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

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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).

In this article we prove a maximal L p -regularity result for stochastic convolutions, which extends Krylov's basic mixed L p (L q )-inequality for the Laplace operator on R d to large classes of elliptic operators, both on R d and on bounded domains in R d with various boundary conditions. Our method of proof is based on McIntosh's H ∞functional calculus, R-boundedness techniques and sharp L p (L q )square function estimates for stochastic integrals in L q -spaces. Under an additional invertibility assumption on A, a maximal space-time L p -regularity result is obtained as well.

We consider function spaces of Besov, Triebel-Lizorkin, Bessel-potential and Sobolev type on R d , equipped with power weights w(x) = |x| γ , γ > −d. We prove two-weight Sobolev embeddings for these spaces. Moreover, we precisely characterize for which parameters the embeddings hold. The proofs are presented in such a way that they also hold for vector-valued functions.2000 Mathematics Subject Classification. 46E35, 46E40.

Abstract. In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of nonadapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.

Abstract. We prove maximal L p -regularity for the stochastic evolution equation dU (t) + AU (t) dt = F (t, U (t)) dt + B(t, U (t)) dW H (t), t ∈ [0, T ], U (0) = u 0 , under the assumption that A is a sectorial operator with a bounded H ∞ -calculus of angle less than 1. Introduction. Maximal L p -regularity techniques have been pivotal in much of the recent progress in the theory of parabolic evolution equations (see [2,22,25,54,76,86] and the references therein). Among other things, such techniques provide a systematic and powerful tool to study nonlinear and time-dependent parabolic problems.For stochastic parabolic evolution equations, maximal L p -regularity results have been obtained previously by Krylov for second order problems on R d [44,46,47,48,49], by Kim for second order problems on bounded domains in R d [43], and by Mikulevicius and Rozovskii for Navier-Stokes equations [63]. A systematic theory of maximal L p -regularity for stochastic evolution equations, however, based on abstract operator-theoretic properties of the operators governing the equation, has yet to be developed. A first step towards such a theory has been taken in our recent paper [68], where it was shown that if A is a sectorial operator with a bounded H ∞ -calculus of angle <