On the trace embedding and its applications to evolution equations

In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity...

Singular stochastic integral operators

We introduce Calderón-Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove L ^p-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the...

Cylindrical continuous martingales and stochastic integration in infinite dimensions

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration...

On the ℓs-boundedness of a family of integral operators

In this paper we prove an ℓs-boundedness result for integral operators with operator-valued kernels. The proofs are based on extrapolation techniques with weights due to Rubio de Francia. The results will be applied by the first and third author in a subsequent paper where a new approach to maximal Lp-regularity for parabolic problems with time...

Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time

In this paper we study maximal Lp-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the Lp-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time...

Ito's formula in UMD Banach spaces and regularity of solution of the Zakai equation

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...

Continuous local martingales and stochastic integration in UMD Banach spaces

Recently, van Neerven, Weis and the author, constructed a theory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integrator is an arbitrary real-valued continuous local martingale. We...

Conditions for stochastic integrability in UMD Banach spaces

Stochastic integration in UMD Banach spaces

Stochastic integration in Banach spaces and applications to parabolic evolution equations

Stochastic partial differential equations (SPDEs) of evolution type are usually modelled as ordinary stochastic differential equations (SDEs) in an infinite-dimensional state space. In many examples such as the stochastic heat and wave equation, this viewpoint may lead to existence and uniqueness results and regularity properties. To model the...

Randomized UMD Banach spaces and decoupling

In this paper we prove the equivalence of decoupling inequalities for stochastic integrals and one-sided randomized versions of the UMD property of a Banach space as introduced by Garling.