In this paper we construct a theory of stochastic integration of processes with values in $\mathcal{L}(H,E)$, where $H$ is a separable Hilbert space and $E$ is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an $H$-cylindrical Brownian motion. Our approach is based on a two-sided $L^p$-decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of $\mathcal{L}(H,E)$-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the It\^{o} isometry, the Burkholder--Davis--Gundy inequalities, and the representation theorem for Brownian martingales.Comment: Published at http://dx.doi.org/10.1214/009117906000001006 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equationwhere A generates an analytic C 0 -semigroup on a UMD Banach space E and W H is a cylindrical Brownian motion with values in a Hilbert space H . We prove that if the mappings F : [0, T ] × E → E and B : [0, T ] × E → L(H, E) satisfy suitable Lipschitz conditions and u 0 is F 0 -measurable and bounded, then this problem has a unique mild solution, which has trajectories in C λ ([0, T ]; D((−A) θ ))) provided λ 0 and θ 0 satisfy λ + θ < 1 2 . Various extensions are given and the results are applied to parabolic stochastic partial differential equations. addresses: J.M.A.M.vanNeerven@tudelft.nl (J.M.A.M. van Neerven), mark@profsonline.nl (M.C. Veraar), Lutz.Weis@math.uni-karlsruhe.de (L. Weis).

Abstract. Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ : (0, T ) → L(H, E) with respect to a cylindrical Wiener process {W H (t)} t∈ [0,T ] . The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution.In this paper we construct a theory of stochastic integration of operatorvalued functions with respect to a cylindrical Wiener process. The range space of the operators is allowed to be an arbitrary real Banach space E. A stochastic integral of this type can be used for solving the linear stochastic Cauchy problemHere, A is the infinitesimal generator of a strongly continuous semigroup {S(t)} t≥0 of bounded linear operators on E, the operator B is a bounded linear operator from a separable real Hilbert space H into E, and {W H (t)} t∈ [0,T ] 2000 Mathematics Subject Classification: Primary 60H05; Secondary 28C20, 35R15, 47D06, 60H15.Key words and phrases: stochastic integration in Banach spaces, Pettis integral, Gaussian covariance operator, Gaussian series, cylindrical noise, convergence theorems, stochastic evolution equations.The first named author gratefully acknowledges the support by a 'VIDI subsidie' in the 'Vernieuwingsimpuls' programme of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281. The second named author was supported by grants from the Volkswagenstiftung (I/78593) and the Deutsche Forschungsgemeinschaft (We 2847/1-1).[131] It is well known that the integral on the right hand side can be interpreted as an Itô stochastic integral if E is a Hilbert space. A comprehensive theory of abstract stochastic differential equations in Hilbert spaces is presented in the monograph by Da Prato and Zabczyk [5]. More generally the integral can be defined for spaces E with martingale type 2. This has been worked out by Brzeźniak [2]. Examples of martingale type 2 spaces are Hilbert spaces and the Lebesgue spaces L p (µ) with p ∈ [2, ∞).In both settings, the integral is defined for step functions first, and then for general functions by a limiting argument. Such an argument depends on the availability of a priori estimates for the integrals of the approximating step functions, and the martingale type 2 property is precisely designed to provide such estimates.Without special assumptions on the geometry of the underl...

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Itô formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation. (Z. Brzeźniak), j.m.a.m.vanneerven@tudelft.nl (J.M.A.M. van Neerven), m.c.veraar@tudelft.nl (M.C. Veraar), lutz.weis@mathematik.uni-karlsruhe.de (L. Weis).

Let E be a separable real Banach space and let Q # L(E*, E) be positive and symmetric. Let S=[S(t)] t 0 be a C 0 -semigroup on E We study the relations between the reproducing kernel Hilbert spaces associated with the operators Q t := t 0 S(s) QS*(s) ds. Under the assumption that these operators are the covariances of centered Gaussian measures + t on E, we also study equivalence + t t+ s for different values of s and t, and we calculate their Radon Nikodym derivatives. Academic Press INTRODUCTIONIn this paper we investigate the reproducing kernel Hilbert spaces and Gaussian measures associated with a nonsymmetric Ornstein Uhlenbeck semigroup on a separable real Banach space E. This study is usually carried out in a Hilbert space setting, and one of the motivations of this paper was to see to what extent the theory can be extended to the Banach space setting.The main difference between the Banach space and the Hilbert space situation is that the covariance operator of a Gaussian measure on a Banach space E is a positive symmetric operator Q (the precise definitions are given in Section 1) from the dual E* into E, rather than an operator on E. Thus, in contrast to the Hilbert space situation, it is no longer possible to represent the reproducing kernel Hilbert space H associated with Q as H=Im Q 1Â2. When working in a Banach space setting, any reference to the operator Q 1Â2 therefore has to be avoided. This turns out article no. FU973237 495

We study space-time Hölder regularity of the solutions of the linear stochastic Cauchy problem dU t = AU t dt + dW t t ∈ 0 T U 0 = 0 where A is the generator of an analytic semigroup on a Banach space E and W is an E-valued Brownian motion. When −A admits a -bounded H -calculus, the solution is shown to have maximal regularity in the sense that U has a modification with paths in L 2 0 T −A 1 2. The results are applied to prove optimal and maximal Hölder space-time regularity for second-order parabolic stochastic partial differential equations.

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Let E be a real Banach space with property (α) and let W be an E-valued Brownian motion with distribution . We show that a function :is stochastically integrable with respect to W if and only if -almost all orbits x are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of -measurable linear extensions of γ -radonifying operators with values in spaces of γ -radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion.