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Ornstein-Uhlenbeck Equations with time-dependent coefficients and Levy Noise in finite and infinite dimensions PDF

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Ornstein-Uhlenbe k Equations with 9 0 time-dependent oe(cid:30) ients 0 2 and Lévy Noise in (cid:28)nite and in(cid:28)nite dimensions n a Florian Knäble J 9 Universität Bielefeld 1 E-mail: f.knaeblegooglemail. om ] R January 2009 P . h Abstra t t a We solve a time-dependent linear SPDE with additive Lévy noise in m the mild and weak sense. Existen e of a generalized invariant measure [ fortheasso iated transitionsemigroLu2pisestablishedandthegeneratoris hara terizedonthe orresponding -spa e. Thesquare(cid:28)eldoperatoris 1 al ulated, allowing to derive a Poin aré and a Harna kinequality. v 7 8 8 1 Introdu tion 2 . Re ently, there seems to be a growing interest in the study of semilinear non- 1 0 autonomous sto hasti evolution equations in in(cid:28)nite dimensional spa es, see 9 e.g. [20℄ and the referen es therein. In the so- alled semigroup approa hwhi h 0 we also pursue in this paper, this is of ourse losely onne ted to solving the : v underlyingnon-autonomousCau hyproblem, overedbythetheoryofevolution i semigroups or evolution families, see e.g. [16℄. So, let us onsider the following X equation: dX = (A(t)X +f(t))dt+B(t)dZ r t t t a X(s) = x (1) (cid:26) H A :H H t onaHilbertspa e ,where → arelinearoperatorsandall oe(cid:30) ients are T-periodi . Fromamathemati alpointofview it isnaturalto study (1) withan additional Lips hitz non-linearity, be ause this ase an still be overed if one atta ks equation (1) using ontra tion methods (e.g. Pi ard-Lindelöf iteration). Our main interest, however, in the linear ase (1), is to exploit a lot of expli it formulae (e.g. for the transition semigroup, whi h are in fa t time-dependent versionsofgeneralizedMehlerformulae)togetaspre iseaspossibleinformation about the solution and the orresponding Kolmogorov equations. Con erning A t the time-dependent (possibly unbounded) operators , for simpli ity, we only 1 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations onsider the ase of a ommon domain. The general ase is te hni ally harder, butweHexp=e RtdourreZsultstoholdtrue. Thiswillbethesubje toffurtherstudy. t For and ad-dimensionalBrownianmotion,(1)wasstudiedinten- sivelyin[7℄. Inspiredbytheirpaper,ourwork onsistsprimarilyingeneralizing H Z t theirresultstothe ase in(cid:28)nite-dimensionaland aLévypro ess. Anumber of our arguments are adapted from [7℄, although the Lévy setting for es us to work more heavily with Fourier transforms and the in(cid:28)nite dimensional setting requires extra are. Nevertheless, we su eed in proving the following results: existen eanduniquenessofthesolution,expli it al ulationofits hara teristi fun tion, proof of the Chapman-Kolmogorov equations, existen e and unique- ness of an evolution system of measures, existen e and form of the generator (in luding a result on its spe trum), pre ise form of the square (cid:28)eld operator, proof of a Poin aré and a Harna k inequality. In the autonomous ase, the semigroups asso iated to this kind of equation areknownasthe generalized Mehlersemigroups mentioned aboveand they are already well understood. Invariant measures are established in [5℄ and [10℄, generators are examined in [11℄ and the square (cid:28)eld operator is identi(cid:28)ed in [12℄. The paper is organized as follows. In Se tion 2 we shortly review the Lévy- Ito de omposition and the Lévy-Khin hine representation. In Se tion 3 we develop the ne essarytheory of integration to givesense to our solution, whose existen e is established in a rather standard way. Then, in Se tion 4, we al- ulate the Fourier transforms of oursolutions and thus determine its transition semigroupexpli itly, whi h is a two-parametersemigroup, sin e the equationis non-autonomous. In this ase the on ept of an invariant measure has to be generalized to allow for a whole olle tion of measures - a so- alled evolution system of measures- whi h are invariant in an appropriatesense. We prove the A t existen e of su h a system under the ondition that the generate an expo- nentially stable semigroup and provided that the Lévy symbol is su(cid:30) iently smooth. In ontrastto the Gaussiansetting, the onstru tionoflimit measures is more deli ate in our ase. Then (as usual) we turn the problem into an autonomous one by enlarging the state spa e, allowing for a one-parameter semigroup. Via the evolution system of measuresand thanks to the periodi ity of the oe(cid:30) ients weareable to onstru t a unique invariant measure for this semigroup as in [7℄. Thus we L2 an introdu e the -spa e with respe t to the invariant measure where the semigroup is then shown to be strongly ontinuous. Se tion5isdedi atedtoananalysisofthegenerator. Weidentifyanexpli it domain of uniqueness for the generator and its a tion on it, thus identifying it as an pseudo-di(cid:27)erential operator in in(cid:28)nitely many variables. Subsequently, in Se tion 6, we establish the form of its square (cid:28)eld operator whi h yields a generalizationtothe ru ial(cid:16)integrationbypartsformula(cid:17) in[7℄. Thenweprove an estimate for the square (cid:28)eld operator, that allows us to obtain a Poin aré and a Harna k inequality for our semigroup. A knowledgement I wish to thank Prof. Dr. Rö kner for introdu ing me to this interesting subje t and for many helpful suggestions. 2 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations 2 Lévy Pro esses H , := In the following let be a separable Hilbert spa e with s alar produ t h· ·i , := H H H h· ·i and norm k·k k·k . An -valued sto hasti pro ess L adapted to a ( ) t t 0 (cid:28)ltration F ≥ is alled a Lévy pro ess if it has independent and stationary P(L = 0) = 1 in remAents,(His)sto hasti ally ontinu0ou/sA,¯and we have . We say that ∈B is bounded below if ∈ N(t,A) A t We denote by the (random) number of "jumps of size " up to time , N(t,A):= 0 s t∆L A s that is ard{ ≤ ≤ | ∈ } A N(t,A) M(A) If isMbo(uAn)d:e=dEbe[Nlow(1,,tAh)e]n is a Poisson pro ess, with intensity , where . M H De(cid:28)nition 2.1 A measure on with : min(1, x 2)M(dx)< k k ∞ ZH\{0} is alled a Lévy measure. L H Theorem 2.2 (Lévy-Ito De omposition) If is an -valuedLévy pro ess, b H R W H W R R there is a drift ve tor ∈ , a -Wiener pro ess on , su h that is N (A) A t independent of for any that is bounded below and we have: L =bt+W (t)+ x(N (dx) tM(dx))+ xN (dx) t R t t − Zkxk<1 Zkxk≥1 N L M t where is the Poisson random measure asso iated to , and the orre- sponding Lévy measure. (cid:4) Proof See e.g. [1℄ Theorem 4.1 L H Theorem 2.3 (Lévy-Khin hine Representation) If isan -valuedLévy pro ess with LévEy-[eItihoLtd,ue i]om=peotsλi(tui)on as in 2.2, then its hara teristi fun tion takes the form: and 1 λ(u)=i b,u u,Ru + ei u,x 1 i u,x χ M(dx) h i− 2h i ZH/{0}h h i− − h i {kxk≤1}i (2) (cid:4) Proof See [13℄ Theorem 5.7.3 Sin e a measure is hara terized by its Fourier transform we will say that a µ [b,R,M] measure is asso iated to a triple if its hara teristi exponent has the form (2). Remark 2.4 A tually the Lévy-Khin hine representation holds not only for Lévy pro esses but for any in(cid:28)nitely divisible random variable. (See [18℄ for 3 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations an a ount of in(cid:28)nite divisibility) Moreover, Lévy pro esses and in(cid:28)nitely divis- ible measures an be brought in a one to one orresponden e. In parti ular the onverse of 2.3 is true: any fun tion of the form 1 exp i b,u u,Ru + ei u,x 1 i u,x χ M(dx) ( h i− 2h i ZH/{0}h h i− − h i {kxk≤1}i ) is the hara teristi fun tion of a measure. 3 Solving the generalized Ornstein-Uhlenbe k equa- tion 3.1 Sto hasti Integration with respe t to Lévy martin- gale measures In this subse tion we follow [2℄, where the proofs of all results an be found. H 1 Let be R the ring of all Borel subsets of the unit ball of whi h arebounded below. H De(cid:28)nMit:ioRn 3.1 A LéΩvy mHartingale measure on a Hilbert spa e is a set fun - + 1 tion ×R × → satisfying: M(0,A)=0 A 1 • almost surely for all ∈R M(t, )=0 • ∅ almost surely M(t,A B) = M(t,A)+M(t,B) t • almost surely we have: ∪ for all and all A B 1 disjoint , ∈R M(t,A) A t 0 1 • { ≥ } is a square-integrable martingale for ea h ∈R A B = M(t,A) M(t,B) t 0 t 0 • if ∩ ∅ { ≥ } and { ≥ } are orthogonal, that is: M(t,A),M(t,B) A,B 1 h i is a real-valued martingale for every ∈R sup E[ M(t,A) 2] A (S ) < n N n • { k k | ∈B } ∞ for every ∈ A A (S ) j j n • for everjy sequen e dliem reasinEg[tMo t(hte,Aem)p2ty]<set su h that ⊂ B j j for all we have: →∞ k k ∞ s < t A M(t,A) M(s,A) 1 • for every and every ∈ R we have that − is s independent of F M(t,A) = xN˜ (dx) H Proposition 3.2 A t is a Lévy martingale measure on A 1 for every ∈R . R Similarly as a Wiener pro ess is hara terized by its ovarian e operator, we andes ribethe ovarian estru tureof aLévymartingalemeasureby afamily 1 of operators parametrized by our ring R . 4 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations Proposition 3.3 E[ M(t,A),v 2]=t v,T v A |h i| h i t 0, v H A T 1 A for all ≥ ∈ ∈R , where the operators are given by T v := T vν(dx) T v := x,v x A A x and x h i . R We will establish only a limited theory of integration, as for our purposes it will be su(cid:30) ient to integrate deterministi operator valued fun tions. We do not even need them to depend on the jump size. The pro edure is the same as for Brownian motion, so let us introdu e the spa e of our integrands, the approximatingsimple fun tions, and state howthe integralis de(cid:28)ned forthem. M([s,t],A):=M(t,A) M(s,A) For onvenien e, we set − . H ′ De(cid:28)nition 3.4 Let be another real separable Hilbert spa e. 2 := 2(T ,T ) R : [T ,T ] (H,U) R + + Let H H − be the spa e of all − → L su h that is strongly measurable and we have: 1 T+ 2 R := tr(R(t)T R (t))ν(dx)ds < 2 x ∗ k kH ZT− Zkxk<1 ! ∞ R 2 Let S be the spa e of all ∈H su h that n R= R χ χ i (ti,ti+1] A i=0 X T = t < t < ... < t = T n N R 0 1 n+1 + i where − for some ∈ , where ea h ∈ (H,H ) A ′ L and where ∈R R , t [T ,T ] + For ea h ∈S ∈ − de(cid:28)ne the sto hasti integral as follows: n I (R):= R M([t t,t t],A) t i i i+1 ∧ ∧ i=0 X 2 Proposition 3.5 The spa e H with inner produ t T+ R,U := tr(R(t)T U (t))ν(dx)ds x ∗ h i ZT− Zkxk<1 is a Hilbert spa e. 2 Proposition 3.6 The spa e S is dense in H . R :E[I (R)]=0 t Proposition 3.7 We have for any ∈S and t E[ I (R) 2]= (R(s)T R (s))ν(dx)ds = χ R(t) 2 k t k ZT−ZAtr x ∗ k [T−,t] kH2 t I : L2(Ω, ,P;H) t So for (cid:28)xed, S → F is an isometry. I 2 t So we an isometri ally extend the operator from S to its losure H . 5 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations 3.2 Sto hasti Convolution We want to give meaning to the integral t X := U(t,r)B(r)dL(r) U,B Zs L H whi h we will all a sto hasti onvolution. Here is a -valued Lévy pro ess U(t,r) (H),B(r) (H) s r t and we have ∈ L ∈ L ∀ ≤ ≤ . In anti ipation of the assumptions in se tion 4 we will pose the following onditions: sup B(r) < • r∈Rk kL(H) ∞ M >0,ω >0 U(t,r) <Me ω(t r) (H) − − • there is su h that : k kL r B(r) r U(t,r) t • 7→ is measurable and 7→ is measurable for any (cid:28)xed U B Proposition 3.8 If and are as above, the sto hasti onvolution exists in the following sense: t U(t,r)B(r)dL(r) Zs t t = U(t,r)B(r)b dr+ U(t,r)B(r)x N (dx) r Zs Zs Zkxk≥1 (3) t t + U(t,r)B(r)dW (r)+ U(t,r)B(r)x N˜ (dx) Q r Zs Zs Zkxk<1 U(t,s)= Proof The proof is analogous to the one of Theorem 6 in [2℄ where S(t s) S − for a strongly ontinuous semigroup : The (cid:28)rst term in (3) is well de(cid:28)ned as a simple Bo hner integral, and the se ond as a (cid:28)nite random sum. For the other two terms, it is straightforward to he k under our assumptions, (cid:4) that the integrands are su h that the respe tive isometries apply. 3.3 Existen e of the Mild Solution In the following we will have to deal with a non-autonomous abstra t Cau hy problem- non-autonomousmeanswearenot in the frameworkofstrongly on- tinuous semigroups anymore. This implies in parti ular, that we have no easy hara terizationofwell-posednessinthesenseoftheHille-Yosidatheoremavail- able. There are di(cid:27)erent, yet te hni al, approa hes (see [15℄ and the referen es therein for a re ent overview), but sin e this subje t is not in the primary in- terest of our work, we assume that the problem is well posed. This is losely related to the notion of evolution semigroups. Our de(cid:28)nition is taken from [4℄ We onsider the following non-autonomous generalisation of the Langevin equation: dX = (A(t)X +f(t))dt+B(t)dL t t t X(s) = x (4) (cid:26) 6 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations B : R (H) wfh:eRre H → L is Lst(rto)ngly oHntinuous and bounded in operator noAr(mt), → is ontinuous, is an -valued Lévy-pro ess and where the H D(A) aAre:lRinearDo(pAe)ratorHs on with ommon domain and × → is su h that we an solve the asso iated non-autonomous abstra t Cau hy problem dX = (A(t)X +f(t))dt t t X(s) = x (5) (cid:26) a ording to the following de(cid:28)nitions: H De(cid:28)nition 3.9 An exponentially bounded evolution family on is a two pa- U(t,s) H t s rameterfamily { } ≥ of boundedlinearoperators on su h thatwe have: U(s,s)=Id U(t,s)U(s,r)=U(t,r) r s t (i) and whenever ≤ ≤ x H (t,s) U(t,s)x s t (ii) for ea h ∈ , 7→ is ontinuous on ≤ M >0 ω >0 U(t,s) Me ω(t s) , s t − − (iii) there is and su h that : k k≤ ≤ Assumption 3.10 There is a unique solution to (5) given by an exponentially U(t,s) bounded evolution family so that the solution takes the form: t X =U(t,s)x+ U(t,r)f(r)dr t Zs Moreover, we assume that : d U(t,s)x=A(t)U(t,s)x dt A t Remark 3.11 Note that in the (cid:28)nite dimensional ase, where ea h is au- tomati ally bounded we get the existen e of an evolution family that solves (5), t A t under the reasonable assumption that 7→ is ontinuous and bounded in the operator norm, by solving the following matrix-valued ODE: ∂ U(t,s)=A(t)U(t,s) ∂t U(s,s)=Id (cid:26) (t,M) A(t)M Existen eanduniquenessareassuredsin e 7→ isglobally Lips hitz M in . This result even holds in in(cid:28)nite dimensions, see [6℄. De(cid:28)nition 3.12 Given assumption 3.10 we all the pro ess: t t X(t,s,x)=U(t,s)x+ U(t,r)f(r)dr+ U(t,r)B(r)dL r Zs Zs a mild solution for (4). 7 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations 3.4 Existen e of the Weak Solution Wehave alledtheaboveexpressionamildsolution,thoughthereisnoobvious relation to the equation yet. Now, we will show that our andidate solution a tually solves our equation in the weak sense. The following de(cid:28)nition makes this pre ise, but (cid:28)rst we need to strengthen our assumption on erning the A(t) ommon domain of the a little: A (t) ∗ Assumption 3.13 Werequirethattheadjointoperators alsohavea om- t D(A ) ∗ mon domain independent of whi h we will denote by . Furthermore, we D(A ) H ∗ assume that is dense in and that we have: d U (t,s)y =U (t,s)A y dt ∗ ∗ ∗t y D(A ) ∗ for every ∈ . H X t De(cid:28)nition 3.14 An -valued pro ess is alled a weak solution for (4) if y D(A ) ∗ for every ∈ we have: t t t X ,y = x,y + X ,A y dr+ f(r),y dr+ B (r)ydL h t i h i h r ∗r i h i ∗ r (6) Zs Zs Zs (B (r)y)(h) := B (r)y,h B (r)y (H,R) ∗ ∗ ∗ Here h i so that ∈ L and the integral is well de(cid:28)ned, sin e 1 B y 2 (t s)sup B(r) 2 y 2 T2e 2ν(dx)< k ∗ kH2 ≤ − rk k k k k kxk<1k x kk ∞. P R X t Theorem 3.15 The mild solution from de(cid:28)nition 3.12 is also a weak solu- tion for (4). Moreover, it is the only weak solution. Proof Analogous to [2℄ Theorem 7. After some relatively straightforward al- ulations the problem is redu ed to proving the following equality: t t r t U(t,r)B(r)dL ,y = U(r,u)B(u)dL ,A y dr+ B (u)ydL r u ∗r ∗ u (cid:28)Zs (cid:29) Zs (cid:28)Zs (cid:29) Zs (7) and we will do so with the help of two lemmas. (M, ,µ) Proposition 3.16 [sto hasti Fubini℄ Let be M a measure spa e with µ G2(M) (H,H ) R ′ (cid:28)nite. By denote the spa e of all L - valued mappings on [s,t] M (r,m) R(r,m)y y H R 2× s:=u htthat R(r,m7→)T21 2ν(dx)iµs(dmmea)sdurr<able for ea h ∈ and k kG2(M) s Mk x k ∞ Then we have: R Rt t R(u,m)dL µ(dm)= R(u,m)µ(dm) dL u u ZM(cid:18)Zs (cid:19) Zs (cid:18)ZM (cid:19) Proof see [2℄ Theorem 5 8 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations R 2 y H Lemma 3.17 Let be ∈H and ∈ . Then we have: t t R(r)dL ,y = R (r)ydL r ∗ r (cid:28)Zs (cid:29) Zs (cid:4) Proof see [2℄ Theorem 4 Now we are able to (cid:28)nish our proof of 3.15: t r t r U(r,u)B(u)dLu,A∗ry dr 3=.17 B∗(u)U∗(r,u)A∗ry dLu dr Zs (cid:28)Zs (cid:29) Zs (cid:18)Zs (cid:19) t t t t d 3=.16 B∗(u)U∗(r,u)A∗ry dr dLu = B∗(u) drU∗(r,u)y dr dLu Zs (cid:18)Zu (cid:19) Zs (cid:18) Zu (cid:19) t t = B∗(u)[U∗(t,u) Id]y dLu 3=.17 [U(t,u) Id]B(u)dLu,y − − Zs (cid:28)Zs (cid:29) t t = U(t,u)B(u)dL ,y B(u)dL ,y u u − (cid:28)Zs (cid:29) (cid:28)Zs (cid:29) (cid:4) and that is pre isely what we had to show. 4 Semigroup and Invariant Measure T >0 Assumption 4.1 From now on, we assume that there exists su h that A,f B T the oe(cid:30) ients and in (4) are -periodi . Re all that the weak solution for (4) takes the following form: t t X(t,s,x)=U(t,s)x+ U(t,r)f(r)dr+ U(t,r)B(r)dL r Zs Zs As opposed to the Gaussian ase we are no longer able to give an easy X(t,s,x) representationofthelawof ,butwe an al ulateitsFouriertransform: Lemma 4.2 ( hara teristi fun tion) E[exp(i h,X(t,s,x) )]= h i t t exp i h,U(t,s)x+ U(t,r)f(r)dr exp λ(B (r)U (t,r)h)dr ∗ ∗ (cid:26) (cid:28) Zs (cid:29)(cid:27) (cid:26)Zs (cid:27) λ L where is the Lévy symbol of . Proof : Straightforward,byusingtheisometriestoapproximatethesto hasti integral by a sum, and then using independen e of in rements and the Lévy- (cid:4) Khin hin formula. Details are in luded in the appendix. The following lemma is a straightforward generalization of the standard monotone lass theorem. 9 F. Knäble Non-Autonomous Ornstein-Uhlenbe k Equations Lemma 4.3 ( omplex monotone lasses) Let H be a omplex ve tor spa e of omplex-valued bounded fun tions, that ontains the onstants and is losed under omponentwise monotone onvergen e. Let M⊂H be losed under mul- σ( ) tipli ation and omplex onjugation. Then, all bounded M -measurable fun - tions belong to H. The last and the next result in ombination will be parti ularly useful: := ei h,x ,h H h i Lemma 4.4 The fun tions M { ∈ } form a omplex multipli a- σ H tive systems that generates the Borel -algebra of . Proof It isobviousthat M is losedunder multipli ation and omplex onju- gation. σ( ) = (H) To show that indeed M B we make use of the following lemma: (see [19℄ page 108) X Lemma 4.5 A ountable family of real-valued fun tions on a Polish spa e X X separating the points of already generates the Borel-sigma-algebra of . Our ountable family will be {fn,k(x) := sin(hn1ek,xi)}k,n∈N ⊂ M where {ek} H is an orthonormal basis of . S1ine e,xth)e sine fun tion is inje tHive in a neighfborhoodof zero, and the fun tions hn k i separatethepointsof ,sodothe n,k. Asrealandimaginarypartsof fun tionsinM,itis lear,thatthesigma-algebrageneratedbythemisin luded σ( ) (cid:4) in M . Now we will show that our solution indu es a two-parameter semigroup, de(cid:28)ned as follows: f :H C De(cid:28)nition 4.6 Whenever → is measurable and bounded, de(cid:28)ne P(s,t)f(x):=E[f(X(t,s,x))] P(s,t) will be alled the two-parameter semigroup (asso iated to the solution X ). f P(s,t) Lemma 4.7 For as above, we have the following (cid:29)ow property, i.e. satis(cid:28)es the Chapman-Kolmogorov equations: P(r,s)P(s,t)f(x)=P(r,t)f(x) f (x) = ei h,x h h i Proof We will show the equality for the fun tions and extend it with the help of 4.3. First note, that by 4.2 we have t t P(s,t)fh(x)=exp i h,U(t,s)x+ U(t,r)f(r)dr + λ(B∗(r)U∗(t,r)h)dr (cid:26) (cid:28) Zs (cid:29) Zs (cid:27) so that: P(r,s)P(s,t)f (x)=E[P(s,t)f (X(s,r,x)] h h =E[exp i U (t,s)h,X(s,r,x) ] ∗ { h i} t t exp i h, U(t,r)f(r)dr + λ(B (r)U (t,r)h)dr ∗ ∗ × (cid:26) (cid:28) Zs (cid:29) Zs (cid:27) 10

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