Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On the Strongly Supermedian Functions and Kernels

We answer to a question of P.J. Fitzsimmons and R.K. Getoor concerning the characterization of a strongly supermedian function by the property of being universally supermedian. Based on the Revuz correspondence we also prove (alternatively to the probabilistic approach of Fitzsimmons and Getoor) that the Radon–Nikodym derivative between the Revuz measures of a semiregular excessive kernel and the regular strongly supermedian kernel generating it, does not depend on the kernel. Mathematics Subject Classifications (2000) Primary 60J45, 31D05; Secondary 60J35, 60J40.     

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

The perturbation of the generator of a Borel right process by a signed measure is investigated, using probabilistic and analytic potential theoretical methods. We establish a Feynman-Kac formula associated with measures charging no polar set and belonging to an extended Kato class. A main tool of this approach is the validity of a Khas’minskii Lemma for Stieltjes exponentials of positive left continuous additive functionals.      

Intensity enhancement of the C5+ Balmer radiation excited by capillary discharge pumping

The intensity of the 18.2 nm Balmer α-transition in C⁵⁺ excited in a capillary discharge using alumina and polyacetal tubes was studied. For discharge currents of up to 80 kA in tubes filled with C₂H₂, intense radiation from the excitation of C⁵⁺ ions and from the recombination of C⁶⁺ ions was observed. With increasing length of the discharge, the intensity in the falling edge of the recombination pulse rises faster than proportional. In contrast to previous investigations, gain by stimulated emission is excluded. The enhancement is ascribed to an optical guiding of the XUV radiation in the dense plasma created by ablation from the tube walls. Received: 1 April 1999 / Revised version: 22 July 1999 / Published online: 30 November 1999     

Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

On the Tightness of Capacities Associated with Sub-Markovian Resolvents

This paper investigates the tightness property of the capacityinduced by the reduction operator with respect to the resolventof a right Markov process. Tightness is verified in two particularsituations: under the ‘weak duality hypothesis’,and if a substitute for ‘the axiom of polarity for thedual theory’ holds. In the second context, the quasi-continuityproperty for the excessive functions is derived. These are extensionsand improvements of results of Lyons and Röckner, Ma andRöckner, Le Jan, and Fitzsimmons, mainly obtained in thecontext of Dirichlet forms. 2000 Mathematics Subject Classification31C25, 60J45 (primary), 31C15, 60J40 (secondary).     

Realization, Operation and Results of the Multichannel Far-Infrared Polarimeter on RFX

This paper illustrates the realisation and testing of the multichannel far-infrared polarimeter for the RFX machine, a plasma confinement experiment with the Reversed Field Pinch configuration. The diagnostic is described and the its main sections are outlined: the laser system operating at = 118.8m, the FIR optical line, the beam splitter and the beam detection units, the signal processing electronics and the motion electronics for the calibration halfwave plates. Emphasis is placed on the work performed on the polarimeter to make it operational. We illustrate the preliminary tests of the diagnostic equipment and the activity performed to bring the diagnostic into reliable operation, in particular the reduction of the noise produced by magnetically induced mechanical vibrations during machine shots. An example of the first Faraday angle measurements performed on five chords are presented and discussed. The measured Faraday angle was also compared to a theoretically calculated value, based on a simple plasma model, with satisfactory agreement between measured and predicted data.     

Fine Densities for Excessive Measures and the Revuz Correspondence

Suppose that U is the resolvent of a Borel right process on a Lusin space X. If is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure with the Radon–Nikodym derivative d/d possesses a finely continuous version. (Fitzsimmons and Fitzsimmons and Getoor gave a probabilistic approach for this result.) We extend essentially a technique initiated by Mokobodzki and deepened by Feyel. The result allows us to establish a Revuz type formula involving the fine versions, and to study the Revuz correspondence between the -finite measures charging no set that is both -polar and -negligible (U being the potential component of ) and the strongly supermedian kernels on X. This is an analytic version of a result of Azéma, Fitzsimmons and Dellacherie, Maisonneuve and Meyer, in terms of additive functionals or homogeneous random measures. Finally we give an application to the context of the semi-Dirichlet forms, covering a recent result of Fitzsimmons.     

Aspects of Number Theory in Terms of Potential Theory

We develop the Potential Theory on the set N^* of all natural numbers 2 associated with the kernel V (resp. V ^*) given by

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