically integrable, and thus we can define Z t Ψ( s ) d B ( s ) := Z T [0 ,t ] ( s )Ψ( s ) d B ( s ) for all t ∈ [0 , T ] . The integral process (cid:0) R t Ψ( s ) d B ( s ) : t ∈ [0 , T ] (cid:1) is continuous in p -th meanfor each p >
1. In order to see that let t n → t as n → ∞ for t n > t andlet Q ( n )Ψ denote the covariance operator of the Gaussian random variable R t n t Ψ( s ) d B ( s ). It follows for each v ∗ ∈ V ∗ that h Q ( n )Ψ v ∗ , v ∗ i = (cid:13)(cid:13) K ∗ ( [ t,t n ] ( · ) i ∗ Ψ ∗ ( · ) v ∗ ) (cid:13)(cid:13) L = (cid:13)(cid:13) [ t,t n ] ( · ) i ∗ Ψ ∗ ( · ) v ∗ (cid:13)(cid:13) M . Each f ∈ c M satisfies (cid:13)(cid:13) [ t,t n ] ( · ) f (cid:13)(cid:13) M → t n → t which follows from (3.9)in case H > and from results in [25, Ch.13.3] in case H < , see alsoProposition 3.2. Consequently, we obtain that h Q ( n )Ψ v ∗ , v ∗ i → t n → t and we can conclude as in the proof of Corollary 2.8 in [30] that the integralprocess is continuous in p -th mean.6. The Cauchy problem
In this section, we apply our previous results to consider stochastic evo-lution equations driven by cylindrical fractional Brownian motions of theform d Y ( t ) = AY ( t ) d t + C d B ( t ) , t ∈ (0 , T ] ,Y (0) = y . (6.1)Here B is a cylindrical fBm in a separable Banach space U , A is a generatorof a strongly continuous semigroup ( S ( t ) , t >
0) in a separable Banach space V and C is an operator in L ( U, V ). The initial condition y is an elementin V .The paths of a solution exhibit some kind of regularity, which is weakerthan P -a.s. Bochner integrable paths: Definition 6.1. A V -valued stochastic process ( X ( t ) : t ∈ [0 , T ]) is called weakly Bochner regular if for every sequence ( H n ) n ∈ N of continuous func-tions H n : [0 , T ] → V ∗ it satisfies: sup t ∈ [0 ,T ] k H n ( t ) k → ⇒ Z T |h X ( t ) , H n k ( t ) i| dt → P -a.s. for k → ∞ , for a subsequence ( H n k ) k ∈ N of ( H n ) n ∈ N . Definition 6.2.
A stochastic process ( Y ( t ) : t ∈ [0 , T ]) in V is called a weaksolution of (6.1) if it is weakly Bochner regular and for every v ∗ ∈ D ( A ∗ ) and t ∈ [0 , T ] we have P -a.s., h Y ( t ) , v ∗ i = h y , v ∗ i + Z t h Y ( s ) , A ∗ v ∗ i d s + B ( t )( C ∗ v ∗ ) . (6.2) YLINDRICAL FRACTIONAL BROWNIAN MOTION 21
From a proper integration theory we can expect that if the convolutedsemigroup S ( t − · ) C is integrable for all t ∈ [0 , T ] then a weak solutionof (6.1) exists and can be represented by the usual variation of constantsformula. Theorem 6.3.
Assume that S ( · ) C is in I . Then the following are equiva-lent: (a) the Cauchy problem (6.1) has a weak solution Y ; (b) the mapping S ( · ) C is stochastically integrable.In this situation the solution ( Y ( t ) : t ∈ [0 , T ]) can be represented by Y ( t ) = S ( t ) y + Z t S ( t − s ) C d B ( s ) for all t ∈ [0 , T ] . (6.3) Proof. (b) ⇒ (a): Proposition 3.2 guarantees for each t ∈ [0 , T ] that themapping [0 ,t ] S ( t − · ) C is in I and that there exists a constant c t > (cid:13)(cid:13) [0 ,t ] i ∗ C ∗ S ∗ ( t − · ) v ∗ (cid:13)(cid:13) M c t k i ∗ C ∗ S ∗ ( · ) v ∗ k M for all v ∗ ∈ V ∗ . Thus Corollary 5.6 guarantees that [0 ,t ] S ( t −· ) C is stochastically integrable,which enables to define the stochastic integral X ( t ) := Z t S ( t − s ) C d B ( s ) for all t ∈ [0 , T ] . It follows from representation (5.3) that the real valued stochastic process( h X ( t ) , v ∗ i : t ∈ [0 , T ]) is adapted for each v ∗ ∈ V ∗ . Pettis’ measurabilitytheorem implies that X := ( X ( t ) : t ∈ [0 , T ]) is adapted.By linearity we can assume that y = 0. The stochastic Fubini theoremfor real valued fBm implies for each v ∗ ∈ D ( A ∗ ) and t ∈ [0 , T ], that Z t h X ( s ) , A ∗ v ∗ i d s = ∞ X k =1 Z t Z s h S ( s − r ) Cie k , A ∗ v ∗ i d b k ( r ) d s = ∞ X k =1 Z t Z tr h S ( s − r ) Cie k , A ∗ v ∗ i d s d b k ( r )= ∞ X k =1 Z t h S ( t − r ) Cie k − Cie k , v ∗ i d b k ( r )= h Z t S ( t − r ) C d B ( r ) , v ∗ i − ∞ X k =1 h ie k , C ∗ v ∗ i b k ( t )= h X ( t ) , v ∗ i − B ( t )( C ∗ v ∗ ) , which shows that the process X satisfies (6.2). In order to show that X isweakly Bochner regular define Ψ t := [0 ,t ] ( · ) S ( t − · ) C for each t ∈ [0 , T ].Note that Proposition 3.2 guarantees that there exists a constant c t > (cid:13)(cid:13) Γ ∗ Ψ t v ∗ (cid:13)(cid:13) L = (cid:13)(cid:13) [0 ,t ] ( · ) i ∗ C ∗ S ∗ ( t − · ) v ∗ (cid:13)(cid:13) M c t k i ∗ C ∗ S ∗ ( · ) v ∗ k M for every v ∗ ∈ V ∗ . Since the derivation of the constant c t in [25, Ch.13.3]shows that sup t ∈ [0 ,T ] c t < ∞ , the uniform boundedness principle implies that sup t ∈ [0 ,T ] (cid:13)(cid:13) Γ ∗ Ψ t (cid:13)(cid:13) V ∗ → L < ∞ . Thus for a sequence ( H n ) n ∈ N of continuousmappings H n : [0 , T ] → V ∗ we obtain E "(cid:12)(cid:12)(cid:12)(cid:12)Z T h X ( t ) , H n ( t ) i d t (cid:12)(cid:12)(cid:12)(cid:12) T Z T E h | Z Ψ t H n ( t ) | i d t = T Z T (cid:13)(cid:13) Γ ∗ Ψ t H n ( t ) (cid:13)(cid:13) d t T sup t ∈ [0 ,T ] (cid:13)(cid:13) Γ ∗ Ψ t (cid:13)(cid:13) V ∗ → L sup t ∈ [0 ,T ] k H n ( t ) k , which shows the weak Bochner regularity.(a) ⇒ (b): by applying Itˆo’s formula for real valued fBm, see e.g. [5, Thm6.3.1], one deduces for every continuously differentiable function f : [0 , T ] → R and real valued fBm b Z T f ′ ( s ) b ( s ) d s = f ( T ) b ( T ) − Z T f ( s ) d b ( s ) P -a.s., (6.4)where the integral on the right-hand side can be understood as a Wienerintegral, since f is deterministic. Let Y be a weak solution of (6.1) anddenote by A ⊙ the part of A ∗ in D ( A ∗ ). Then D ( A ⊙ ) is a weak ∗ -sequentiallydense subspace of V ∗ . From the integration by parts formula (6.4) it followsas in the proof of Theorem 7.1 in [30] that h Y ( T ) , v ∗ i = Z Ψ v ∗ for all v ∗ ∈ D ( A ⊙ ) , (6.5)where Z Ψ denotes the cylindrical integral of Ψ := S ( T − · ) C . It remainsto show that (6.5) holds for all v ∗ ∈ V ∗ , for which we mainly follow thearguments of the proof of Theorem 2.3 in [30]. Observe that the randomvariable Y ( T ) is Gaussian since the right hand side in (6.5) is Gaussian foreach v ∗ ∈ D ( A ⊙ ) and Gaussian distributions are closed under weak limits.Let R : V ∗ → V and Q : V ∗ → V ∗∗ denote the covariance operators of Y ( T )and Z Ψ , respectively. Since R is the covariance operator of a Gaussianmeasure there exists a Hilbert space H which is continuously embedded bya γ -radonifying mapping j : H → V such that R = jj ∗ . Equality (6.5)implies h Rv ∗ , v ∗ i = h Qv ∗ , v ∗ i for all v ∗ ∈ D ( A ⊙ ) . (6.6)Let ( v ∗ n ) n ∈ N be a sequence in D ( A ⊙ ) converging weakly ∗ to v ∗ in V ∗ . Thuslim n →∞ j ∗ v ∗ n = j ∗ v ∗ weakly in H since H is a Hilbert space and j ∗ is weak ∗ continuous. As a consequence of the Hahn-Banach theorem one can con-struct a convex combination w ∗ n of the v ∗ n such that lim n →∞ j ∗ w ∗ n = j ∗ v ∗ strongly in H and lim n →∞ w ∗ n = v ∗ weakly ∗ in V ∗ . Since w ∗ m − w ∗ n is in D ( A ⊙ ) for all m, n ∈ N , inequality (6.6) implies k i ∗ C ∗ S ∗ ( T − · )( w ∗ m − w ∗ n ) k M = k K ∗ ( i ∗ C ∗ S ∗ ( T − · )( w ∗ m − w ∗ n )) k L = h Q ( w ∗ m − w ∗ n ) , w ∗ m − w ∗ n i = h R ( w ∗ m − w ∗ n ) , w ∗ m − w ∗ n i = k j ∗ ( w ∗ m − w ∗ n ) k H → m, n → ∞ . YLINDRICAL FRACTIONAL BROWNIAN MOTION 23
Thus, (cid:0) i ∗ C ∗ S ∗ ( T − · ) w ∗ n (cid:1) n ∈ N is a Cauchy sequence in M and therefore itconverges to some g ∈ M . By the same arguments as in the proof of Proposi-tion 5.1 it follows that there is a subsequence such that lim k →∞ i ∗ C ∗ S ∗ ( T − s ) w ∗ n k = g ( s ) for Lebesgue almost all s ∈ [0 , T ]. On the other hand, theweak ∗ convergence of ( w ∗ n k ) k ∈ N implies that lim k →∞ h i ∗ C ∗ S ∗ ( T − s ) w ∗ n k , x i = h i ∗ C ∗ S ∗ ( T − s ) v ∗ , x i for all x ∈ X and s ∈ [0 , T ], which yields g = i ∗ C ∗ S ∗ ( T −· ) v ∗ . It follows that h Rw ∗ n k , w ∗ n k i = (cid:13)(cid:13) i ∗ C ∗ S ∗ ( t − · ) w ∗ n k (cid:13)(cid:13) M → k i ∗ C ∗ S ∗ ( t − · ) v ∗ k M = h Qv ∗ , v ∗ i , as k → ∞ . Therefore the covariance operators R and Q coincide on V ∗ ,which yields that the cylindrical distribution of Z Ψ extends to a Radonmeasure. (cid:3) If H > then the first condition in Theorem 6.3, i.e. S ( · ) C ∈ I , issatisfied for every strongly continuous semigroup as L ([0 , T ]; X ) ⊆ c M . If H < this condition is not obvious but an important case is covered by thefollowing result. Proposition 6.4.
Let
H < . If ( S ( t ) , t > is an analytic semigroup ofnegative type, then the mapping S ( · ) C is in I .Proof. Define for arbitrary f ∈ L ([0 , T ]; X ) the function G f ( s ) (6.7):= b H f ( s )( T − s ) − H + (cid:18) − H (cid:19) s − H Z Ts s H − f ( s ) − t H − f ( t )( t − s ) − H d t ! for all s ∈ (0 , T ]. It follows from (3.8) that a function f ∈ L ([0 , T ]; X ) isin M if and only G f ∈ L ([0 , T ]; X ), in which case k f k M = k G f k L . Bythe same computations as in the proof of [21, Le.11.7] one derives that thereexists a constant c > Z T k G f ( s ) k d s c Z T k f ( s ) k ( T − s ) − H d s + Z T k f ( s ) k s − H d s + Z T Z Ts k f ( t ) − f ( s ) k ( t − s ) − H d t ! d s ! . (6.8)We check that each term on the right hand side of (6.8) is finite for f = i ∗ C ∗ S ∗ ( · ) v ∗ , v ∗ ∈ V ∗ . The growth bound of the semigroup guarantees thatthere exist some constants β, c > k S ( s ) k c e − βs for all s ∈ [0 , T ] . It is immediate that the first two integrals on the right hand side in (6.8)are finite since 1 − H <
1. In order to estimate the last term, recall that,as S is analytic, there exists for each α > c > < s t we have k S ( t ) − S ( s ) k = k ( S ( t − s ) − Id) S ( s ) k c ( t − s ) α s − α e − βs . Fix some α ∈ ( − H, ). The third summand on the right hand side of(6.8) can be estimated by Z T Z Ts k i ∗ C ∗ S ∗ ( t ) v ∗ − i ∗ C ∗ S ∗ ( s ) v ∗ k ( t − s ) − H d t ! d s ( c k i k k C k k v ∗ k ) Z T Z Ts e − βs s α ( t − s ) α ( t − s ) − H d t ! d s ( c k i k k C k k v ∗ k ) (cid:18)Z T e − βs s α d s (cid:19) (cid:18)Z T t − H − α d t (cid:19) ( c k i k k C k k v ∗ k ) (2 β ) α − Γ(1 − α ) T H + α − ) , which completes the proof. (cid:3) Another example of a semigroup satisfying S ( · ) C in I is considered in Sec-tion 7. Further examples can be derived using the known fact that the spaceof H¨older continuous functions of index larger than − H is continuouslyembedded in M . Example 6.5. If V is a Hilbert space then a function Ψ ∈ I is stochasti-cally integrable if and only if Γ Ψ is Hilbert-Schmidt, according to Theorem5.5. Thus, if ( f k ) k ∈ N denotes an orthonormal basis of V , the function Ψ isstochastically integrable if and only if the adjoint operator Γ ∗ Ψ is Hilbert-Schmidt, that is ∞ X k =1 k Γ ∗ Ψ f k k L = ∞ X k =1 k K ∗ ( i ∗ Ψ ∗ ( · ) f k ) k L = ∞ X k =1 k i ∗ Ψ ∗ ( · ) f k k M < ∞ . In the case
H > we obtain that there exists a weak solution of (6.1) if ∞ X k =1 Z T Z T k i ∗ C ∗ S ∗ ( s ) f k k k i ∗ C ∗ S ∗ ( t ) f k k | s − t | H − d s d t < ∞ . For the case
H < assume that the semigroup ( S ( t ) , t >
0) is analytic andof negative type. A similar calculation as in the proof of Proposition 6.4shows that if there exists a constant α ∈ ( − H, ) such Z T k S ( s ) Ci k HS s α d s < ∞ , then S ( · ) C is stochastically integrable. Here k·k HS denotes the Hilbert-Schmidt norm of an operator U : L ([0 , T ]; X ) → V .7. Example: the stochastic heat equation
As an example we consider a self-adjoint generator A of a semigroup( S ( t ) , t >
0) in a separable Hilbert space V such that there exists an or-thonormal basis ( e k ) k ∈ N of V satisfying Ae k = − λ k e k for some λ k > k ∈ N and λ k → ∞ as k → ∞ . Thus the semigroup satisfies S ( t ) e k = e − λ k t e k for all t > k ∈ N . YLINDRICAL FRACTIONAL BROWNIAN MOTION 25
A specific instance is the Laplace operator with Dirichlet boundary condi-tions on L ( D ; R ) for a set D ∈ B ( R n ). We assume that C = Id and weidentify the dual space V ∗ with V , i.e. we consider the Cauchy problemd Y ( t ) = AY ( t ) d t + d B ( t ) for all t ∈ [0 , T ] . (7.1)The system (7.1) is perturbed by a cylindrical fBm B in V which is inde-pendent along the orthonormal basis ( e k ) k ∈ N of eigenvectors e k of A , thatis we consider the cylindrical fBm ( B ( t ) : t >
0) in V from Example 4.4: B ( t ) v = ∞ X k =1 h ie k , v i b k ( t ) for all v ∈ V, t > , where ( b k ) k ∈ N is a sequence of independent, real valued standard fBms ofHurst parameter H ∈ (0 ,
1) and the embedding i : V → V is defined by iv = ∞ X k =1 q k h e k , v i e k for a sequence ( q k ) k ∈ N ⊆ R satisfying sup k | q k | < ∞ . Note that in this case X = V . Theorem 7.1.
Let A be a self-adjoint generator satisfying the conditionsdescribed above. If ∞ X k =1 q k λ Hk < ∞ , then equation (7.1) has a weak solution ( Y ( t ) : t ∈ [0 , T ]) in V . The solutioncan be represented by the variation of constants formula (6.3) .Proof. Note that in this situation we have i ∗ S ∗ ( t ) e k = q k e − λ k t e k for each k ∈ N and t ∈ [0 , T ]. (7.2)According to Theorem 5.5 and Theorem 6.3 we have to establish that S ( · )is in I and the operator Γ : L ([0 , T ]; V ) → V defined by h Γ f, v i = Z T (cid:2) K ∗ (cid:0) i ∗ S ∗ ( · ) v (cid:1) ( s ) , f ( s ) (cid:3) d s for all f ∈ L ([0 , T ]; V ) , v ∈ V is γ -radonifying. Since V is a separable Hilbert space, the operator Γ is γ -radonifying if and only if it is Hilbert-Schmidt.If H > then S ( · ) is in I and equality (7.2) yields for each k ∈ N k i ∗ S ∗ ( · ) e k k c M = H (2 H − Z T Z T k i ∗ S ∗ ( t ) e k k k i ∗ S ∗ ( s ) e k k | s − t | H − d s d t = H (2 H − q k Z T Z T e − λ k t e − λ k s | s − t | H − d s d t. (7.3) The iterated integral can be estimated by Z T e − λ k t Z T e − λ k s | s − t | H − d s d t = 2 Z T e − λ k t Z t e − λ k s | s − t | H − d s d t = 2 Z T e − λ k t Z t e λ k s s H − d s d t = 2 Z T e λ k s s H − Z Ts e − λ k t d t d s λ k Z T e − λ k s s H − d s λ Hk Γ(2 H − . (7.4)Since inequality (3.11) guarantees that there exists a constant c > ∞ X k =1 k Γ ∗ e k k L = ∞ X k =1 (cid:13)(cid:13) K ∗ (cid:0) i ∗ S ∗ ( · ) e k (cid:1)(cid:13)(cid:13) L c ∞ X k =1 k i ∗ S ∗ ( · ) e k k c M , we can conclude from (7.3) and (7.4) that Γ ∗ and thus Γ are Hilbert-Schmidtoperators.If H < Proposition 6.4 guarantees that S ( · ) is in I . As in the proof ofProposition 6.4 it follows that there exists a constant c > k ∈ N k K ∗ ( i ∗ S ∗ ( · ) e k ) k L c Z T k i ∗ S ∗ ( s ) e k k ( T − s ) − H d s + Z T k i ∗ S ∗ ( s ) e k k s − H d s + Z T Z Ts k i ∗ S ∗ ( t ) e k − i ∗ S ∗ ( s ) e k k ( t − s ) − H d t ! d s ! . (7.5)Equality (7.2) implies for the first integral the estimate Z T k i ∗ S ∗ ( s ) e k k ( T − s ) − H d s q k Z T e − λ k s ( T − s ) − H d s = q k (2 λ k ) H Z λ k T e − s (2 λ k T − s ) − H d s q k (2 λ k ) H (cid:18) H (cid:19) . (7.6)Here, the estimate of the integral follows from the fact that if 2 λ k T Z λ k T e − s (2 λ k T − s ) − H d s Z λ k T λ k T − s ) − H d s H , and if 2 λ k T > Z λ k T e − s (2 λ k T − s ) − H d s Z λ k T − e − s d s + Z λ k T λ k T − (2 λ k T − s ) H − d s H .
YLINDRICAL FRACTIONAL BROWNIAN MOTION 27
The second integral in (7.5) can be bounded by Z T k i ∗ S ∗ ( s ) e k k s − H d s q k Z T e − λ k s s − H d s Γ(2 H ) q k (2 λ k ) H . (7.7)Another application of equality (7.2) yields for the third term in (7.5) Z T Z Ts k i ∗ S ∗ ( t ) e k − i ∗ S ∗ ( s ) e k k ( t − s ) − H d t ! d s = q k Z T Z Ts (cid:12)(cid:12) e − λ k t − e − λ k s (cid:12)(cid:12) ( t − s ) − H d t ! d s = q k Z T e − λ k s (cid:18)Z T − s − e − λ k t t − H d t (cid:19) d s. (7.8)Applying the changes of variables λ k s = x and λ k t = y yields Z T e − λ k s (cid:18)Z T − s − e − λ k t t − H d t (cid:19) d s = 1 λ Hk Z λ k T e − x Z λ k T − x − e − y y − H d y ! d x = 1 λ Hk Z λ k T e − λ k T − x ) Z x − e − y y − H d y ! d x λ Hk c , (7.9)where c > H but not on λ k . Thefiniteness of the constant c and its independence of λ k follow from thefollowing three estimates: Z e − λ k T − x ) Z x − e − y y − H d y ! d x Z Z − e − y y − H d y ! d x, Z λ k T e − λ k T − x ) Z − e − y y − H d y ! d x H + ) Z λ k T e − λ k T − x ) d x, Z λ k T e − λ k T − x ) Z x − e − y y − H d y ! d x H − ) Z λ k T e − λ k T − x ) x H − d x H − ) Z λ k T e − λ k T − x ) d x. By applying the estimates (7.6)–(7.9) to (7.5), it follows that there exists aconstant c > k Γ ∗ e k k L = k K ∗ ( i ∗ S ∗ ( · ) e k ) k L c q k λ Hk for all k ∈ N . As before we can conclude that Γ is Hilbert-Schmidt. (cid:3)
Consider now the special case of the heat equation with Dirichlet bound-ary conditions driven by a cylindrical fractional noise with independent com-ponents, that is with Q = Id. In this case q k ≡ λ k ∼ k /n so that the condition for the existence of aweak solution becomes the well known n/ < H <
1. This result is in linewith the literature, see for example [7, 12, 16].
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Department of Mathematics, King’s College, Lon-don WC2R 2LS, United Kingdom
E-mail address , E. Issoglio: [email protected]
E-mail address , M. Riedle:, M. Riedle: