Existence,uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Summary A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupR _t are obtained. We show thatR _t is a compactC ₀-semigroup in all Sobolev spacesW ^n,p which are built on its invariant measure . Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inL ^p() spaces and spacesW ^1,p . As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to . It is shown also that the density of this measure with respect to is inL ^p() for allp1.This work was done during the first author's stay at UNSW supported by ARC Grant 150.346 and the second author's stay at ód University supported by KBN Grant 2.1020.91.01     

Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

We study regularity properties for invariant measures of semilinear diffusions in a separable Hilbert space. Based on a pathwise estimate for the underlying stochastic convolution, we prove a priori estimates on such invariant measures. As an application, we combine such estimates with a new technique to prove the L¹-uniqueness of the induced Kolmogorov operator, defined on a space of cylindrical functions. Finally, examples of stochastic Burgers equations and thin-film growth models are given to illustrate our abstract result.     

Existence and uniqueness of invariant measures for a class of transition semigroups on Hilbert spaces

We prove a smoothing property and the irreducibility of transition semigroups corresponding to a class of semilinear stochastic equations on a separable Hilbert space H. Existence and uniqueness of invariant measures are discussed as well.     

On calibrated representations and the Plancherel Theorem for affine Hecke algebras

This paper has two main purposes. Firstly, we generalise Ram’s combinatorial construction of calibrated representations of the affine Hecke algebra to the multi-parameter case (including the non-reduced BC _n case). We then derive the Plancherel formulae for all rank 1 and rank 2 affine Hecke algebras, using our calibrated representations to construct all representations involved.     

On the index of invariant subspaces in Hilbert spaces of vector-valued analytic functions

Let H be a Hilbert space of analytic functions on the unit disc D with ‖Mz‖?1, where Mz denotes the operator of multiplication by the identity function on D. Under certain conditions on H it has been shown by Aleman, Richter and Sundberg that all invariant subspaces have index 1 if and only if for all f∈H, f?0 [A. Aleman, S. Richter, C. Sundberg, Analytic contractions and non-tangential limits, Trans. Amer. Math. Soc. 359 (7) (2007) 3369-3407]. We show that the natural counterpart to this statement in Hilbert spaces of Cn-valued analytic functions is false and prove a correct generalization of the theorem. In doing so we obtain new information on the boundary behavior of functions in such spaces, thereby improving the main result of [M. Carlsson, Boundary behavior in Hilbert spaces of vector-valued analytic functions, J. Funct. Anal. 247 (1) (2007)].     

Rigged Hilbert spaces and contractive families of Hilbert spaces

The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved that, when it exists, this rigged Hilbert space is the same as the canonical rigged Hilbert space associated to a family of closable operators in the central Hilbert space.     

Sub-Bergman Hilbert spaces

In this paper we analyze sub-Bergman Hilbert spaces in the unit disk associated with finite Blaschke products. We also analyze their analogues in weighted Bergman spaces.     

The Goldstine Theorem for asymmetric normed linear spaces

It is well known that if (X,q) is an asymmetric normed linear space, then the function qs defined on X by qs(x)=max{q(x),q(−x)}, is a norm on the linear space X. However, the lack of symmetry in the definition of the asymmetric norm q yields an algebraic asymmetry in the dual space of (X,q). This fact establishes a significant difference with the standard results on duality that hold in the case of locally convex spaces. In this paper we study some aspects of a reflexivity theory in the setting of asymmetric normed linear spaces. In particular, we obtain a version of the Goldstine Theorem to these spaces which is applied to prove, among other results, a characterization of reflexive asymmetric normed linear spaces.     

The comparison theorem for Hilbert spaces of entire functions

A generalization of the Sturm comparison theorem is obtained for formally self-adjoint ordinary differential operators of finite order given in canonical form. The result is stated within the vector theory of Hilbert spaces of entire functions when the coefficient space is a finite-dimensional vector space.     

Ultrametric spaces and related Hilbert spaces

We present simple proofs of the possibility of embedding ultrametric spaces in Hilbert spaces. The main part of the paper deals with ultrametric spaces that we call totally infinite spaces. Related Hilbert spaces, automorphisms of totally infinite spaces, and the corresponding linear operators are considered. Transplated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 223–237, August, 1997. Translated by V. E. Nazaikinskii     

A Hilbert Basis Theorem for Quantum Groups

A noncommutative version of the Hilbert basis theorem is usedto show that certain R-symmetric algebras S_R(V) are Noetherian.This result applies in particular to the coordinate ring ofquantum matrices A_R(V) associated with an R-matrix R operatingon the tensor square of a vector space V, to show that, undera natural set of hypotheses on R, the algebra A_R(V) is Noetherianand its augmentation ideal has a polynormal set of generators.As a corollary we deduce that these properties hold for thegeneric quantized function algebras R_q[G] over any field ofcharacteristic zero, for G an arbitrary connected, simply connected,semisimple group over C. That R_q[G] is Noetherian recovers aresult due to Joseph [10], with a different proof.1991 MathematicsSubject Classification 17B37, 16P40.