Lucian Beznea
University of Bucharest
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Archive | 2004
Lucian Beznea; Nicu Boboc
Introduction. 1: Excessive Functions. 1.1. Sub-Markovian resolvent of kernels. 1.2. Basics on excessive functions. 1.3. Fine topology. 1.4. Excessive measures. 1.5. Ray topology and compactification. 1.6. The reduction operation and the associated capacities. 1.7. Polar and semipolar sets. Nearly measurable functions. 1.8. Probabilistic interpretations: Sub-Markovian resolvents and right processes. 2: Cones of Potentials and H Cones. 2.1. Basics on cones of potentials and H-cones. 2.2. sigma-Balayages on cones of potentials. 2.3. Balayages on H-cones. 2.4. Quasi bounded, subtractive and regular elements of a cone of potentials. 3: Fine Potential Theoretical Techniques. 3.1. Cones of potentials associated with a sub-Markovian resolvent. 3.2. Regular excessive functions, fine carrier and semipolarity. 3.3. Representation of balayages on excessive measures. 3.4. Quasi bounded, regular and subtractive excessive measures. 3.5. Tightness for sub-Markovian resolvents. 3.6. Localization in excessive functions and excessive measures. 3.7. Probabilistic interpretations: Continuous additive functionals and standardness. 4: Strongly Supermedian Functions and Kernels. 4.1. Supermedian functionals. 4.2. Supermedian lambda-quasi kernels. 4.3. Strongly supermedian functions. 4.4. Fine densities. 4.5. Probabilistic interpretations: Homogeneous random measures. 5: Subordinate Resolvents. 5.1. Weak subordination operators. 5.2. Inverse subordination. 5.3. Probabilistic interpretations: Multiplicative functionals. 6: Revuz Correspondence. 6.1. Revuz measures. 6.2. Hypothesis (i) of Hunt. 6.3. Smooth measures and sub-Markovian resolvents. 6.4. Measure perturbation of sub-Markovian resolvents. 6.5. Probabilistic interpretations: Positive left additive functionals. 7: Resolvents under Weak Duality Hypothesis. 7.1. Weak duality hypothesis. 7.2. Natural potential kernels and the Revuz correspondence. 7.3. Smooth and cosmooth measures. 7.4. Subordinate resolvents in weak duality. 7.5. Semi-Dirichlet forms. 7.6. Weak duality induced by a semi-Dirichlet form. 7.7. Probabilistic interpretations: Multiplicative functionals in weak duality. A. Appendix: A.1. Complements on measure theory, kernels, Choquet boundary and capacity. A.2. Complements on right processes. A.3. Cones of potentials and H-cones. A.4. Basics on coercive closed bilinear forms. Notes. Bibliography. Index.
Potential Analysis | 1996
Lucian Beznea; Nicu Boboc
We study the quasi-boundedness and subtractivity in a general frame of cones of potentials (more precisely in H-cones). Particularly we show that the subtractive elements are strongly related to the existence of recurrent balayages. In the special case of excessive measures we improve results of P. J. Fitzsimmons and R. K. Getoor from [13], obtained with probabilistic methods.
Archive | 1994
Lucian Beznea; Nicu Boboc
We show that the balayage operation on measurable sets exists in the class of all universally measurable excessive functions, giving an analytic version of the well-known fundamantal result of G. A. Hunt and C. T. Shih. The quasi-continuous elements in H-cones are introduced and characterized. (In the classical situation these are the countable sums of bounded continuous potentials with compact carrier.) In the particular case of excessive measures we prove that the quasi-continuous elements are exactly the potentials of measures which does not charge the semi-polar sets.
Journal of Mathematical Analysis and Applications | 2011
Lucian Beznea; Gerald Trutnau
Abstract We obtain a criterion for the quasi-regularity of generalized (non-sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. The second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels.
Potential Analysis | 1999
Lucian Beznea; Nicu Boboc
We study the strongly supermedian functions and the supermedian functionals associated with a submarkovian resolvent on a Lusin space. We extend results of D. Feyel and J. F. Mertens to a general right process. The existence of a positive potential replaces the usual hypothesis that the process possesses left limits in the space.
Potential Analysis | 2004
Lucian Beznea; Nicu Boboc
Suppose that U is the resolvent of a Borel right process on a Lusin space X. If ξ is a U-excessive measure on X then we show by analytical methods that for every U-excessive measure η with η≪ξ the Radon–Nikodym derivative dη/dξ 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 σ-finite measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential component of ξ) 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.
Potential Analysis | 2001
Lucian Beznea; Nicu Boboc
In the context of a transient Borel right Markov process with a fixed excessive measure ξ, we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no ξ-semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential part of ξ), answering to a problem of Revuz.
Potential Analysis | 1997
Lucian Beznea; Nicu Boboc
If Exc is the set of all excessive measures associated with a submarkovian resolvent on a Lusin measurable space and B is a balayage on Exc then we show that for any m∈Exc there exists a basic set A (determined up to a m-polar set) such that Bξ=(BA)*ξ for any ξ∈ Exc, ξ ≪ m. The m-quasi-Lindelöf property (for the fine topology) holds iff for any B there exists the smallest basic set A as above. We characterize the case when any B is representable i.e. there exists a basic set such that B=(BA)* on Exc.
Archive | 2014
Lucian Beznea; Iulian Cîmpean
We prove the Bochner-Kolmogorov theorem on the existence of the limit of projective systems of second countable Hausdorff (non-metrizable) spaces with tight probabilities, such that the projection mappings are merely measurable functions. Our direct and transparent approach (using Lusin’s theorem) should be compared with the previous work where the spaces are assumed metrizable and the main idea was to reduce the general context to a regular one via some isomorphisms. The motivation of the revisit of this classical result is an application to the construction of the continuous time fragmentation processes and related branching processes, based on a measurable identification between the space of all fragmentation sizes considered by J. Bertoin and the limit of a projective system of spaces of finite configurations.
Infinite Dimensional Analysis, Quantum Probability and Related Topics | 2006
Lucian Beznea; Nicu Boboc
We show that the right (Markov) process associated with a quasi-regular semi-Dirichlet form has a dual process, proving that the weak duality hypothesis is satisfied.