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For a resistance form \({(X, \mathcal{D}(\varepsilon),\varepsilon)}\) and a point \({x_0 \in X}\) as boundary, on the space \({X_0:=X {\setminus}\{x_0\}}\) we consider the Dirichlet space \({\mathcal{D}_{x_0}:=\{f\in\mathcal{D}(\varepsilon)\, |\, f(x_0)=0\}}\) and we develop a good potential theory. For any finely open subset D of X0 we consider a localized resistance form (\({\mathcal{D}_{X_0 {\setminus} D},\varepsilon_{D}}\)) where \({\mathcal{D}_{X_0 {\setminus} D}:=\{f\in\mathcal{D}_{x_0}\, |\, f=0}\) on \({X_0 {\setminus} D\},\, \varepsilon_D(f,g):=\varepsilon(f,g)}\) for all \({f,g\in\mathcal{D}_{X_0 {\setminus} D}}\). The main result is the equivalence between the local property of the resistance form and the sheaf property for the excessive elements on finely open sets.

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.

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.

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 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.

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.

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 Azema, 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.

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.

If S is an H-cone and P:S→S is a localizable dilation operator on S (i.e., P is additive increasing, contractive, continuous in order from below and s∧(Ps+t−Pt+Pf)∈S, s,t∈S,f∈(S−S)+), then it is proved that its adjoint P*:S*→S* (i.e., P*μ=μ○P) is also a localizable dilation operator. This is an improvement of a result obtained by G. Mokobodzki in the frame of excessive functions.

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

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.

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ. With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ–polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azema [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process.

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.

Resume Nous montrons qu'il existe une correspondance entre les mesures σ-finies qui ne chargent pas les ensembles ξ-polaires et ρ-negligeables (resp. ξ-polaire, ξ-semi-polaires) et les noyaux fortement surmedians reguliers (resp. noyaux excessifs semi-reguliers, naturels, reguliers), ou ξ = ρ ∘ U + h est une mesure excessive par rapport a une resolvante U d'un processus droit borelien. Dans cette correspondance l'hypothese (B) de Hunt signifie que tout noyau excessif semi-regulier est naturel. Ce sont des versions analytiques et des extensions, des resultats probabilistes obtenus par Azema, Dellacherie-Maisonneuve-Meyer, Fitzsimmons, Getoor-Sharpe.

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″.

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-Lindelof 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.

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.

In the frame of standard H-cones of functions (the cone of all excessive functions with respect to a submarkovian resolvent of kernels with reference measure on a measurable space) on a Green set we show that the cofine closure of the complement of an absorbent set in coabsorbent. We obtain different characterizations concerning the parabolicity, ellipticity and quasiellipticity in terms of the Green function. We also show that these notions are the same in the direct and the dual theory.

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.

The chapter deals with the weak duality between two sub-Markovian resolvents on a Lusin topological space, with respect to a given measure m. This frame covers the probabilistic context of two Borel right processes in weak duality, which will be presented in Section 7.7. The main results of the first three sections are related to the coincidence of the m-semipolar and m-cosemipolar sets, the Revuz correspondence generated by the measures charging no cofinely open m-polar set. Section 7.4 develops, under the weak duality hypothesis, the subordination procedures from Chapter 5. In Section 7.5 we expose the preliminary results on the semi-Dirichlet forms, starting with a sub-Markovian resolvent of kernels satisfying the strong sector condition. Section 7.6 relates the transient semi-Dirichlet forms with the weak duality as it is presented in the first part of the chapter.

We investigate in this chapter the strongly supermedian functions and kernels. The notion of strongly supermedian function has been introduced by J.F. Mertens [Mer 73] in the frame of a right process; see also [Mer 74]. Later on P.A. Meyer showed that these functions may be characterized in terms of the excessive functions, and clarified some of Mertens’ results from the analytical point of view (cf. [Me 73a], [Me 73b] and [MeTr 73]). Section 4.1 presents the λ-supermedian functionals and their representations as λ-strongly supermedian functions. It is also obtained the λ-quasi Lindelof property for the fine topology. Section 4.2 exposes the regular λ-strongly supermedian functions and the regular supermedian λ-quasi kernels. Section 4.3 is devoted to the study of the strongly supermedian functions and kernels. It is obtained the Mertens decomposition of a finite strongly supermedian function as a sum of an excessive function and a regular strongly supermedian one. Relevant results on the regular strongly supermedian functions are also presented: the fine carrier and the associated strongly supermedian kernels. In Section 4.4 it is proved the remarkable fact if η, ξ are two excessive measures, η ≪ ξ, then the Radon-Nikodym derivative dη/dξ has a ξ-fine version, i.e. a ξ-version which is finite and finely continuous outside a set that is ξ-polar and ρ-negligible, ρ ◦ U being the potential component of ξ. Section 4.5 interprets the strongly supermedian functions and their properties in the probabilistic frame of the right processes, in terms of the homogeneous random measures.

This chapter presents further properties of the excessive functions and measures, in connection with the cone of potentials and H-cone structures. In Section 3.2 we investigate the regular excessive functions, their fine carriers and the associated regular excessive kernels. The λ-semipolar and λ-mince sets are characterized by means of such regular excessive kernels. In Section 3.3 we discuss the representation of the balayages on the excessive measures as the duals of the balayages on the excessive functions. In Section 3.4 we study the quasi bounded, regular and subtractive excessive measures. Section 3.5 presents a tightness criterion for the capacities induced by the reduction operation. In Section 3.6 we expose the localization on a finely open set of the excessive functions and measures. Section 3.7 contains some probabilistic interpretations of the regular excessive kernels in terms of the continuous additive functionals, and of the tightness of the capacities in terms of the standardness of the process.

This chapter is a rather special part of the book. It contains the basic elements concerning the abstract structures of cone of potential and H-cones. The prototype for the cone of potentials (resp. H-cones) will be the convex cone of all U-excessive functions (resp. of all U-excessive measures) which are finite U-a.e. (resp. σ-finite), where U is a given sub-Markovian resolvent of kernels on a Radon measurable space.

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.

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.

The behaviour of the balayage on the complement of a fine open set is investigated near an irregular boundary point. We extend also the classical Harnack inequality. We generalise essentially, in the frame of excessive functions, results of M. Brelot, E. Smyrnelis, H. Bauer, W. Hansen and I. Netuka obtained in the context of harmonic spaces.