A new approach to stochastic evolution equations with adapted drift

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 non-adapted 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.     

A Lie-Trotter product formula for Ornstein-Uhlenbeck semigroups in infinite dimensions

We prove a Lie-Trotter product formula for the Ornstein--Uhlenbeck semigroup associated with the stochastic linear Cauchy problem Here A is the generator of a C₀-semigroup on a separable real Banach space E and is an E-valued Brownian motion.     

On the method of Da Prato and Debussche for the 3D stochastic Navier Stokes equations

3D stochastic Navier-Stokes equations with a suitable nondegenerate noise are considered. Following a method introduced by Da Prato and Debussche, it is proved that every Markov process associated to the equations has a Strong Feller like continuity property with respect to initial conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The resolvent average for positive semidefinite matrices

We define a new average - termed the resolvent average - for positive semidefinite matrices. For positive definite matrices, the resolvent average enjoys self-duality and it interpolates between the harmonic and the arithmetic averages, which it approaches when taking appropriate limits. We compare the resolvent average to the geometric mean. Some applications to matrix functions are also given.     

Invariant measure for the stochastic Ginzburg Landau equation

The existence of martingale solutions and stationary solutions of stochastic Ginzburg-Landau equations under general hypothesizes on the dimension, the non linear term and the added noise is investigated. With a few more assumptions, it is established that the transition semi-group is well defined and that the stationary martingale solution yields the existence of an invariant measure. Moreover this invariant measure is shown to be unique.     

Invariant measures for stochastic evolution equations of pure jump type

In this paper, we obtain a characterization of invariant measures of stochastic evolution equations and stochastic partial differential equations of pure jump type. As an application, it is shown that the equation has a unique invariant probability measure under some reasonable conditions.     

On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs

We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes with weights.     

Existence for a degenerate diffusion problem with a nonlinear operator

We study a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. We establish existence, uniqueness and regularity results. An application arising in the theory of water diffusion in porous media is discussed as an example.      

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

Stochastic partial differential equations in M-type 2 Banach spaces

We study abstract stochastic evolution equations in M-type 2 Banach spaces. Applications to stochastic partial differential equations inL ^p spaces withp2 are given. For example, solutions of such equations are Hölder continuous in the space variables.The author is an Alexander von Humboldt Stiftung fellow     

Regularity of invariant measures for a class of perturbed Ornstein-Uhlenbeck operators

In the framework of [5] we prove regularity of invariant measures for a class of Ornstein-Uhlenbeck operators perturbed by a drift which is not necessarily bounded or Lipschitz continuous. Regularity here means that is absolutely continuous with respect to the Gaussian invariant measure of the unperturbed operator with the square root of the Radon-Nikodym density in the corresponding Sobolev space of order 1.Partially supported by the International Science Foundation (Grant M 38000), the Russian Foundation of Fundamental Research (Grant 94-01-01556), and EC-Science Project SC1^*CT92-0784.Partially supported by the Italian National Project MURST Problemi nonlineari nell'AnalisiPartially supported by the DFG(SFB-256-Bonn, SFB-343-Bielefeld) and EC-Science Project SC1^*CT92-0784.     

2D backward stochastic Navier-Stokes equations with nonlinear forcing

The paper is concerned with the existence and uniqueness of a strong solution to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear forcing, driven by a Brownian motion. We use the spectral approximation and the truncation and variational techniques. The methodology features an interactive analysis on the basis of the regularity of the deterministic Navier-Stokes dynamics and the stochastic properties of the Itô-type diffusion processes.     

Structural reducts and the full implies strong problem

Let be a finite algebra. We show that if there exists a particular dualising alter ego that satisfies a weak form of injectivity, then the notions of full duality and strong duality for are equivalent. Presented by R. Willard. Received August 30, 2005; accepted in final form August 1, 2006.     

Gauss inequalities on ordered linear spaces

Markov inequalities on ordered linear spaces are tightened through the α-unimodality of the corresponding measures. Modality indices are studied for various induced measures, including the singular values of a random matrix and the periodogram of a time series. These tools support a detailed study of linear inference and the ordering of random matrices, to include fixed and random designs and probability bounds on their comparative efficiencies. Other applications include probability bounds on quadratic forms and of order statistics on Rⁿ, on periodograms in the analysis of time series, and on run-length distributions in multivariate statistical process control. Connections to other topics in applied probability and statistics are noted.     

Approximating a Feller Semigroup by Using the Yosida Approximation of the Symbol of its Generator

We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (T_t)_t≥0 then the Feller semigroups (T_t^(v))_t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (T_t)_t≥0 as ν →∞.     

Topo-canonical completions of closure algebras and Heyting algebras

We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces. Received January 20, 2006; accepted in final form September 12, 2006.     

On a backward Cauchy problem associated with continuous spectrum operator

The nonlinear backward Cauchy problem

u_t+Au(t)=f(u(t)),u(T)=φ,     

On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

We extend to infinite dimensions an explicit formula of Chill, Fašangová, Metafune, and Pallara for the optimal angle of analyticity of analytic Ornstein-Uhlenbeck semigroups. The main ingredient is an abstract representation of the Ornstein-Uhlenbeck operator in divergence form. The authors are supported by the ‘VIDI subsidie’ 639.032.201 of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281. Received: 28 June 2006 Revised: 5 January 2007     

Discrete wick calculus and stochastic functional equations

We develop Wick calculus over finite probability spaces and prove that there is a one-to-one correspondence between the solutions of Wick stochastic functional equations and the solutions of the deterministic functional equations obtained by turning off the noise. We also point out some possible applications to ordinary and partial stochastic differential equations.This research is supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den Norske Stats Oljeselskap a.s. (STATOIL).     

The endomorphism spectrum of an ordered set

Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n – 1) ^{n – 1} implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n ²+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)n}, it follows thatc ₁ n ²h(n)c ₂ n ⁿ⁺¹ for somec ₁ andc ₂. The lower bound disproves the conjecture thath(n)2n. It is shown that if |P| – 1 spec(P) thenP has a retract of size |P| – 1 but that for all there is a bipartite ordered set with spec(P) = {|P| – 2, |P| – 4, ...} which has no proper retract of size|P| – . The case of reflexive graphs is also treated.Partially supported by a grant from the NSERC.Partially supported by a grant from the NSERC.