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).     

Extreme operator-valued continuous maps

Let ℒ(H) denote the space of operators on a Hilbert spaceH. We show that the extreme points of the unit ball of the space of continuous functionsC(K, ℒ(H)) (K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dimH<∞ and cardK<∞, (b) exposed if and only ifH is separable andK carries a strictly positive measure.     

On a class of hyperbolic 3-orbifolds of small volume and small heegaard genus associated to 2-bridge links

We consider a class of hyperbolic 3-orbifoldsO(α/β); the underlying topological space of such an orbifold is the 3-sphere and the singular set is obtained by adding the two standard (upper and lower) unknotting tunnels to a 2-bridge linkL(α/β) (and associating branching order two to both unknotting tunnels). These 3-orbifolds are extremal with respect to the notion of Heegaard genus or Heegaard number of 3-orbifolds; it is to be expected that they are also extremal with respect to the volume, that is the smallest volume hyperbolic 3-orbifolds should belong to this or some closely related class. We show that an orbifoldO(α/β) has a uniqueD ₂-covering by an orbifoldℒ _n(α/β) wose space is the 3-sphere and whose singular set is the same 2-bridge linkL(α/β) used for the construction ofO(α/β); moreoverO(α/β) is hyperbolic if and only ifℒ _n(α/β) is hyperbolic. As the volumes of the orbifoldsℒ _n(α/β) are known resp. can be computed, this allows to compute the volumes of the orbifoldsO(α/β). The problem of computation of volumes remains open for some closely related classes of 3-orbifolds which are also extremal with respect to the Heegaard genus (for example associating a branching order bigger than two to one or both unknotting tunnels).     

Some generalizations on the boundedness of bilinear operators

For 0<p<∞, let H^p(R ⁿ) denote the Lebesgue space for p>1 and the Hardy space for p ≤1. In this paper, the authors study H^p(R ⁿ)×H^q(R ⁿ)→H^r(R ⁿ) mapping properties of bilinear operators given by finite sums of the products of the standard fractional integrals or the standard fractional integral with the Calderón-Zygmund operator. The authors prove that such mapping properties hold if and only if these operators satisfy certain cancellation conditions. Supported by the NNSF and the National Education Comittee of China.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Onsager-Machlup functional for the fractional Brownian motion

 Let X be the solution of the stochastic differential equation where B ^H is a fractional Brownian motion with Hurst parameter H. In this paper we compute the Onsager-Machlup functional of X for the supremum norm and H?lder norms of order β with in the case and for H?lder norms of order β with when . Received: 16 July 2001 / Revised version: 12 March 2002 / Published online: 10 September 2002     

The Order Properties of r-compact Operators on Banach Lattices

We present here that F（E,F）, the space of all r-compact operators from E into F, is a generalised sublattice of L^r（E, F） for arbitary Banach lattices E and F, and that the characterization of the regular norm on F（E, F） is order continuous. Some conditions for F（E, F） to be a KB-space or a band in .L（E, F） are also provided.     

Boundary-Crossing Identities for Diffusions Having the Time-Inversion Property

We review and study a one-parameter family of functional transformations, denoted by (S ^(β)) _β∈ℝ, which, in the case β<0, provides a path realization of bridges associated to the family of diffusion processes enjoying the time-inversion property. This family includes Brownian motions, Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary-crossing times for these diffusions over a given function f to those over the image of f by the mapping S ^(β), for some fixed β∈ℝ. We give some new examples of boundary-crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.     

Integration questions related to fractional Brownian motion

Let {B _H (u)} _{u ∈ℝ} be a fractional Brownian motion (fBm) with index H∈(0, 1) and (B _H ) be the closure in L ²(Ω) of the span Sp(B _H ) of the increments of fBm B _H . It is well-known that, when B _H = B _1/2 is the usual Brownian motion (Bm), an element X∈(B _1/2) can be characterized by a unique function f _X ∈L ²(ℝ), in which case one writes X in an integral form as X = ∫_ℝ f _X (u)dB _1/2(u). From a different, though equivalent, perspective, the space L ²(ℝ) forms a class of integrands for the integral on the real line with respect to Bm B _1/2. In this work we explore whether a similar characterization of elements of (B _H ) can be obtained when H∈ (0, 1/2) or H∈ (1/2, 1). Since it is natural to define the integral of an elementary function f = ∑ _{k =1} ⁿ f _k 1 _[uk,uk+1) by ∑ _{k =1} ⁿ f _k (B _H (u _{k +1}) −B _H (u _k )), we want the spaces of integrands to contain elementary functions. These classes of integrands are inner product spaces. If the space of integrands is not complete, then it characterizes only a strict subset of (B _H ). When 0<H<1/2, by using the moving average representation of fBm B _H , we construct a complete space of integrands. When 1/2<H<1, however, an analogous construction leads to a space of integrands which is not complete. When 0<H<1/2 or 1/2<H<1, we also consider a number of other spaces of integrands. While smaller and henceincomplete, they form a natural choice and are convenient to workwith. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm. Received: 9 August 1999 / Revised version: 10 January 2000 / Published online: 18 September 2000     

A Decomposition Theorem for Compact-Valued Henstock Integral

We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.     

A fragmentation process connected to Brownian motion

Let (B _s , s≥ 0) be a standard Brownian motion and T ₁ its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ ^(t) of the Brownian motion with drift t, B ^(t) _s = B _s + ts, and the decreasing sequence F(t) = (F ₁(t), F ₂(t), …) of lengths of the intervals of the random partition of [0, T ₁] induced by ℒ ^(t) . The main result of this work is that (F(t), t≥ 0) is a fragmentation process, in the sense that for 0 ≤t < t′, F(t′) is obtained from F(t) by breaking randomly into pieces each component of F(t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3]. Received: 19 February 1999 / Revised version: 17 September 1999 /?Published online: 31 May 2000     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

We study the asymptotic behaviour of solutions of the stochastic abstract Cauchy problem $$ \left\{ {\begin{array}{*{20}l} {dU\left( t \right) = AU\left( t \right)dt + BdW_H \left( t \right),\quad t \geqslant 0,} \hfill\ {U\left( 0 \right) = 0,} \hfill\ \end{array}} \right. $$ where A is the generator of a C₀-semigroup on a Banach space E, W_H is a cylindrical Brownian motion over a separable Hilbert space H, and $$ B \in \user1{\mathscr L}\left( {H,E} \right) $$ is a bounded operator. Assuming the existence of a solution U, we prove that a unique invariant measure exists if the resolvent R(λ, A) is R-bounded in the right half-plane {Reλ > 0}, and that conversely the existence of an invariant measure implies the R-boundedness of R(λ, A)B in every half-plane properly contained in {Re λ > 0}. We study various abscissae related to the above problem and show, among other things, that the abscissa of R-boundedness of the resolvent of A coincides with the abscissa corresponding to the existence of invariant measures for all γ -radonifying operators B provided the latter abscissa is finite. For Hilbert spaces E this result reduces to the Gearhart-Herbst-Prüss theorem. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?^*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L ^p (E), where 1 < p <  +∞. We study the same problem for the operator Δ _M  + V in L ^p (M), where 1 < p < ∞, Δ _M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.     

Ergodicity of cylinder flows arising from irregularities of distribution

LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1 _[0,β] (x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals. This paper was prepared while I was very graciously hosted by the Centro de Investigacion y Estudios Avanzados, Mexico City.     

Continuity and equicontinuity of semigroups on norming dual pairs

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C _b (E) which is continuous with respect to the strict topology β ₀ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β ₀-continuous semigroups on C _b (E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β ₀-continuous.     

Chaos decomposition of multiple fractional integrals and applications

Chaos decomposition of multiple integrals with respect to fractional Brownian motion (with H > 1/2) is given. Conversely the chaos components are expressed in terms of the multiple fractional integrals. Tensor product integrals are introduced and series expansions in those are considered. Strong laws for fractional Brownian motion are proved as an application of multiple fractional integrals. Received: 22 September 1998 / Revised version: 20 April 1999     

The chromatic number of the product of two 4-chromatic graphs is 4

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs. This research was supported by NSERC grant A7213     

Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals

Let dγ(x) ≡ π ^−n/2 e ^{−|x| 2} dx for all x ∈ ℝ ⁿ be the Gauss measure on ℝ ⁿ . In this paper, the authors establish the characterizations of the space BMO(γ) of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order β is bounded from L ^p (γ) to L ^p/(1−pβ)(γ), or from the Hardy space H ¹(γ) of Mauceri and Meda to L ^1/(1−β)(γ) or from L ^1/β (γ) to BMO(γ), where β ∈ (0, 1) and p ∈ (1, 1/β).