o f P r o b a b i l i t y Vol. 15 (2010), Paper no. 50, pages Journal URL ejpecp/

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1 E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol. 15 (1), Paer no. 5, ages Journal URL htt:// ejec/ Exonential Estimates for Stochastic Convolutions in -smooth Banach Saces Jan Seidler Institute of Information Theory and Automation Academy of Science of the Czech Reublic Pod vodárenskou věží Praha 8, Czech Reublic seidler@utia.cas.cz htt://simu9.utia.cas.cz/seidler Abstract. Shar constants in a (one-sided) Burkholder-Davis-Gundy tye estimate for stochastic integrals in a -smooth Banach sace are found. As a consequence, exonential tail estimates for stochastic convolutions are obtained via Zygmund s extraolation theorem. Key words: stochastic integrals in -smooth Banach saces, Burkholder- Davis-Gundy inequality, exonential tail estimates, stochastic convolutions. AMS Classification: 6H15. Submitted to EJP on Aril 13, 1, final version acceted Setember 6, 1. This research was suorted by the GAČR Grant No. 1/7/

2 . Introduction. The role layed by stochastic convolutions in the semigrou theory of (semilinear) stochastic artial differential equations is well understood nowadays, see for examle the monograhs [7] or [6] for a systematic exosition. If E and H are searable Hilbert saces, W a (cylindrical) Wiener rocess on H, (e At ) a C -semigrou on E and ψ a rogressively measurable rocess with values in a suitable sace of linear oerators from H to E, then the stochastic convolution is a random rocess of the form J t (ψ) = e A(t s) ψ(s)dw(s), t. Unfortunately, J(ψ) is a not a (local) martingale, and its basic roerties are far from being obvious. In articular, to establish L -estimates for J(ψ) that would relace martingale moment inequalities which are no longer available requires new ideas. One may consult the above quoted books and references therein to get acquainted with methods develoed to surmount this roblem. In the aer [4] we showed that L -estimates of J(ψ) with shar constants (that is, constants which grow as 1/ as ) lead in a very simle way, via Zygmund s extraolation theorem, to exonential tail estimates { P su t T } e A(t s) ψ s dw s ε k 1 e k ε, ε. (.1) Moreover, it was roved that the constants in L -estimates for J(ψ) are asymtotically equivalent as to the constants C in the Burkholder-Davis-Gundy inequality su L M t E C M 1/ (.) (Ω) t L (Ω) for E-valued continuous martingales. For real-valued continuous martingales at least three different roofs that C = O( 1/ ),, (.3) are available (cf. [8], Theorem 3.1; [1], Proosition 4.; [3], Theorem 1); (.3) may be easily extended to E-valued continuous martingales by using a general rocedure roosed in [14] (see [4] for details). To obtain finer regularity results for solutions to stochastic artial differential equations it is often advantageous to abandon Hilbertian setting and work in a -smooth Banach sace E, tyically, in E = L q (µ) for some q, see [3] for a ioneering work in this direction. (The first to develo stochastic integration theory in -smooth Banach saces was A. Neidhardt in his Ph.D. thesis[17] which, however, has been never ublished. From several available exositions, we shall use the aer [18] since the theory is resented there in a form suitable for our uroses.) In a -smooth Banach sace E, no estimate like (.) is known for general continuous 1557

3 E-valued martingales, nevertheless a closely related inequality may be roved in a articular case of stochastic integrals driven by a Wiener rocess. However, the methods used to derive (.3) in the Hilbertian case cease to be alicable in -smooth saces. The main result of the resent aer is a roof that (.3) remains valid for the constant C aearing in the Burkholder-Gundy-Davis tye inequality for stochastic integrals in a -smooth Banach sace E. Our aroach is based on the Rosenthal inequality for discrete-time martingales, which we use in a form that was obtained in [13] for real-valued martingales and in [] for E-valued rocesses. Exonential tail estimates (.1) for stochastic convolutions in E then follow in a very straightforward manner. To state our results recisely, we have to introduce some notation and recall a few results. Hereafter, will be a searable -smooth Banach sace, Υ a searable real Hilbert sace, and W a cylindrical Wiener rocess on Υ with a covariance oerator Q L(Υ), defined on a stochastic basis (Ω, F,(F t ),P). Let us denote by U the sace RngQ 1/ equied with the norm x U = Q 1/ x Υ, where Q 1/ denotes the inverse to the restriction of the oerator Q 1/ to (KerQ 1/ ). The sace of all γ-radonifying oerators from some Hilbert sace H to will be denoted by γ(h,) and its norm by γ(h,). We shall write simly γ instead of γ(u,) if there is no danger of confusion. By Γ(U,) we shall denote the set of all rogressively measurable γ(u, )-valued stochastic rocesses ψ satisfying ψ γ L loc (R +) almost surely, that is, ψ(s) γ ds < for all t P-almost surely. Let Fin(Υ,) denote the sace of all finite dimensional oerators from Υ to, i.e., B : Υ belongs to Fin(Υ,) if and only if B = K k=1 h k x k for some h k Υ and x k. If B has this reresentation then BW t is a well-defined random variable in, namely K BW t = W t (h k )x k. k=1 We shall write B(W t W s ) instead of BW t BW s. We shall use reeatedly the following estimate (see e.g. [18], Lemma 3.3): for every > there exists a constant α < such that E B(W t W s ) α (t s) / BQ 1/ γ(υ,) (.4) for all B Fin(Υ,) and t s. If = then α = 1 and equality holds in (.4). Finally, the norm of the sace L (Ω) will be denoted by. It is known (see again e.g. [18]) that for any ψ Γ(U,) a stochastic integral I t (ψ) = ψ(s)dw(s), t, 1558

4 is well-defined and I (ψ) is a local martingale in having continuous trajectories. Moreover, the following form of a (one-sided) Burkholder-Davis-Gundy inequality holds true: for any > a constant C < may be found such that E su t T ( / ψ s dw s CE ψ s γ ds) (.5) for all T > and all ψ Γ(U,). Several roofs of (.5) are available in the literature. A roof using the Itô formula is rovided in [4], Theorem 3.1, under an additional hyothesis uon smoothness of the norm. The constant C obtained in [4] deends on constants aearing in this extra hyothesis. If = L q, q, then C = O() as may be exected with reference to the emloyed method. (Note that a related result including a Burkholder-tye inequality for stochastic convolutions was roved by the same argument already in [5], Theorem 1.1. The estimate (.5) is also stated in [3], but a comlete roof does not seem to be given there.) Ondreját s roof ([18], 5) is based on good λ-inequalities and, according to [19], it again leads to a constant C = O(),. These O()-estimates are not sufficient for our uroses. The first roof of (.5), u to our knowledge, was given by E. Dettweiler in [1], Theorem 4. (cf. also [9], Theorem 1.4). His roof uses the same basic tool as we do in the resent aer a martingale version of Rosenthal s inequality is alied to a discrete aroximation of a stochastic integral in and so it is ossible that it may yield a constant C with a desired growth. However, Dettweiler works in a rather general setting and it seems difficult to figure out a recise value of the constant he obtained. Therefore, we aim at roviding an alternativeroofof(.5)leadingtoaconstantwithanotimalgrowthc = O( ). Acknowledgements. Thanks are due to Martin Ondreját who offered valuable comments on a reliminary version of this aer. 1. The main result. We shall rove the following refinement of the inequality (.5): Theorem 1.1. Let be a searable -smooth Banach sace, W a cylindrical Q-Wiener rocess on a real searable Hilbert sace Υ and U = RngQ 1/. There exists a constant Π <, deendent only on (, ), such that su t τ ψ(s)dw(s) Π ( τ 1/ ψ(s) γ ds) holds whenever >, τ is a stoing time and ψ is in Γ(U,). (1.1) The estimate (.5) for ],] follows from (1.1) by a standard alication of Lenglart s inequality. However, we shall use imlicitly the = case of (.5), since it is hidden in the construction of a stochastic integral in [18] we rely on. This might be avoided by a small modification of the roof, but this oint is unimortant for the aims of our aer. The roof of Theorem 1.1 is deferred to Section

5 . Alications to stochastic convolutions. Let(e At )beac -semigrou on, the stochastic convolution is defined by J t (ψ) = e A(t s) ψ(s)dw(s), t, for rogressively measurable rocesses ψ satisfying e A(t s) ψ(s) ds < for all t P-almost surely. (.1) γ Our goal is to rove exonential L -integrability of J(ψ) by combining Theorem 1.1 with Zygmund s extraolation theorem. This is rather straightforward for contractive and ositive (i.e., ositivity reserving)semigrousonanl q -sace, sincethenwemayassdirectlyfromstochastic integrals to stochastic convolutions using Fendler s theorem on isometric dilations (see [1], Section 4, for details). Assume that (M, M,µ) is a searable measure sace, q [, [, and = L q (µ), hence is a searable -smooth sace. Suose further that (e At ) is a strongly continuous semigrou of ositive contractions on L q (µ). Then [1], Proosition 4.1, together with Theorem 1.1 imly that su J t (ψ) Lq Π ( (µ) t T ψ(s) ) 1/ γ(u,l q (µ)) ds (.) holds for all >, T R + and every stochastic rocess ψ Γ(U,L q (µ)), Π being the constant from Theorem 1.1. Let us fix T >, denote by Dom(J) the cone of rogressively measurable rocesses in L (Ω;L ((,T);γ(U,L q (µ)))) and define a maing J : Dom(J) L (Ω), ψ su J t (ψ) Lq (µ). t T The maing J is sublinear, ositive homogeneous and, by (.), satisfies J(ψ) L (Ω) Π ψ L (Ω;L ((,T);γ(U,L q (µ)))), >. From Zygmund s theorem (see [6], Theorem II.4.41; the easy extension to vectorvalued functions may be found in [4], Theorem A.1) we get the following result: Theorem.1. Let (M, M,µ) be a searable measure sace, q [, [, T R +, and (e At ) a ositive contractive C -semigrou on L q (µ). There exist constants K < and λ > such that λ su e A(t s) ψ(s)dw(s) t T Eex L q (µ) ψ L K ((,T);γ(U,L q (µ))) 156

6 holds for every rogressively measurable ψ L (Ω;L ((,T);γ(U,L q (µ)))). In articular, we obtain an exonential tail estimate: { P su t T } ) e A(t s) ψ(s)dw(s) ε Kex ( λε L q (µ) κ (.3) for all ε > and every rocess ψ Γ(U,L q (µ)) satisfying ess su Ω ψ κ. γ(u,l q (µ)) In this form, the estimate (.3) seems to be new, but a closely related result was obtained in [5], Theorem 1.3, by a comletely different (and more comlicated) method based on the Itô formula. Analogously we may roceed if (e At ) is a semigrou whose (negative) generator has a bounded H -calculus of angle less than π since then again a suitable dilation theorem is available. Theorem.. Theorem.1 remains true under the following hyotheses: (M, M,µ) is a searable measure sace, q [, [, T R +, and (e At ) is a C - semigrou on L q (µ) with an infinitesimal generator A such that A has a bounded H -calculus of H -angle ω H ( A) < π. Theorems.1 and. are closely related but indeendent: A has a bounded H -calculus if A generates a ositive contractive C -semigrou on L q (µ), however, without an analyticity assumtion it may haen that ω H ( A) > π. (Cf. Corollaries 1.15 and 1.16 in [15].) The roof of Theorem. is deferred to the end of Section. If dilation results are not available, we have to resort to the factorization method (cf. [7], where this method is thoroughly dealt with), which rovides L -estimates of stochastic convolutions for > without any additional hyotheses on the semigrou. Instead of a Burkholder-tye estimate (.) one gets, however, only a weaker inequality E su t T e A(t s) ψ(s)dw(s) D E ψ(s) ds. (.4) γ If isahilbertsace,itwasshownin[4],theorem.3,thatd = O( ),, so from (.4) exonential L -integrability of stochastic convolutions may be easily inferred. The roof remains essentially the same for -smooth Banach saces, only the Burkholder-Davis-Gundy inequality must be relaced with Theorem 1.1. Let us denote by γ, the norm of the Banach sace L ([,T] Ω;γ(U,)), that is, ψ γ, = esssu ψ γ(u,). (s,ω) [,T] Ω 1561

7 Proceeding as in the roof of Theorem 1.1 in [4] we arrive at the following result: Theorem.3. Let (e At ) be a C -semigrou on a searable -smooth Banach sace and T R +. There exist constants K 1 < and λ 1 > such that Eex ( λ 1 ψ γ, su t T e A(t s) ψ(s)dw(s) for every rogressively measurable ψ L ([,T] Ω;γ(U,)). ) K 1 Due to Theorem.3, exonential tail estimates (.3) hold for all ε > and any rocess ψ Γ(U,) satisfying ess suψ γ(u,) κ. [,T] Ω The stochastic convolution J(ψ) is well defined under the assumtion (.1), but all results hitherto considered deended on a more stringent hyothesis ψ Γ(U, ). NowweshowthatTheorem.3remainsvalidforrocesseswhichareonly L(U,)- valued, rovided the semigrou (e At ) consists of radonifying oerators. Let us denote by L s (U,) the sace of all bounded linear oerators from U to endowed with the strong oerator toology and by ST, the sace of all rogressively measurable maings ψ : [,T] Ω L s (U,) such that [ψ], = esssu ψ(t,ω) L(U,) <. (t,ω) [,T] Ω (Let us recall that ψ is rogressively measurable as an L s (U,)-valued rocess if and only if ψu is a rogressively measurable -valued rocess for all u U. Owing to searability, ψ L(U,) is then a real-valued rogressively measurable function.) We shall denote by P γ the (Banach) oerator ideal of γ-summing oerators. (Recall that for Banach saces E and F the sace P γ (E,F) consists of all B L(E,F) such that su n 1 n E g k Bx k k=1 F < for all weakly -summable sequences {x i } in E and a sequence {g i } of indeendent standard Gaussian random variables. If E is a searable Hilbert sace and F does not contain a coy of c, then P γ (E,F) = γ(e,f), cf. e.g. [18], Proosition.4.) Mimicking the roof of Theorem 1.3 from [4] we obtain: Theorem.4. Let (e At ) be a C -semigrou on a searable -smooth Banach sace and T R +. Let there exist γ ],1[ such that t γ e At dt <. (.5) P γ (,) 156

8 Then there exist constants K < and λ > such that ( λ t Eex su e A(t s) ψ(s)dw(s) holds for all ψ S T,. [ψ], t T ) K We shall return to the assumtion (.5) in Section 4, but let us indicate here that the stochastic convolution J(ψ) is well defined under the hyotheses of Theorem.4. The sace being reflexive surely does not contain a coy of c, thus e A(t s) ψ s γ(u,) = e A(t s) ψ s Pγ (U,) e A(t s) Pγ (,) ψ s L(U,). If we suose that E ψ s L(U,) ds < for some, then J t (ψ) is well defined for almost all t [,T] and a standard alication of the factorization rocedure yields that J(ψ) has a continuous modification for > /γ. Processes ψ corresonding to diffusion terms in standard stochastic artial differential equations are often only L s (U,)-valued, but they may take values in γ(u,y) for some suersace Y of. This situation is covered by the following result, the roof of which is almost identical with that of the receding theorem. Theorem.5. Let (e At ) be a C -semigrou on a searable -smooth Banach sace and T R +. Suose that there exists a searable Banach sace Y such that is embedded in Y, the oerators e At, t >, may be extended to oerators in L(Y,) and t γ e At dt < (.6) L(Y,) for some γ ],1[. Then there exist constants K 3 < and λ 3 > such that λ 3 su e A(t s) ψ(s)dw(s) t T Eex ψ K 3 L ([,T] Ω;γ(U,Y)) holds for all rogressively measurable ψ L ([,T] Ω;γ(U,Y)). Remark.1. In an imortant articular case when there exists a γ-radonifying embedding j : U Y we may aly Theorem.5 to rocesses jψ with ψ ST,U obtaining ( λ 4 t ) Eex su e A(t s) ψ(s)dw(s) K 4 [ψ],u t T 1563

9 for some K 4 <, λ 4 > and any ψ S T,U. Other results from [4], as estimates over unbounded time intervals or estimates in norms of interolation saces between and Dom(A) may extended to -smooth saces in the same way. We shall not dwell on it, since no new ingredients besides Theorem 1.1 are needed. Proof of Theorem.. Suose that (M, M,µ) is a searable measure sace, q [, [, and (e At ) is a C -semigrou on L q (µ) with an infinitesimal generator A. Assume further that A has a bounded H -calculus of the H -angle ω H ( A) < π. According to Corollary 5.4 in [11] there exist a searable measure sace (N, N,ν), a closed subsace Y of L q (ν), an isomorhism J : L q (µ) Y, a bounded rojection P : L q (ν) Y and a bounded grou (B t ) on L q (ν) such that the diagram Y J L q (µ) L q (ν) e At L q (µ) B t L q (ν) P J Y commutes for all t, that is, Je At = PB t J, t. Let us set δ = J 1 L(Y,L q (µ)) su t PB t L(L q (ν),y), ζ = su B t J L(L q (µ),l q (ν)) t and take T R +, > and ψ Γ(U,L q (µ)) arbitrary. Since L q (ν) is a searable -smooth sace, Theorem 1.1 is alicable and we get su t T e A(t s) ψ(s)dw(s) Lq (µ) = su t J 1 Je A(t s) ψ(s)dw(s) t T L q (µ) J 1 su t Je A(t s) ψ(s)dw(s) t T Y = J 1 su t PB t s Jψ(s)dW(s) t T Y J 1 su t PB t L(L q (ν),y) B s Jψ(s)dW(s) t T δ su t B s Jψ(s)dW(s) t T L q (ν) δπ ( B s Jψ(s) ) 1/ ds. γ(u,l q (ν)) Lq (ν) 1564

10 Let {g n } be a sequence of indeendent standard Gaussian random variables and {k n } an orthonormal basis of U. By the definition of the norm in γ(u,l q (ν)) we have B s Jψ(s) γ(u,l q (ν)) = E g n B s Jψ(s)k n Whence we obtain that su t e A(t s) ψ(s)dw(s) t T n ζ E g n ψ(s)k n n Lq (µ) L q (µ) L q (ν) = E B s J n = ζ ψ(s) γ(u,l q (µ)). g n ψ(s)k n L q (ν) δζπ ( ) 1/ ψ(s) γ(u,l q (µ)) ds holds for every T >, > and ψ Γ(U,L q (µ)) and thus the extraolation theorem may be used exactly in the same manner as in the roof of Theorem.1. Q.E.D. 3. Proof of Theorem 1.1. We begin with a well known lemma on aroximation with simle rocesses. For reader s convenience, we shall sketch its roof since only the case = is treated exlicitly in [18]. Let us recall that a rocess in Γ(U,) is called simle, if it is of the form ψ(s,ω) = n 1 i= j=1 m 1 Fij (ω)a ij 1 ]ti,t i+1 ](s), s T, ω Ω, (3.1) where = { = t < t 1 < < t n = T} is a artition of the interval [,T], A ij Fin(Υ,), i =,...,n 1, j = 1,...,m, are finite dimensional oerators, and {F ij, 1 j m} F ti is a system of disjoint sets for each i =,...,n 1. Lemma 3.1. Let, T >, and let ψ Γ(U,) be such that ( / E ψ(s) γ ds) <. Then a sequence {ψ n } n 1 of simle rocesses may be found such that ( / lim E ψ n (s) ψ(s) γ ds) =. n Proof. Lemma3.4from[18]shownthatthereexistsasequenceF n : γ(u,) Fin(Υ,), n 1, of Borel maings satisfying F n (y) y γ ց as n for all y γ(u,), RngF n havingcardinalitynforalln 1. Inarticular, RngF 1 = {Z} for some Z Fin(Υ,). Since F n (y) y γ F 1 (y) y γ Z γ + y γ, y γ(u,), 1565

11 the dominated convergence theorem imlies that hence also F n (ψ(s)) ψ(s) γ ds n P-almost surely, ( / E F n (ψ(s)) ψ(s) γ ds). n Obviously, for any n fixed we can find rogressively measurable sets M 1,...,M r and oerators A 1,...,A r Fin(Υ,U) such that F n (ψ(s,ω)) = r A i 1 Mi (s,ω), s [,T], ω Ω. i=1 A standard result yields real-valued simle rocesses m i,k satisfying lim E 1 Mi (s) m i,k (s) ds = k and the roof of Lemma 3.1 may be comleted easily. Q.E.D. Proof of Theorem 1.1. As usual, it suffices to rove (1.1) only for finite deterministic times τ = T R +. Indeed, assume that the estimate su I t (ψ) Π ( 1/ ψ(s) γ ds) t T has been established for all T >. Then (1.1) holds for τ = + by the monotone convergence theorem and for any stoing time λ we have I t (ψ) = sui t λ (ψ) ( = sui t 1[,λ[ ψ ) t t su t λ (cf. [18], Lemma 3.6), which roves our claim. So, let us fix > and T > arbitrarily. First, we shall rove (1.1) for simle rocesses. Suose that ψ is a simle rocess of the form (3.1) for some artition = { = t < t 1 < < t n = T} of the interval [,T]. We shall denote by n( ) = max t i+1 t i i n 1 mesh of the artition. The estimate of the -th moment of I T (ψ) will be deduced from an estimate for discrete time martingales due to I. Pinelis. Towards this end, let us set f =, f k = k 1 i= j=1 m 1 Fij A ij (W(t i+1 ) W(t i )), 1 k n, 1566

12 and note that f n = I T (ψ). Further, set d k = f k f k 1, 1 k n, and f i = F ti, i n. Then (f k,f k ) is a martingale in, let us denote by ( n s n (f) = E ( d k ) ) 1/ fk 1 k=1 its conditional square function. According to [], Theorem 4.1, we have max f k Λ max d k +Λ s n (f) 1 k n 1 k n (3.) for a constant Λ deending only on (, ) (hence, in articular, indeendent of both and (f k )). With our choice of (f k ), let us estimate the terms on the right hand side of (3.). First, s n (f) ( n 1 = E E ( d k+1 ) ) / f k = E = E = E = E = E k= ( n 1 k= ( n 1 ( m E j=1 1 Fkj A kj (W(t k+1 ) W(t k )) F tk )) / ( m E 1 Fkj A kj (W(t k+1 ) W(t k )) )) / F tk k= ( n 1 k= j=1 ( n 1 k= j=1 ( n 1 k= j=1 j=1 m 1 Fkj E ( A kj (W(t k+1 ) W(t k )) ) ) / F tk m 1 Fkj E A kj (W(t k+1 ) W(t k )) ) / m ( / = E ψ(s) γ ds), 1 Fkj A kj Q 1/ ) / γ(υ,) (t k+1 t k ) where we used disjointness of F k1,...,f km in the third equality and indeendence of increments of A kj W in the fifth one. Secondly, max d ( ) k = E max d k+1 = E max 1 k n k n 1 n 1 E d k+1 k= k n 1 d k

13 Therefore, we get m E 1 Fkj A kj (W(t k+1 ) W(t k )) n 1 = = = k= n 1 k= j=1 n 1 k= j=1 α j=1 m E {1 Fkj A kj (W(t k+1 ) W(t k )) } m P(F kj )E A kj (W(t k+1 ) W(t k )) n 1 k= j=1 n 1 αn( ) 1 E = α n( ) 1 E I T (ψ) Λα n( ) 1 1 m (t k+1 t k ) / P(F kj ) A kj Q 1/ γ(υ,) k= j=1 ψ s γ ds m 1 Fkj A kj Q 1/ γ(υ,) (t k+1 t k ) ψ s γ ds. 1/ 1 +Λ ( 1/ ψ s γ ds). (3.3) Let σ = { = s < s 1 < < s N = T} be a artition of [,T] finer than, that is, there exist r(i) such that s r(i) = t i, i =,...,n. Let us define a simle function ψ by a formula ψ(s,ω) = N 1 r= j=1 m 1 Hrj (ω)ãrj1 ]sr,s r+1 ](s), s T, ω Ω, where H rj = F ij, Ã rj = A ij for r(i) r < r(i + 1), i =,...,n 1. Reeating the roof of (3.3) we obtain I T ( ψ) Λα n(σ) 1 1 However, we lainly have ψ s γ ds 1/ 1 +Λ ( 1/ ψ s γ ds). I T (ψ) = I T ( ψ), ψ s q γ ds = ψ q γ ds P-almost surely for each q by the very definition of ψ, so I T (ψ) Λα n(σ) 1 1 ψ s γ ds 1/ 1 +Λ ( 1/ ψ s γ ds). (3.4) 1568

14 For any ε > we may choose a artition σ with mesh n(σ) < ε, thus (3.4) imlies I T (ψ) Λ ( 1/ ψ s γ ds). The rocess ( I (ψ) ) is a continuous submartingale, hence from the Doob inequality we get ( su I t (ψ) I T (ψ) t T 1) Λ ( 1/ ψ γ ds). Setting Π = Λ we conclude that Theorem 1.1 holds true for simle rocesses ψ. Finally, take ψ Γ(U,) arbitrary. We may assume that ( / E ψ(s) γ ds) <, since otherwise there is nothing to rove. By Lemma 3.1 there exist simle rocesses ψ n, n N, such that consequently, ( / lim E ψ n (s) ψ(s) γ ds) =, (3.5) n ( / ( / lim E ψ n (s) γ ds) = E ψ(s) γ ds). n The construction of a stochastic integral using a Burkholder-tye inequality for =, asresentedin[18], 3, imliesthat, byassingtoasubsequenceifnecessary, we may obtain su t T I t (ψ n ) n from (3.5). Hence the Fatou lemma yields su I t (ψ) t T E su I t (ψ) liminf E su I t (ψ n ) t T n t T ( Π / liminf n E ( = Π / E ψ(s) γ ds P-almost surely ) / ψ n (s) γ ds ) / 1569

15 and the roof of Theorem 1.1 follows. Q.E.D. 4. A note on Theorems.4 and.5. Now we turn to a simle examle illustrating sufficient conditions for alicability of Theorems.4 and.5. Such conditions are essentially well known and may be found in literature on stochastic artial differential equations, nevertheless we hoe that our argument, based on the oerator ideals theory, may be of indeendent interest and may rovide some new insights. First, let us consider the assumtion (.5). If (e At ) is a bounded analytic semigrou on a Banach sace such that belongs to the resolvent set of its generator, then from [], Corollary 4.(b), we know that e At Pγ (,) = O(t θ ), t +, rovided that ( A) θ P γ (,). Hence to check (.5) in this case it suffices to find θ ], 1 [ such that ( A) θ P γ (,). Let G R n be a bounded domain with a smooth boundary, set = L q (G) for some q and Dom(A) = W 1,q (G) W,q (G), Au = u for u Dom(A). (4.1) As ( A) θ : Dom(( A) θ ) is an isomorhism and Dom(( A) θ ) W θ,q (G), we get (.5) if the embedding W θ,q (G) L q (G) is γ-summing. For u,v [1, ] and an arbitrary oerator ideal A set σ G (A,u,v) = inf { λ > ; ι A(W λ,u (G),L v (G)) }, where W λ,u are the usual Sobolev-Slobodeckii saces and ι : W λ,u (G) L v (G) is the embedding oerator (which is well defined for λ > n( 1 u 1 v ) ). H. König roved (see [1], Theorem.7.4) that 1 n σ G(A,u,v) = λ(a,u,v)+ 1 u 1 v, whenever A is a quasi-banach oerator ideal and λ(a,u,v) denotes the limit order of A (cf. [1], 14.4 for the definition of the limit order). By [16], Theorem 8, or [], Proosition 3.1, if u,v [, ] then λ(p γ,u,v) = 1 v. Therefore, the embedding W θ,q (G) L q (G) is γ-summing if ( θ > n λ(p γ,q,q)+ 1 q 1 ) = n q q. We see that, for our choice of and A, the assumtion (.5) is satisfied rovided q > n. In alications to stochastic artial differential equations, usually Υ = L (G) and ψ corresonds to a multilication oerator on U. Such oerators may belong 157

16 to L(U,L q (G)) if the covariance oerator Q is nice, e.g. RngQ 1/ L (G), but if the driving rocess is a standard cylindrical Wiener rocess one has U = Υ = L (G) and even bounded diffusion coefficients define multilication oerators only on L (G). In this case, one may try to aly Theorem.5 (or better, Remark.1). Let Y be the comletion of = L q (G) with resect to the norm ( A) θ for some θ ],1/[, then (.6) is satisfied. Theorem.5 will be alicable if L (G) is embedded into Y and the embedding oerator in γ-radonifying. For an oerator ideal A let us denote by A dual its dual ideal, that is By [1], Proosition , A dual (E,F) = { T L(E,F); T A(F,E ) }. λ(a dual,u,v) = λ(a,v,u ) holds for any oerator ideal A and u,v [1, ], where u, v denote the dual exonents. Since Y = W θ,q (G), the embedding L (G) Y is γ-summing rovided that the embedding W θ,q (G) L (G) belongs to P dual γ, which by König s theorem holds true if ( θ > n λ(p dual γ,q,)+ 1 q 1 ( = n λ(p γ,,q)+ 1 q 1 ) ( 1 = n q + 1 q ) 1 = n. ) Therefore, if = L q (G) for some q and A is defined by (4.1), Theorem.5 is alicable rovided n = 1. A note added in roof. This aer having been submitted, I got acquainted with a rerint [5] where the roblem of finding an otimal constant in (.5) is addressed in the articular case = L q from a very different oint of view. References [1] M. T. Barlow, M. Yor: Semi-martingale inequalities via the Garsia-Rodemich- Rumsey lemma, and alications to local times, J. Funct. Anal. 49(198), [] S. Blunck, L. Weis: Oerator theoretic roerties of semigrous in terms of their generators, Studia Math. 146(1), [3] Z. Brzeźniak: On stochastic convolution in Banach saces and alications, Stochastics Stochastics Re. 61(1997),

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