L´evy Processes and Infinitely Divisible Measures in the Dual of a Nuclear Space. C. A. Fonseca-Mora Escuela de Matem´atica, Universidad de Costa Rica, San Jos´e, 11501-2060, Costa Rica. 7 E-mail: [email protected] 1 0 2 n Abstract a J LetΦbeanuclearspaceandletΦ′ denoteitsstrongdual. Inthisworkweestablishtheone- 3 β to-one correspondence between infinitely divisible measures on Φ′ and L´evy processes taking 2 β ] vaanlduetsheinexΦis′βt.enMceoorefocva`edrl,a`gweveprrsoiovnestfhoer ΦL´e′v-vya-IlutˆoeddeLc´eovmypporsoictieosnse,st.hAe Lch´eavrya-cKtehriinztacthioinnefoforrLm´euvlya R β measuresonΦ′ isalsoestablished. Finally,weprovethe L´evy-Khintchineformulaforinfinitely β P divisible measures on Φ′ . . β h 2010 Mathematics Subject Classification: 60B11, 60G51, 60E07,60G20. t a Key words and phrases: L´evy processes, infinitely divisible measures, cylindrical L´evy pro- m cesses,dualofanuclearspace,L´evy-Itˆodecomposition,L´evy-Khintchineformula,L´evymeasure. [ 1 1 Introduction v 0 This work is concerned with the study of L´evy processes and infinitely divisible measures on 3 the dual of a nuclear space. 6 A L´evy process is essentially a stochastic processes with independent and stationary incre- 6 ments. In the case of the dual of a nuclear space, the study of some specific classes of L´evy 0 . processes, in particular of Wiener processes and stochastic analysis defined with respect to 1 these processes received considerable attention during the decades of 1980s and 1990s (see e.g. 0 7 [5,15,18]). However,totheextentofourknowledgetheonlypreviousworkonthestudyofthe 1 properties of general additive (and hence L´evy)processes in the dual of some classes of nuclear : spaces was carried out by U¨stu¨nel in [33]. Also, cone-additive processes in the dual of some v i particular Fr´echet nuclear spaces were studied in [22]. X On the other hand, an infinitely divisible measure is a probability measure which has a r convolution nth root for every natural n. Properties of infinitely divisible measures defined on a locally convex spaces were explored by several authors during the decades of 1960s and 1970s (see e.g. [8, 9, 10, 31]). Nevertheles, the author of this article is not aware of any work that studies the correspondencebetweenL´evyprocessesandinfinitely divisible measuresinthe dual of a general nuclear space. It is for the above reasons that the aim of this paper is to gain some deeper understanding on the properties of L´evy processes that takes values in the strong dual Φ′ of a generalnuclear β space Φ, and their relationship with the infinitely divisible measures defined on Φ′. Our main β motivation is to begin with a systematic study of L´evyprocesses on the dual of a nuclear space which could lead to the introduction of stochastic integrals and SPDEs driven by L´evy noise in Φ′. Some work into this direction was carried out by the author in [11]. β WestartinSection2withsomepreliminaryresultsonnuclearspaces,cylindricalandstochas- ticprocessesandRadonmeasuresonthedualofanuclearspace. Then,inSection3.1weutilise 1 someresultsofSiebert[29,30]tostudytheproblemofembeddingagiveninfinitedivisiblemea- sureµintoacontinuousconvolutionsemigroupofofprobabilitymeasuresΦ′ . Later,inSection β 3.2 by using recent results in [12] that provides conditions for a cylindrical process to have a ca`dla`g version(known as regularizationtheorems), we provide conditions for the existence of a ca`dla`g L´evy version to a given cylindrical L´evy process in Φ′ or to a Φ′ -valued L´evy process. β In particular we show that if the space Φ is nuclear and barrelled, then every L´evy process in Φ′ has a ca`dla`g version that is also a L´evy process. β In Section 3.3 we proceed to prove the one-to-one correspondence between L´evy processes and infinitely divisible measures on Φ′. Here it is important to remark that the standard β argument to prove the correspondence that works on finite dimensions (see e.g. Chapter 2 in [26])doesnotworkinourcontextastheKolmogorovextensiontheoremisnotapplicableonthe dual of a general nuclear space. To overcome this situation we use a projective system version of the Kolmogorovextension theorem (see [24], Theorem1.3.4) to show a generaltheorem that guarantee the existence of a cylindrical L´evy process L whose cylindrical distributions extends foreachtime t to the measureµ ofthe continuousconvolutionsemigroup{µ } in whichthe t t t≥0 given infinitely divisible measure µ can be embedded. Then, for this cylindrical process L we usetheresultsinSection3.2toshowtheexistenceofaΦ′ -valuedca`dla`gL´evyprocessL˜ thatis β a version of L, and hence the probability distribution of L˜ coincides with µ and then we have 1 the correspondence. In Section 3.4 we review some properties of Wiener processes in Φ′. β After study in Sections 4.1 and 4.2 the basic properties of Poisson integrals defined by Poisson random measures on Φ′, in Sections 4.3 and 4.4 we investigate the properties of the β L´evymeasuresonΦ′ . Inparticular,wewillshowthatL´evymeasuresonΦ′ arecharacterizedby β β ansquareintegrabilitypropertyexpressedintermsofthenormρ′ofaHilbertspacecontinuously embedded in the dual space Φ′ . Moreover, our characterization generalizes, in the context of β the dual of a nuclear space, the characterization for the L´evy measure of an infinitely divisible measures obtained by Dettweiler in [8] for the case of complete Badrikian spaces. Later,weproceedtoproveinSection4.5theso-calledL´evy-Itˆodecompositionforthepathsof aΦ′ -valuedL´evyprocess. Morespecifically,weshowthataΦ′ -valuedL´evyprocessL={L } β β t t≥0 has a decomposition of the form (see Theorem 4.23): L =tm+W + fN(t,df)+ fN(t,df), ∀t≥0, t t ZBρ′(1) ZBρ′(1)c wherem∈Φ′ ,ρ′isthenormassociatedtoethesquareintegrabilitypropertyoftheL´evymeasure β ν of L and Bρ′(1) is the unit ball of ρ′, {Wt}t≥0 is a Wiener process taking values in a Hilbert space continuously embedded in the dual space Φ′ , fN(t,df):t≥0 is a mean-zero, β Bρ′(1) square integrable, ca`dla`g L´evy process taking valuesninRa Hilbert space continouously embedded in the dual space Φ′ (small jumps part) and fN(t,dfe):t≥0 is a Φ′ -valued ca`dla`g β Bρ′(1)c β L´evyprocessdefinedbymeansofaPoissonintnegRralwithrespecttothePooissonrandommeasure N of the L´evy process L (large jumps part). Our L´evy-Itˆo decomposition improves the decomposition proved by U¨stu¨nel in [33] in two directions. First, for our decomposition we only assume that the space Φ is nuclear and we do notassumeanypropertyonthedualspaceW,thisincontrasttothedecompositionin[33]where Φisassumedtobeseparable,completeandnuclear,andΦ′ isassumedtobeSuslinandnuclear. β Second,wehaveobtainedamuchsimpleranddetailedcharacterizationofthecomponentsofthe decomposition than in [33]. In particular, contrary to the decomposition in [33] we have been abletoshowtheindependence ofalltherandomcomponentsinourdecomposition. Thismakes our decomposition more suitable to for example introduce stochastic integrals with respect to L´evyprocesses. As a consequenceofourproofofthe L´evy-Itˆodecomposition,we provea L´evy- Khintchine formula for the characteristic function of a Φ′ -valued L´evy process (see Theorem β 4.24). Finally, byusingthe one-to-onecorrespondencebetweenL´evyprocessesandinfinitely divis- iblemeasures,inSection5weprovetheL´evy-Khintchineformulaforthecharacteristicfunction of an infinitely divisible measure on Φ′ (see Theorem5.1). More specifically, we provethat the β 2 characteristic function µ of an infinitely divisible measure µ on Φ′ is of the form: β b 1 µ(φ)=exp im[φ]− Q(φ)2+ eif[φ]−1−if[φ] (f) ν(df) , ∀φ∈Φ, " 2 ZΦ′β(cid:16) 1Bρ′(1) (cid:17) # wherebm ∈ Φ′ , Q is a continuous Hilbertian semi-norm on Φ, and ν is a L´evy measure on Φ′ β β withcorrespondingHilbertiannormρ′. HereitisimportanttoremarkthatourL´evy-Khintchine formula works in a case that is not covered by the formula proved by Dettweiler in [8] because our dual space is not assumed to be a complete Badrikian space as in [8]. 2 Preliminaries 2.1 Nuclear Spaces And Its Strong Dual Inthissectionweintroduceournotationandreviewsomeofthekeyconceptsonnuclearspaces and its dual space that we will need throughout this paper. For more information see [27, 32]. Let Φ be a locally convex space (over R or C). If each bounded and closed subset of Φ is complete, then Φ is said to be quasi-complete. On the other hand, the space Φ called a barrelled space if every convex, balanced, absorbing and closed subset of Φ (i.e. a barrel) is a neighborhood of zero. If p is a continuous semi-norm on Φ and r > 0, the closed ball of radius r of p given by B (r) = {φ∈Φ:p(φ)≤r} is a closed, convex, balanced neighborhood of zero in Φ. A p continuous semi-norm (respectively a norm) p on Φ is called Hilbertian if p(φ)2 = Q(φ,φ), for all φ ∈ Φ, where Q is a symmetric, non-negative bilinear form (respectively inner product) on Φ×Φ. LetΦ be the Hilbert space thatcorrespondsto the completionofthe pre-Hilbertspace p (Φ/ker(p),p˜), where p˜(φ+ker(p))=p(φ) for each φ∈Φ. The quotient map Φ→Φ/ker(p) has an unique continuous linear extension i :Φ→Φ . p p LetqbeanothercontinuousHilbertiansemi-normonΦforwhichp≤q. Inthiscase,ker(q)⊆ ker(p). Moreover,the inclusion map from Φ/ker(q) into Φ/ker(p) is linear and continuous, and therefore it has a unique continuous extension i : Φ → Φ . Furthermore, we have the p,q q p following relation: i =i ◦i . p p,q q We denote by Φ′ the topological dual of Φ and by f[φ] the canonical pairing of elements f ∈Φ′, φ∈Φ. We denote byΦ′ the dualspaceΦ′ equipped withits strong topology β, i.e. β is β thetopologyonΦ′ generatedbythefamilyofsemi-norms{η },whereforeachB ⊆Φ′ bounded B we have η (f) = sup{|f[φ]| : φ ∈ B} for all f ∈ Φ′. If p is a continuous Hilbertian semi-norm B on Φ, then we denote by Φ′ the Hilbert space dual to Φ . The dual norm p′ on Φ′ is given by p p p p′(f) = sup{|f[φ]| : φ ∈ B (1)} for all f ∈ Φ′. Moreover, the dual operator i′ corresponds to p p p the canonical inclusion from Φ′ into Φ′ and it is linear and continuous. p β Let p and q be continuous Hilbertian semi-norms on Φ such that p ≤ q. The space of continuouslinearoperators(respectivelyHilbert-Schmidtoperators)fromΦ intoΦ isdenoted q p by L(Φ ,Φ ) (respectively L (Φ ,Φ )) and the operator norm (respectively Hilbert-Schmidt q p 2 q p norm) is denote by ||·|| (respectively ||·|| ). We employ an analogous notation L(Φq,Φp) L2(Φq,Φp) for operators between the dual spaces Φ′ and Φ′. p q Among the many equivalent definitions of a nuclear space (see [23, 32]), the following is the most useful for our purposes. Definition 2.1. A(Hausdorff)locallyconvexspace(Φ,T)iscallednuclear ifits topologyT is generated by a family Π of Hilbertian semi-norms such that for each p∈ Π there exists q ∈Π, satisfying p≤q and the canonical inclusion i :Φ →Φ is Hilbert-Schmidt. p,q q p Someexamplesofnuclearspacesarethefollowing(see[32],Chapter51and[23],Chapter6): S(Rd), S′(Rd), C∞(K) (K: compact subset of Rd); C∞(X), E(X) := C∞(X), E′(X), D(X), c c D′(X) (X: open subset of Rd). Let Φ be a nuclear space. If p is a continuous Hilbertian semi-norm on Φ, then the Hilbert space Φ is separable (see [23], Proposition 4.4.9 and Theorem 4.4.10, p.82). Now, let {p } p n n∈N 3 be an increasing sequence of continuous Hilbertian semi-norms on (Φ,T). We denote by θ the locally convex topology on Φ generated by the family {p } . The topology θ is weaker than n n∈N T. We will call θ a weaker countably Hilbertian topology on Φ and we denote by Φ the space θ (Φ,θ). ThespaceΦ isaseparablepseudo-metrizable(notnecessarilyHausdorff)locallyconvex θ space and its dual space satisfies Φ′ = Φ′ (see [12], Proposition 2.4). We denote the θ n∈N pn completion of Φ by Φ and its strong dual by (Φ )′ . θ θ S θ β f f 2.2 Cylindrical and Stochastic Processes Unless otherwise specified, in this section Φ will always denote a nuclear space over R. Let (Ω,F,P) be a complete probability space. We denote by L0(Ω,F,P) the space of equivalence classes of real-valued random variables defined on (Ω,F,P). We always consider thespaceL0(Ω,F,P)equippedwiththetopologyofconvergenceinprobabilityandinthiscase it is a complete, metrizable, topological vector space. For two Borel measures µ and ν on Φ′, we denote by µ∗ν their convolution. Recall that β µ∗ν(A)= (x+y)µ(dx)ν(dy), for any A∈B(Φ′). Denote ν∗n =ν∗···∗ν (n-times) Φ′×Φ′1A β and we use the convention ν0 =δ , where δ denotes the Dirac measure on Φ′ for f ∈Φ′. R 0 f β A Borelmeasure µ on Φ′ is calleda Radon measure if for every Γ∈B(Φ′) and ǫ>0, there β β exist a compact set K ⊆Γ such that µ(Γ\K )<ǫ. In generalnot every Borelmeasure on Φ is ǫ ǫ Radon. We denote by Mb (Φ′ )andby M1(Φ′ ) the spaces ofallboundedRadonmeasuresand R β R β of all Radon probability measures on Φ′ . A subset M ⊆ Mb(Φ′ ) is called uniformly tight if β R β (i) sup{µ(Φ′ ) : µ∈M}<∞, and (ii) for every ǫ>0 there exist a compact K ⊆Φ′ such that β β µ(Kc) < ǫ for all µ ∈ M. Also, a subset M ⊆ Mb (Φ′ ) is called shift tight if for every µ ∈ M R β there exists f ∈Φ′ such that {µ∗δ :µ∈M} is uniformly tight. µ β fµ For any n∈N and any φ ,...,φ ∈Φ, we define a linear map π :Φ′ →Rn by 1 n φ1,...,φn π (f)=(f[φ ],...,f[φ ]), ∀f ∈Φ′. (2.1) φ1,...,φn 1 n The map π is clearly linear and continuous. Let M be a subset of Φ. A subset of Φ′ of φ1,...,φn the form Z(φ ,...,φ ;A)={f ∈Φ′ : (f[φ ],...,f[φ ])∈A}=π−1 (A) (2.2) 1 n 1 n φ1,...,φn where n ∈ N, φ ,...,φ ∈ M and A ∈ B(Rn) is called a cylindrical set based on M. The 1 n set of all the cylindrical sets based on M is denoted by Z(Φ′,M). It is an algebra but if M is a finite set then it is a σ-algebra. The σ-algebra generated by Z(Φ′,M) is denoted by C(Φ′,M) and it is called the cylindrical σ-algebra with respect to (Φ′,M). If M =Φ, we write Z(Φ′) = Z(Φ′,Φ) and C(Φ′) = C(Φ′,Φ). One can easily see from (2.2) that Z(Φ′ ) ⊆ B(Φ′). β β Therefore, C(Φ′ )⊆B(Φ′). β β A function µ:Z(Φ′)→[0,∞] is called a cylindrical measure on Φ′, if for each finite subset M ⊆Φ′ therestrictionofµtoC(Φ′,M)isameasure. Acylindricalmeasureµissaidtobefinite if µ(Φ′) < ∞ and a cylindrical probability measure if µ(Φ′) = 1. The complex-valued function µ:Φ→C defined by ∞ b µ(φ)= eif[φ]µ(df)= eizµ (dz), ∀φ∈Φ, φ ZΦ′ Z−∞ where for each φ ∈ Φ,bµ := µ◦π−1, is called the characteristic function of µ. In general, a φ φ cylindrical measure on Φ′ does not extend to a Borel measure on Φ′ . However, necessary and β sufficientconditionsforthiscanbe givenintermsofthe continuityofitscharacteristicfunction by means of the Minlos theorem (see [7], Theorem III.1.3, p.88). A cylindrical random variable in Φ′ is a linear map X : Φ → L0(Ω,F,P). If Z = Z(φ ,...,φ ;A) is a cylindrical set, for φ ,...,φ ∈Φ and A∈B(Rn), let 1 n 1 n µ (Z):=P((X(φ ),...,X(φ ))∈A)=P◦X−1◦π−1 (A). X 1 n φ1,...,φn 4 Themapµ isacylindricalprobabilitymeasureonΦ′ anditiscalledthecylindrical distribution X ofX. Conversely,toeverycylindricalprobabilitymeasureµonΦ′thereisacanonicalcylindrical random variable for which µ is its cylindrical distribution (see [28], p.256-8). If X is a cylindrical random variable in Φ′, the characteristic function of X is defined to be the characteristic function µ : Φ → C of its cylindrical distribution µ . Therefore, X X µ (φ) = EeiX(φ), ∀φ ∈ Φ. Also, we say that X is n-integrable if E(|X(φ)|n) < ∞, ∀φ ∈ Φ, X and has zero mean if E(X(φ))=0, ∀φ∈Φ. b Let X be a Φ′ -valued random variable, i.e. X : Ω → Φ′ is a F/B(Φ′)-measurable map. b β β β We denote by µ the distribution of X, i.e. µ (Γ)=P(X ∈Γ), ∀Γ∈B(Φ′), and it is a Borel X X β probability measure on Φ′ . For each φ∈Φ we denote by X[φ] the real-valuedrandomvariable β definedbyX[φ](ω):=X(ω)[φ],forallω ∈Ω. Then,themappingφ7→X[φ]definesacylindrical random variable. Therefore, the above concepts of characteristic function and integrability can be analogously defined for Φ′ -valued random variables in terms of the cylindrical random β variable they determines. If X is a cylindrical random variable in Φ′, a Φ′ -valued random variable Y is a called a β version of X if for every φ∈Φ, X(φ)= Y[φ] P-a.e. The following results establish alternative characterizationsfor regular random variables. AΦ′ -valuedrandomvariableX iscalledregular ifthereexistsaweakercountablyHilbertian β topology θ on Φ such that P(ω :X(ω)∈Φ′)=1. θ Theorem 2.2 ([12], Theorem 2.9). Let X be a Φ′-valued random variable. Consider the β statements: (1) X is regular. (2) The map X :Φ→L0(Ω,F,P), φ7→X[φ] is continuous. (3) The distribution µ of X is a Radon probability measure. X Then, (1)⇔(2) and (2)⇒(3). Moreover, if Φ is barrelled, we have (3)⇒(1). LetJ =[0,∞)orJ =[0,T]forsomeT >0. WesaythatX ={X } isacylindricalprocess t t∈J in Φ′ if X is a cylindrical random variable, for each t ∈ J. Clearly, any Φ′ -valued stochastic t β processesX ={X } defines acylindricalprocessunderthe prescription: X[φ]={X [φ]} , t t∈J t t∈J for each φ∈Φ. We will say that it is the cylindrical process determined by X. A Φ′-valued processes Y = {Y } is said to be a Φ′ -valued version of the cylindrical β t t∈J β process X ={X } on Φ′ if for each t∈J, Y is a Φ′ -valued version of X . t t∈J t β t LetX ={X } beaΦ′ -valuedprocess. WesaythatX iscontinuous (respectivelyc`adla`g) t t∈J β if for P-a.e. ω ∈ Ω, the sample paths t 7→ X (w) ∈ Φ′ of X are continuous (respectively right- t β continuous with left limits). On the other hand, we say that X is regular if for every t∈J, X t is a regular random variable. The following two results contains some useful properties of Φ′ - β valued regular processes. For proofs see Chapter 1 in [11]. Proposition2.3. LetX ={X } andY ={Y } beΦ′ -valuedregular stochasticprocesses t t∈J t t∈J β suchthatforeachφ∈Φ,X[φ]={X [φ]} isaversionofY ={Y [φ]} . ThenX isaversion t t∈J t t∈J of Y. Furthermore, if X and Y are right-continuous then they are indistinguishable processes. Proposition 2.4. Let X1 = X1 , ..., Xk = Xk be Φ′ - valued regular processes. t t∈J t t∈J β Then, X1,...,Xk areindependentifandonlyiffor alln∈Nandφ ,...,φ ∈Φ, theRn-valued (cid:8) (cid:9) (cid:8) (cid:9) 1 n processes {(Xj[φ ],...,Xj[φ ]):t∈J}, j =1,...,k, are independent. t 1 t n The followingsequence of results offersan extensionofMinlos’theoremto the moregeneral case of cylindrical stochastic processes defined on Φ. Here it is important to remark that equicontinuityofa family ofcylindricalrandomvariablesis equivalentto equicontinuityatzero of its characteristic functions (see [34], Proposition IV.3.4). We start with one of the main tools we have at our disposal and that plays a fundamental role throughout this work. It establishes conditions for a cylindrical stochastic process in Φ′ to have a regular continuous or ca`dla`g version. Theorem 2.5 (RegularizationTheorem; [12], Theorem3.2). Let X ={X } be a cylindrical t t≥0 process in Φ′ satisfying: 5 (1) For each φ ∈ Φ, the real-valued process X(φ) = {X (φ)} has a continuous (respectively t t≥0 c`adla`g) version. (2) For every T > 0, the family {X : t ∈ [0,T]} of linear maps from Φ into L0(Ω,F,P) is t equicontinuous. Then, there exists a countably Hilbertian topology ϑ on Φ and a (Φ )′ -valued continuous X ϑX β (respectively c`adla`g) process Y = {Y } , such that for every φ ∈ Φ, Y[φ] = {Y [φ]} is a t t≥0 t t≥0 version of X(φ) = {X (φ)} . Moreover, Y is a Φ′ -valued, regular,gcontinuous (respectively t t≥0 β c`adla`g) version of X that is unique up to indistinguishable versions. The following result is a particular case of the regularization theorem that establish condi- tions for the existence of a regularcontinuousor ca`dla`gversionwith finite moments and taking values in one of the Hilbert spaces Φ′. q Theorem 2.6 ([12], Theorem 4.3). Let X ={X } be a cylindrical process in Φ′ satisfying: t t≥0 (1) For each φ ∈ Φ, the real-valued process X(φ) = {X (φ)} has a continuous (respectively t t≥0 c`adla`g) version. (2) There exists n ∈N and a continuous Hilbertian semi-norm ̺ on Φ such that for all T >0 there exists C(T)>0 such that E sup |X (φ)|n ≤C(T)̺(φ)n, ∀φ∈Φ. (2.3) t t∈[0,T] ! Then, there exists a continuous Hilbertian semi-norm q on Φ, ̺ ≤ q, such that i is Hilbert- ̺,q Schmidt and there exists a Φ′-valued continuous (respectively c`adla`g) process Y = {Y } , q t t≥0 satisfying: (a) For every φ∈Φ, Y[φ]={Y [φ]} is a version of X(φ)={X (φ)} , t t≥0 t t≥0 (b) For every T >0, E sup q′(Y )n <∞. t∈[0,T] t Furthermore, Y is a Φ′ -(cid:16)valued continuous(cid:17)(respectively c`adla`g) version of X that is unique up β to indistinguishable versions. ThefollowingisaconverseoftheregularizationtheoremwhenΦisabarrellednuclearspace. Theorem 2.7. Let Φ be a barrelled nuclear space and L={L } , be a cylindrical process in t t≥0 Φ′. Suppose that for every t≥0 the cylindrical probability distribution of L can be extended to t a Radon probability measure µ on Φ′ such that for every T > 0 the family {µ : t ∈ [0,T]} Lt β Lt is uniformly tight. Then, for every T > 0 the family of linear maps {L : t ∈ [0,T]} is t equicontinuous. Proof. Let T > 0 and ǫ > 0. First, because the family {µ : t ∈ [0,T]} is uniformly tight, Lt there exists a compact K ⊆Φ′ such that µ (Kc)<ǫ for all t∈[0,T]. β Lt Now, as K is compact and hence bounded in Φ′ (recall Φ′ is Hausdorff), and because Φ β β is barrelled, then K is a equicontinuous subset of Φ′ (see [27], Theorem IV.5.2, p.141) and consequently the polar K0 of K is a neighborhood of zero of Φ (see [20], Theorem 8.6.4(b), p.246). But as Φ is nuclear, there exists a continuous Hilbertian semi-norm p on Φ such that B (1/ǫ)⊆K0. Therefore,fromthepropertiesofpolarsets(see[20],Chap.8)wehavethatK ⊆ p (K0)0 ⊆Bp′(ǫ):={f ∈Φ′ :p′(f)=supφ∈Bp(1)|f[φ]|≤ǫ}=Bp(1/ǫ)0. Thus, Bp′(ǫ)c ⊆Kc. On the other hand, note that for every φ∈B (1) we have π−1([−ǫ,ǫ]c)={f ∈Φ′ :|f[φ]|> p φ ǫ}⊆Bp′(ǫ)c ={f ∈Φ′ :p′(f)=supφ∈Bp(1)|f[φ]|>ǫ}. Hence, foreveryφ∈B (1) itfollowsfromthe argumentsonthe aboveparagraphsandfrom p the fact that µ is an extension of the cylindrical distribution of L that Lt t P(|Lt(φ)| >ǫ)=P(Lt(φ)∈[−ǫ,ǫ]c)=µLt ◦πφ−1([−ǫ,ǫ]c)≤µLt(Bp′(ǫ)c)≤µLt(Kc)<ǫ, for all t ∈ [0,T]. But because B (1) is a neighborhood of zero of Φ, the above shows that the p family of linear maps {L :t∈[0,T]} is equicontinuous at zero, and hence equicontinuous. (cid:3) t 6 3 L´evy Processes and Infinitely Divisible Measures. InthissectionwestudytherelationshipbetweenL´evyprocessesandinfinitelydivisiblemeasures. The link between these two concepts are the cylindrical L´evy processes and the semigroups of probability measures. 3.1 Infinitely Divisible Measures and Convolution Semigroups in the Strong Dual. LetΨbealocallyconvexspace. Ameasureµ∈M1(Ψ′ )iscalledinfinitely divisible ifforevery R β n∈N there exist a n-th root of µ, i.e. a measure µ ∈M1(Ψ′) such that µ=µ∗n. We denote n R β n by I(Ψ′ ) the set of all infinitely divisible measures on Ψ′. β β A family {µ } ⊆M1(Ψ′) is said to be a convolution semigroup if µ ∗µ =µ for any t t≥0 R β s t s+t s,t ≥ 0 and µ = δ . Moreover, we say that the convolution semigroup is continuous if the 0 0 mapping t7→µ from [0,∞) into M1(Ψ′) is continuous in the weak topology. t R β The following result follows easily form the definition of continuous convolution semigroup. Proposition 3.1. If {µ } is a convolution semigroup in M1(Ψ′ ), then ∀t≥0 µ ∈I(Ψ′ ). t t≥0 R β t β Now, to prove the converse of Proposition 3.1 we will need the following definitions. Let µ be an infinitely divisible measure on Ψ′. We define the root set of µ by β R(µ):= νm :ν ∈M1(Ψ′ ) with νn =µ, 1≤m≤n . R β n≥1 [ (cid:8) (cid:9) We say that µ is root compact if its root set R(µ) is uniformly tight. We areready forthe mainresultof this section. As stated onits proof,the mainarguments are based on several results due to E. Siebert (see [29, 30]). Theorem 3.2. Assume that Ψ is a locally convex space for which Ψ′ is quasi-complete. If β µ ∈ I(Ψ′ ), then there exists a unique continuous convolution semigroup {µ } in M1(Φ′ ) β t t≥0 R β such that µ =µ. 1 Proof. First, as Ψ′ is locally convex and µ∈I(Ψ′ ), there exists a rational continuous convo- β β lution semigroup {ν } in M1(Ψ′ ) such that ν =µ (see [29], Korollar 5.4). t t∈Q∩[0,∞) R β 1 Now,as µ=ν =ν∗q , thenν is arootofµ foreachq ∈N\{0}. But asforp,q ∈N\{0} 1 1/q 1/q we have ν∗p/q =ν∗p , then we have that ν ∈R(µ) for each t∈Q∩[0,1]. 1/q t Onthe otherhand,asµ istight(is Radon)andΨ′ isaquasi-completelocallyconvexspace, β therootsetR(µ)ofµ isuniformlytight(see [29],Satz6.2and6.4). Hence,the set{ν } t t∈Q∩[0,1] is uniformly tight and by Prokhorov’s theorem it is relatively compact. This last property guarantees the existence of a (unique) continuous convolution semigroup {µ } in M1(Φ′ ) t t≥0 R β suchthatν =µ foreacht∈Q∩[0,∞)(see[30],Proposition5.3). Therefore,µ=ν =µ . (cid:3) t t 1 1 The following result will be of great importance in further developments. Lemma 3.3. Assume that Ψ′ is quasi-complete and let {µ } be a continuous convolution β t t≥0 semigroup in M1(Φ′ ). Then, ∀T >0 {µ :t∈[0,T]} is uniformly tight. R β t Proof. Let T > 0. Similar arguments to those used in the proof of Theorem 3.2 shows that {µ } ⊆ R(µ ), and because µ is tight, the root set R(µ) of µ is uniformly tight (see t t∈Q∩[0,T] T T [29], Satz 6.2 and 6.4), and hence {µ } is also uniformly tight. t t∈Q∩[0,T] Now, note that for each r ∈ I∪[0,T] the continuity of the semigroup {µ } shows that t t≥0 µ = lim µ in the weak topology. Therefore, {µ } is in the weak closure r qցr,q∈Q∩[0,T] q t t∈[0,T] {µ } of {µ } . But because the weak closure of an uniformly tight family in t t∈Q∩[0,T] t t∈Q∩[0,T] M1(Φ′ ) is also uniformly tight (see [34], Theorem I.3.5), then it follows that {µ } is R β t t∈Q∩[0,T] uniformly tight and hence {µ } is uniformly tight too. (cid:3) t t∈[0,T] 7 3.2 L´evy Processes and Cylindrical L´evy Processes From now on and unless otherwise specified, Φ will always be a nuclear space over R. We start with our definition of L´evy processes on the dual of a nuclear space. Definition 3.4. A Φ′-valued process L={L } is called a L´evy process if it satisfies: β t t≥0 (1) L =0 a.s. 0 (2) Lhasindependent increments,i.e. foranyn∈N,0≤t <t <···<t <∞theΦ′ -valued 1 2 n β random variables L ,L −L ,...,L −L are independent. t1 t2 t1 tn tn−1 (3) L has stationary increments, i.e. for any 0 ≤ s ≤ t, L − L and L are identically t s t−s distributed. (4) For every t≥0 the distribution µ of L is a Radonmeasure and the mapping t7→µ from t t t [0,∞) into M1(Φ′ ) is continuous at 0 in the weak topology. R β The probability distributions of a Φ′-valued L´evy process satisfy the following properties: β Theorem 3.5. If L = {L } is a L´evy process in Φ′ , the family of probability distributions t t≥0 β {µ } of L is a continuous convolution semigroup in M1(Φ′ ). Moreover, each µ is in- Lt t≥0 R β Lt finitely divisible for every t≥0. Furthermore, if Φ is also a barrelled space, then for each T >0 the family {µ :t∈[0,T]} is uniformly tight. Lt Proof. The semigroup property of {µ } is an easy consequence of the stationary and in- Lt t≥0 dependent increments properties of L. The weak continuity is part of our definition of L´evy process. The fact that each µ is infinitely divisible follows from Proposition3.1. Finally, if Φ Lt is also a barrelled space, then Φ′ is quasi-complete (see [27], Theorem IV.6.1, p.148). Hence, β the uniform tightness of {µ :t∈[0,T]} for each T >0 follows from Lemma 3.3. (cid:3) Lt Following the definition givenin Applebaum and Riedle [2] for cylindricalL´evy processes in Banach spaces, we introduce the following definition. Definition 3.6. AcylindricalprocessL={L } inΦ′ issaidtobe acylindrical L´evy process t t≥0 if ∀n∈N, φ ,...,φ ∈Φ, the Rn-valued process {(L (φ ),...,L (φ ))} is a L´evy process. 1 n t 1 t n t≥0 Lemma 3.7. Every Φ′ -valued L´evy process L={L } determines a cylindrical L´evy process β t t≥0 in Φ′. Proof. Let n ∈ N and φ ,...,φ ∈ Φ. It is clear that (L [φ ],...,L [φ ]) = 0 P-a.e. The 1 n 0 1 0 n fact that {(L [φ ],...,L [φ ])} has stationary and independent increments follows from the t 1 t n t≥0 correspondingpropertiesofLasaΦ′ -valuedprocess(seeProposition2.4). Finally,thestochas- β tic continuity of {(L [φ ],...,L [φ ])} is a consequence of the weak continuity of the map t 1 t n t≥0 t7→µ (see [1], Proposition 1.4.1). (cid:3) t The following result is a converse of Lemma 3.7. Theorem3.8. LetL={L } beacylindricalL´evyprocessinΦ′ suchthatforeveryT >0,the t t≥0 family {L : t ∈[0,T]} of linear maps from Φ into L0(Ω,F,P) is equicontinuous. Then, there t exists a countablyHilbertian topology ϑ on Φ and a (Φ )′ -valued c`adla`g process Y ={Y } , L ϑL β t t≥0 such that for every φ ∈ Φ, Y[φ] = {Y [φ]} is a version of L(φ) = {L (φ)} . Moreover, t t≥0 t t≥0 Y is a Φ′ -valued, regular, c`adla`g L´evy process that isga version of L and that is unique up to β indistinguishable versions. Proof. First, as for each φ ∈ Φ the real-valued process L(φ) = {L (φ)} is a L´evy process, t t≥0 then it has a ca`dla`g version (see Theorem 2.1.8 of Applebaum [1], p.87). Hence, L satisfies all theconditionsoftheregularizationtheorem(Theorem2.5)andthistheoremshowstheexistence of a countably Hilbertian topology ϑ on Φ and a (Φ )′ -valued ca`dla`g process Y = {Y } , L ϑL β t t≥0 such that for every φ∈Φ, Y[φ]={Y [φ]} is a version of L(φ)={L (φ)} . Moreover,it is t t≥0 t t≥0 a consequence of the regularizationtheorem that Y igs a Φ′ -valued, regular,ca`dla`g versionof L β that is unique up to indistinguishable versions. Our next step is to show that Y is a Φ′ -valued L´evy process. First, as Y [φ] = L (φ) = 0 β 0 0 P-a.e. for every φ ∈ Φ, it follows that Y = 0 P-a.e. (Proposition 2.3). Second, as for each 0 8 φ ,...,φ ∈Φ,theRn-valuedprocess{(L (φ ),...,L (φ ))} hasindependentandstationary 1 n t 1 t n t≥0 increments, and because for each t≥0 we have that (L (φ ),...,L (φ ))=(Y [φ ],...,Y [φ ]), P−a.e., t 1 t n t 1 t n then the Rn-valued process{(Y [φ ],...,Y [φ ])} alsohas independent and stationaryincre- t 1 t n t≥0 mentsforeveryφ ,...,φ ∈Φ. Hence,becauseY isaΦ′ -valuedregularprocess,itthenfollows 1 n β from Propositions 2.3 and 2.4 that Y has independent and stationary increments. Now, the fact that Y is a Φ′ -valued regular process and Theorem 2.2 shows that for each β t≥0 the probability distribution µ of Y is a Radon measure. t t Our final step to show that Y is a Φ′-valued L´evy process is to prove that the mapping β t7→µ from [0,∞) into M1(Φ′ ) is continuous in the weak topology. t R β Let t ≥ 0. Our objective is to show that for any net {s } in [0,∞) such that lim s = t α α α we have lim µ = µ in the weak topology on M1(Φ′ ). As convergence of filterbases is only α sα t R β determined by terminal sets, we can choose without loss of generality some sufficiently large T > 0 and consider only nets in [0,T] satisfying {s } such that lim s = t. Let {s } be such α α α α a net. First, as for each φ ∈ Φ, Y[φ] = {Y [φ]} is stochastically continuous, it follows that the t t≥0 family {Y [φ]} converges in probability to Y [φ]. Now, this last property in turns shows that sα t lim µ (φ)=µ (φ) for every φ∈Φ. α sα t Now, for each r ≥ 0 denote by ν the cylindrical distribution of the cylindrical random r variable L . Then, the equicontinuity of the family {L :r ∈[0,T]} of linear maps from Φ into b r b r L0(Ω,F,P) implies that the family of characteristic functions {ν } is equicontinuous at r r∈[0,T] zero. But as for each r ≥ 0, ν (φ) = µ (φ) for all φ ∈ Φ, we then have that the family of r r characteristic functions {µ } of {Y } is equicontinuous at zero. However, as Φ is r r∈[0,T] r r∈[0,T] b a nuclear space the equicontinuity of {µ } at zero implies that {µ } is uniformly b br r∈[0,T] r r∈[0,T] tight (see [7], Lemma III.2.3, p.103-4). This last in turn shows that {µ } is uniformly tight, b sα and by the Prokhorov’s theorem (see [7b], Theorem III.2.1, p.98) the family {µsα} is relatively compact in the weak topology. Because we also have that lim µ (φ)=µ (φ) for every φ∈Φ, α sα t we then conclude that lim µ =µ in the weak topology (see [34], Theorem IV.3.1, p.224-5). α sα t Consequently, the map t 7→ µt is continuous in the weak topolobgy and Ybis a Φ′β-valued L´evy process. (cid:3) An important variation of the above theorem is the following: Theorem 3.9. Let L = {L } be a cylindrical L´evy process in Φ′. Assume that there exist t t≥0 n∈NandacontinuousHilbertian semi-norm̺onΦsuchthatfor allT >0thereisaC(T)>0 such that E sup |L (φ)|n ≤C(T)̺(φ)n, ∀φ∈Φ. t t∈[0,T] ! Then, there exists a continuous Hilbertian semi-norm q on Φ, ̺ ≤ q, such that i is Hilbert- ̺,q Schmidt and there exists a Φ′-valued c`adla`g L´evy process Y ={Y } , satisfying: q t t≥0 (a) For every φ∈Φ, Y[φ]={Y [φ]} is a version of L(φ)={L (φ)} , t t≥0 t t≥0 (b) For every T >0, E sup q′(Y )n <∞. t∈[0,T] t Moreover, Y is a Φ′ -valu(cid:16)ed, regular, c`adla`(cid:17)g version of L that is unique up to indistinguishable β versions. Furthermore, if the real-valued process L(φ) is continuous for each φ∈Φ, then Y can be choose to be continuous in Φ′ and hence in Φ′. q β Proof. The existence of the Φ′-valued ca`dla`g process Y = {Y } satisfying the conditions q t t≥0 in the statement of the theorem follows from Theorem 2.6. Finally, similar arguments to those used in the proof of Theorem 3.8 show that Y is a Φ′-valued L´evy process. (cid:3) q We now provide a sufficient condition for the existence of a ca`dla`g version for a Φ′ -valued β L´evy process. Theorem 3.10. Let L= {L } be a Φ′ -valued L´evy process. Suppose that for every T > 0, t t≥0 β the family {L : t ∈ [0,T]} of linear maps from Φ into L0(Ω,F,P) given by φ 7→ L [φ] is t t 9 equicontinuous. Then, L has a Φ′ -valued, regular, c`adla`g version L˜ = {L˜ } that is also a β t t≥0 L´evy process. Moreover, there exists a countably Hilbertian topology ϑ on Φ such that L˜ is a L (Φ )′ -valued c`adla`g process. ϑL β Proof. First, note that our assumption on L implies that L is regular. This is because for eagch t≥0 the fact that L :Φ→L0(Ω,F,P) is continuous shows that L is a regular random t t variable in Φ′ (Theorem 2.2). β Now,asLisaΦ′-valuedL´evyprocessthecylindricalprocessdeterminedbyLisacylindrical β L´evy process (Lemma 3.7). But from our assumptions on L, this cylindrical L´evy process satisfies the assumptions in Theorem 3.8. Therefore, there exists a Φ′ -valued, regular, ca`dla`g β L´evy process L˜ ={L˜ } , such that for every φ∈Φ, L˜ [φ]=L [φ] P-a.e. for each t≥0. This t t≥0 t t last property together with the fact that both L˜ and L are regular process shows that L˜ is a version of L (Proposition 2.3). Finally, from Theorem 3.8 there exists a countably Hilbertian topology ϑ on Φ such that L˜ is a (Φ )′ -valued ca`dla`g process. (cid:3) L ϑL β Corollary 3.11. If Φ is a barrelled nuclear space and L={L } is a Φ′ -valued L´evy process, g t t≥0 β then L has a Φ′ -valued c`adla`g version satisfying the properties given in Theorem 3.10. β Proof. It follows from Theorem 3.5 that for every T > 0 the family {µ : t ∈ [0,T]} is Lt uniformlytight. Then,itfollowsfromTheorem2.7thatLsatisfiestheassumptionsonTheorem 3.10. Hence, the result follows. (cid:3) Finally, the next result provides sufficient conditions for the existence of a ca´dla´g version that is a L´evy process with finite n-th moment in some of the Hilbert spaces Φ′. q Theorem 3.12. Let L={L } be a Φ′ -valued L´evy process. Assume that there exist n∈N t t≥0 β and a continuous Hilbertian semi-norm ̺ on Φ such that for all T >0 there is a C(T)>0 such that E sup |L [φ]|n ≤C(T)̺(φ)n, ∀φ∈Φ. t t∈[0,T] ! Then, there exists a continuous Hilbertian semi-norm q on Φ, ̺ ≤ q, such that i is Hilbert- ̺,q Schmidt and a Φ′-valued, c`adla`g (continuous if Lis continuous),L´evy process L˜ ={L˜ } that q t t≥0 is a version of L. Moreover, E sup q′(Y )n <∞ ∀T >0. t∈[0,T] t Proof. The proof follows from(cid:16)Theorem 3.9 and(cid:17)similar arguments to those used in the proof of Theorem 3.10. (cid:3) 3.3 Correspondence of L´evy Processes and Infinitely Divisible Measures We have already show in Theorem 3.5 that for every Φ′-valued L´evy process L = {L } the β t t≥0 probability distribution µ of L is infinitely divisible for each t ≥ 0. In this section we will Lt t show that if the space Φ is barrelled and nuclear, to every infinitely divisible measure µ on Φ′ β there corresponds a Φ′ -valued L´evy process L such that µ =µ. β L1 In order to prove our main result (Theorem 3.14), we will need the following theorem that establishes the existence of a cylindrical L´evy process from a given family of cylindrical proba- bility measures with some semigroup properties. We formulate our result in the more general context of Hausdorff locally convex spaces. The definitions of cylindrical probability measure and cylindrical L´evy process are exactly the same to those given in Sections 2.2 and 3.2. Theorem 3.13. Let Ψ be a Hausdorff locally convex space. Let {µ } be a family of cylin- t t≥0 drical measures on Ψ′ such that for every finite collection ψ ,ψ ,...,ψ ∈ Ψ, the family 1 2 n {µ ◦π−1 } isacontinuousconvolutionsemigroupofprobabilitymeasuresonRn. Then, thetreeψx1i,sψt2s,.a..,ψcynlint≥d0rical process L={L } inΨ′ definedonaprobability space(Ω,F,P), such t t≥0 that: (1) For every t≥0, ψ ,ψ ,...,ψ ∈Ψ and Γ∈B(Rn), 1 2 n P((L (ψ ),L (ψ ),...,L (ψ ))∈Γ)=µ ◦π−1 (Γ). t 1 t 2 t n t ψ1,ψ2,...,ψn 10