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We study the Schrodinger equation (q − ℒ)u + µu = f, where ℒ is the generator of a Borel right process and µ is a signed measure on the state space. We prove the existence and uniqueness results in Lp, 1 ⩽ p < ∞. Since we consider measures µ charging no polar set, we have to use new tools: the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation (subordination) operators (in the sense of G. Mokobodzki) induced by the regular strongly supermedian kernels. We extend the results on the Schrodinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on Lp. If the measure µ is positive then the perturbed process solves the martingale problem for ℒ − µ and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having µ as Revuz measure.

We present an analytic version of the following uniqueness problem for Markov processes: if two subprocesses of a given transient, Borel right process have the same excessive functions then they coincide. Our treatment is given in terms of subordinate sub-Markovian resolvents of kernels in the sense of P. A. Mayer and uses essentially the subordination operators introduced by G. Mokobodzki.

We show that every C 0 -resolvent on L p ( E , μ ) , where ( E , B ) is a Lusin measurable space and μ is a σ-finite measure on B , has an associate sufficiently regular Markov process on a (larger) Lusin topological space containing E as a Borel subset. We give general conditions on the resolvent's generator such that the above process lives on E. We present two applications: (i) we settle a question of G. Mokobodzki on the existence of a (Lusin) topology on E having B as Borel σ-algebra such that a given Dirichlet form on L 2 ( E , μ ) becomes quasi-regular; (ii) we solve stochastic differential equations on Hilbert spaces in the sense of a martingale problem. To cite this article: L. Beznea et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).

We give general conditions on a generator of a C-o-semigroup (resp. of a C-o-resolvent) on L-p (E, mu), p >= 1, where E is an arbitrary (Lusin) topological space and mu a sigma-finite measure on its Borel sigma-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.

We give an analytic version of the well known Shih's theorem concerning the Markov processes whose hitting distributions are dominated by those of a given process. The treatment is purely analytic, completely different from Shih's arguments and improves essentially his result (in the case when the given processes are transient

We study the weak duality between two sub-Markovian resolvents of kernels on a Lusin topological space with respect to a given measure m. Our frame covers the probabilistic context of two Borel Markov processes in weak duality. The main results are related to: the coincidence of the m-semipolar and m-cosemipolar sets, the Revuz correspondence between the measures charging no cofinely open m-polar set and the potential kernels associated with the homogeneous random measures of the process, the equivalence between smoothness and cosmoothness (a smooth measure is the Revuz measure of a continuous additive functional). We extend and improve results of R.M. Blumenthal-R.K. Getoor, J. Azema, D. Revuz, J.B. Walsh, R.K. Getoor-M.J. Sharpe.

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.

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 α > 0 with T = αS. Particularly if e′, e″ are two elliptic Dirichlet forms on L2 ( μ ) having the same domain of definition and the same α-excessive (resp. α-coexcessive) elements for any α > 0 then e′ = e″.

We prove that for any semi-Dirichlet form ${\left({\varepsilon,D{\left(\varepsilon\right)}}\right)}$ on a measurable Lusin space E there exists a Lusin topology with the given $\sigma$ -algebra as the Borel $\sigma$ -algebra so that ${\left({\varepsilon,D{\left(\varepsilon\right)}}\right)}$ becomes quasi-regular. However one has to enlarge E by a zero set. More generally a corresponding result for arbitrary $L^p$ -resolvents is proven.

In the frame of Borel right Markov processes, we investigate, following an analytical point of view, the Revuz correspondence between classes of potential kernels and their associated measures, improving upon the results of Revuz, Azema, Getoor and Sharpe, Fitzsimmons, Fitzsimmons and Getoor and Dellacherie, Maisonneuve and Meyer. In the probabilistic approach of the problem, the kernels that occur are the potential operators of different types of homogeneous random measures. We completely characterize the hypothesis (B) of Hunt in terms of Revuz measures.

In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.