Randomized UMD Banach spaces and decoupling inequalities for stochastic integrals

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.

     

Some inequalities for martingales and stochastic convolutions

We give a stopped Doob inequality for a right continuous martingale in Hilbert space,, Using this we obtain inequalities for p-th moments with 0 < p < in terms of the Meyer process and the quadratic variation of the pure jump part. We also consider the convolution of a contraction type semigroup and a right continuous martingale and obtain inequalities similar to those of a martingale     

Stochastic quasilinear evolution equations in UMD Banach spaces

In this work we prove the existence and uniqueness up to a stopping time for the stochastic counterpart of Tosio Kato's quasilinear evolutions in UMD Banach spaces. These class of evolutions are known to cover a large class of physically important nonlinear partial differential equations. Existence of a unique maximal solution as well as an estimate on the probability of positivity of stopping time is obtained. An example of stochastic Euler and Navier–Stokes equation is also given as an application of abstract theory to concrete models.     

On the central limit theorem in Banach spaces

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Stochastic integration for operator valued processes on hilbert spaces and on nuclear spaces

The representation of a nuclear space valued square integrable martingale by means of another nuclear space valued square integrable martingale is given in terms of stochastic inegrals of operator valued processes. The construction of the stochastic integral goes through that of operator valued processes on Hilbert spaces. A new approach is given for the Hilbertian case, so that only the integration of Hilbert-Schmidt operator valued processes is needed to represent square integrable martingales     

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.     

Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces

We define stochastic integrals of Banach valued random functions w.r.t. compensated Poisson random measures. Different notions of stochastic integrals are introduced and sufficient conditions for their existence are established. These generalize, for the case where integration is performed w.r.t. compensated Poisson random measures, the notion of stochastic integrals of real valued random functions introduced in Ikeda and Watanabe (1989) [Stochastic Differential Equations and Diffusion Processes (second edition), North-Holland Mathematical Library, Vol. 24, North Holland Publishing Company, Amsterdam/Oxford/New York.], (in a different way) in Bensoussan and Lions (1982) [Contróle impulsionnel et inquations quasi variationnelles. (French) [Impulse control and quasivariational inequalities] Méthodes Mathématiques de l'Informatique [Mathematical Methods of Information Science], Vol. 11. (Gauthier-Villars, Paris), and Skorohod, A.V. (1965) [Studies in the theory of random processes (Addison-Wesley Publishing Company, Inc, Reading, MA), Translated from the Russian by Scripta Technica, Inc. ], to the case of Banach valued random functions. The relation between these two different notions of stochastic integrals is also discussed here.     

UMD Banach spaces and square functions associated with heat semigroups for Schrödinger,Hermite and Laguerre operators

In this paper we define square functions (also called Littlewood‐Paley‐Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) ‐boundedness properties for the square functions to our Banach valued setting by using γ‐radonifying operators. We also prove that these ‐boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.     

Stochastic evolution equations in UMD Banach spaces

We discuss existence, uniqueness, and space-time Hölder regularity for solutions of the parabolic stochastic evolution equation

     

Banach空间二阶周期边值问题解的存在性

通过构造格林函数,借助Banach空间中不连续增算子的不动点定理,研究了一类Banach空间中二阶微分方程周期边值问题解的存在性.     

Central limit theorem and weak law of large numbers with rates for martingales in Banach spaces

This paper is concerned with large-$\text{O}$ error estimates concerning convergence in distribution as well as norm convergence for Banach space-valued martingale difference sequences. Indeed, two general limit theorems equipped with rates of convergence for such difference sequences are established. Applications of these lead to the central limit theorem and the weak law of large numbers with rates for Banach space-valued martingales.     

Stochastic Integration of Operator-Valued Functions with Respect to Banach Space-Valued Brownian Motion

Let E be a real Banach space with property (α) and let W _Γ be an E-valued Brownian motion with distribution Γ. We show that a function is stochastically integrable with respect to W _Γ if and only if Γ-almost all orbits Ψx are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Γ-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. 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 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).     

Complementation and decompositions in some weakly Lindelöf Banach spaces

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c₀(Γ) or (2) for every c₀(Γ)⊆X has a complemented c₀(E) for an uncountable E⊆Γ or (3) has a decomposition X=A⊕B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P-ideal dichotomy, for every c₀(Γ)⊆C(K) there is a complemented c₀(E) for an uncountable E⊆Γ, which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C(K) for K of height ω+1 where every operator is of the form cI+S for c∈R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K).     

Separable anisotropic differential operators in weighted abstract spaces and applications

In this paper operator-valued multiplier theorems in Banach-valued weighted Lp spaces are studied. Also weighted Sobolev-Lions type spaces are discussed when E₀, E are two Banach spaces and E₀ is continuously and densely embedded on E. Several conditions are found that ensure the continuity of the embedding operators that are optimally regular in these spaces in terms of interpolations of E₀. These results permit us to show the separability of the anisotropic differential operators in an E-valued weighted Lp space. By using these results the maximal regularity of infinite systems of quasi elliptic partial differential equations are established.     

Arithmetic means and invariance principles in stochastic approximation

Summation for the sequence (U _n ) _n N of random variables with values in a real separable Banach space recursively defined by