On principal T-bands in a Banach lattice

It is shown that for every positive order continuous Riesz operatorT, defined on an order complete complex Banach latticeE which is separated by its Köthe dual, there exists a Frobenius decomposition ofE into a countable number of disjoint principalT-bands and a band on whichT is quasi-nilpotent. A basis for the generalized eigenspace ofT pertaining to its maximal eigenvalue is constructed and the positivity properties of its elements are studied. The distinguished eigenvalues ofT are characterized and it is also shown that the theory ofT-bands is symmetric with respect to the duality which exists betweenE and its Köthe dual. This generalizes aspects of work done by H.D. Victory and R.-J. Jang-Lewis.     

Diffusions with singular drift related to wave functions

Summary Schrödinger equations are equivalent to pairs of mutually time-reversed non-linear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space-time Sobolev space.By means of Nagasawa's multiplicative functionalN _s ^t , a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived.The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solutions of a Feynman-Kac integral equation with a locally integrable potential function.     

Convergence rates in the strong laws of asymptotically negatively associated random fields

In this paper, a notion of negative side p-mixing (p -mixing) which can be regardedas asymptotic negative association is defined, and some Rosenthal type inequalities for p -mix-ing random fields are established. The complete convergence and almost sure summability onthe convergence rates with respect to the strong law of large numbers are also discussed for p--mixing random fields. The results obtained extend those for negatively associated sequences andp“ -mixing random fields.     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

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     

Hypersurfaces of prescribed mean curvature in central projection-II

This work, which is a direct continuation of a preceding paper, is dedicated to interior regularity results with respect to a singular variational integral with linear growth. The Dirichlet Problem associated with the corresponding variational problem describes surfaces of prescribed mean curvature that admits of a central projection onto an open domain of the unit sphere Sⁿ. Our results are accomplished by using methods from geometric measure theory together with certain L bounds that we shall prove here as well.Received: 11 February 2003     

<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>-calculus for Products of Non-Commuting operators

It is shown that the product of two sectorial operators A and B admits a bounded H^∞-calculus on a Banach space X provided suitable commutator estimates and Kalton-Weis type assumptions on A and B are satisfied.     

Continuity properties of decomposable probability measures on euclidean spaces

It is shown that every full e^A decomposable probability measure on R^k, where A is a linear operator all of whose eigenvalues have negative real part, is either absolutely continuous with respect to Lebesgue measure or continuous singular with respect to Lebesgue measure. This result is used to characterize the continuity properties of random variables that are limits of solutions of certain stochastic difference equations.     

Diffusion-limited aggregation on a tree

Summary. We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to α ⁻ⁿ , where α < 1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when α≧ 1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that “fingering” occurs. Received: 16 August 1994 / In revised form: 18 March 1996     

Sur la mesure de sortie du super mouvement brownien

Summary We study some properties of the exit measure of super Brownian motion from a smooth domainD inR ^d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of u=u ² inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of u=u ² inD that tends to on an open subsetO of D and to 0 on the complement in D of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

     

Potential Theory of Geometric Stable Processes

In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(|x|^d log ²|x|), while the Lévy density behaves like 1/|x|^d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes. The research of this author is supported in part by MZT grant 0037118 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by a joint US-Croatia grant INT 0302167. The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

On the hellinger square integral with respect to an operator valued measure and stationary processes

A construction of the Hellinger square integral with respect to a semispectral measure in a Banach space B is given. It is proved that the space of values of a B-valued stationary stochastic process is unitarily isomorphic to the space of all B¹-valued measures that are Hellinger square integrable with respect to the spectral measure of the process. Some applications of the above theorem in the prediction theory (especially to interpolation problem) are also considered.     

Strong approximation of semimartingales and statistical processes

Summary As an application of general convergence results for semimartingales, exposed in their book Limit Theorems for Stochastic Processes, Jacod and Shiryaev obtained a fundamental result on the convergence of likelihood ratio processes to a Gaussian limit. We strengthen this result in a quantitative sense and show that versions of the likelihood ratio processes can be defined on the space of the limiting experiment such that we get pathwise almost sure approximations with respect to the uniform metric. The approximations are considered under both sequences of measures, the hypothesisP ⁿ and the alternative . A consequence is e.g. an estimate for the speed of convergence in the Prohorov metric. New approximation techniques for stochastic processes are developed.This article was processed by the author using the L^AT_EX style filepljourIm from Springer-Verlag.     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Nonlinear transformations of the canonical gauss measure on Hilbert space and absolute continuity

The papers of R. Ramer and S. Kusuoka investigate conditions under which the probability measure induced by a nonlinear transformation on abstract Wiener space(,H,B) is absolutely continuous with respect to the abstract Wiener measure. These conditions reveal the importance of the underlying Hilbert spaceH but involve the spaceB in an essential way. The present paper gives conditions solely based onH and takes as its starting point, a nonlinear transformationT=I+F onH. New sufficient conditions for absolute continuity are given which do not seem easily comparable with those of Kusuoka or Ramer but are more general than those of Buckdahn and Enchev. The Ramer-Itô integral occurring in the expression for the Radon-Nikodym derivative is studied in some detail and, in the general context of white noise theory it is shown to be an anticipative stochastic integral which, under a stronger condition on the weak Gateaux derivative of F is directly related to the Ogawa integral.Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL 03 92 G 0008.     

Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyers type

We have obtained the following limit theorem: if a sequence of RCLL supersolutions of a backward stochastic differential equations (BSDE) converges monotonically up to (y _t ) with E[sup _t |y _t |²] < ∞, then (y _t ) itself is a RCLL supersolution of the same BSDE (Theorem 2.4 and 3.6). We apply this result to the following two problems: 1) nonlinear Doob–Meyer Decomposition Theorem. 2) the smallest supersolution of a BSDE with constraints on the solution (y, z). The constraints may be non convex with respect to (y, z) and may be only measurable with respect to the time variable t. this result may be applied to the pricing of hedging contingent claims with constrained portfolios and/or wealth processes. Received: 3 June 1997 / Revised version: 18 January 1998     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

The generalized Banach match-box problem: Application in disc storage management

A computer system manages disc storage of finite capacity c blocks. This storage must be divided among N files in such a way that the expected number of disc accesses accomplished until the necessary reorganization is maximized. Each access to the disc appends a record of a fixed length to the ith file with probability p _i (i=1,h., N). The reorganization is needed when the chosen file has run out of space. It is shown that the above problem is a generalization of Banach's match-box problem known from the probability theory. A detailed separate analysis for the N=2 case and for the multivariate case is performed and some approximate results for large c are given.     

Random sets of finite perimeter

An approach to modelling random sets with locally finite perimeter as random elements in the corresponding subspace of L¹ functions is suggested. A Crofton formula for flat sections of the perimeter is shown. Finally, random processes of particles with finite perimeter are introduced and it is shown that their union sets are random sets with locally finite perimeter.