Uniqueness and absolute continuity of weak solutions for parabolic SPDE's

The paper presents necessary and sufficient conditions for the absolute continuity of measures generated by infinite-dimensional martingale problems. This result is applied to the study of the existence and uniqueness of weak solutions to nonlinear parabolic SPDE's. The paper also addresses the problem of stochastic integration with respect to a martingale in a quasi-complete locally convex topological vector space.This work was partially supported by NSF Grant # DMS-9002997 and ONR Grant # N00014-91-J-1526.     

Hypersurfaces of prescribed Gauß curvature in exterior domains

We prove an existence theorem for convex hypersurfaces of prescribed Gau? curvature in the complement of a compact set in Euclidean space which are close to a cone. Received: 23 February 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

REMPI as a tool for studies of OH radicals in catalytic reactions

2 +? O₂→H₂O on polycrystalline Pt foils has been studied by detection of desorbing OH radicals using the Resonance Enhanced Multiphoton Ionization, REMPI, technique. The measurements were performed at catalyst temperatures of 1000–1400 K and a total pressure below 10^-4 mbar. The studies of OH desorption by REMPI were achieved using a two-photon excitation D²Σ^-–X²Π (1–0), followed by one-photon ionization. The ions were detected in a time-of-flight mass spectrometer, TOF-MS, in order to avoid interference from non-resonantly ionized molecules. By applying TOF-MS, a simultaneous non-resonant ionization and detection of H₂, O₂ and H₂O was achieved. Recorded REMPI spectra were compared with spectra simulated using known molecular constants. The kinetics of the reaction derived from the measurements were compared with what was obtained in earlier LIF detection of OH, performed at higher total reactant pressure using the A–X transition. REMPI TOF-MS is shown to be a complement to LIF for reaction studies below 5×10^-4 mbar total pressure, where LIF is too insensitive for quantification. The reaction kinetics was found to be in agreement with a theoretic model and previous LIF studies. Received: 8 March 1996/Revised version: 4 October 1996     

Regenerative simulation of TES processes

Regenerative simulation has become a familiar and established tool for simulation-based estimation. However, many applications (e.g., traffic in high-speed communications networks) call for autocorrelated stochastic models to which traditional regenerative theory is not directly applicable. Consequently, extensions of regenerative simulation to dependent time series is increasingly gaining in theoretical and practical interest, with Markov chains constituting an important case. Fortunately, a regenerative structure can be identified in Harris-recurrent Markov chains with minor modification, and this structure can be exploited for standard regenerative estimation. In this paper we focus on a versatile class of Harris-recurrent Markov chains, called TES (Transform-Expand-Sample). TES processes can generate a variety of sample paths with arbitrary marginal distributions, and autocorrelation functions with a variety of functional forms (monotone, oscillating and alternating). A practical advantage of TES processes is that they can simultaneously capture the first and second order statistics of empirical sample paths (raw field measurements). Specifically, the TES modeling methodology can simultaneously match the empirical marginal distribution (histogram), as well as approximate the empirical autocorrelation function. We explicitly identify regenerative structures in TES processes and proceed to address efficiency and accuracy issues of prospective simulations. To show the efficacy of our approach, we report on a TES/M/1 case study. In this study, we used the likelihood ratio method to calculate the mean waiting time performance as a function of the regenerative structure and the intrinsic TES parameter controlling burstiness (degree of autocorrelation) in the arrival process. The score function method was used to estimate the corresponding sensitivity (gradient) with respect to the service rate. Finally, we demonstrated the importance of the particular regenerative structure selected in regard to the estimation efficiency and accuracy induced by the regeneration cycle length.     

Large deviation principle for stochastic evolution equations

Summary The large deviation principle obtained by Freidlin and Wentzell for measures associated with finite-dimensional diffusions is extended to measures given by stochastic evolution equations with non-additive random perturbations. The proof of the main result is adopted from the Priouret paper concerning finite-dimensional diffusions. Exponential tail estimates for infinite-dimensional stochastic convolutions are used as main tools.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Betti numbers of one-relator groups

We determine the L ²-Betti numbers of all one-relator groups and all surface-plus-one-relation groups. We also obtain some information about the L ²-cohomology of left-orderable groups, and deduce the non-L ² result that, in any left-orderable group of homological dimension one, all two-generator subgroups are free. Warren Dicks was Funded by the DGI (Spain) through Project BFM2003-06613.     

Dynkin's isomorphism theorem and the Ray-Knight theorems

Summary In the case of diffusions, we show that the isomorphism theorem of Dynkin and the Ray-Knight theorems can be derived from each other. Our proof uses additivity properties of squared Bessel processes and an absolute continuity relation between squared Bessel processes of dimensions one and three.Research supported in part by Air Force Office of Scientific Research (USAFOSR 89-0261)This work was carried out while visiting the Technion