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I: 01, 3-17, LNM 39 (1967)

**AVANISSIAN, Vazgain**

Sur l'harmonicité des fonctions séparément harmoniques (Potential theory)

This paper proves a harmonic version of Hartogs' theorem: separately harmonic functions are jointly harmonic (without any boundedness assumption) using a complex extension procedure. The talk is an extract from the author's original work in*Ann. ENS,* **178**, 1961

Comment: This talk was justified by the current interest of the seminar in doubly excessive functions, see Cairoli 102 in the same volume

Keywords: Doubly harmonic functions

Nature: Exposition

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I: 02, 18-33, LNM 39 (1967)

**CAIROLI, Renzo**

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

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I: 03, 34-51, LNM 39 (1967)

**COURRÈGE, Philippe**

Noyaux de convolution singuliers opérant sur les fonctions höldériennes et noyaux de convolution régularisants (Potential theory)

The Poisson equation $ėlta\,(Uf)=-f$, where $U$ is the Newtonian potential is proved to be true in the strictest sense when $f$ is a Hölder function (while it is not for mere continuous functions). This involves an exposition of singular integral kernels on Hölder spaces

Comment: This talk was a by-product of the extensive work of Courrège, Bony and Priouret on Feller semi-groups on manifolds with boundary (*Ann. Inst. Fourier,* **16**, 1968)

Keywords: Newtonian potential

Nature: Exposition

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I: 07, 163-165, LNM 39 (1967)

**MEYER, Paul-André**

Sur un théorème de Deny (Potential theory, Measure theory)

In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit

Comment: a version of this result in classical potential theory was proved by Deny,*C.R. Acad. Sci.*, **218**, 1944. The cone of excessive functions possesses good compactness properties, discovered by Mokobodzki. See Dellacherie-Meyer, *Probabilités et Potentiel,* end of chapter XII

Keywords: A.e. convergence, Subsequences

Nature: Original

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I: 09, 177-189, LNM 39 (1967)

**WEIL, Michel**

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,*Ill. J. Math.*, **9**, 1965 and *J. Math. Mech.*, **15**, 1966

Keywords: Green potentials, Dual semigroups

Nature: Exposition

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II: 10, 171-174, LNM 51 (1968)

**MEYER, Paul-André**

Les résolvantes fortement fellériennes d'après Mokobodzki (Potential theory)

On a compact space, a submarkov kernel $N$ has the strong Feller property if it maps Borel bounded functions into continuous functions, and the stronger Feller property if the mapping $x\rightarrow \epsilon_x N$ is continuous in the norm topology of measures. It is proved that the product of two strong Feller kernels is stronger Feller, and as a consequence if the kernels of are resolvent are strong Feller they are automatically stronger Feller

Comment: This follows from a result on weakly compact operators on continuous functions due to Grothendieck (*Canadian J. Math.*, **5**, 1953). Mokobodzki's proof is less general (it uses positivity) but very simple. This result is rather useful

Keywords: Resolvents, Strong Feller properties

Nature: Exposition

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II: 11, 175-199, LNM 51 (1968)

**MEYER, Paul-André**

Compactifications associées à une résolvante (Potential theory)

Let $E$ be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps $C_k$ (the continuous functions with compact support) into continuous bounded functions. Let $F$ be a compact space containing $E$ as a dense subset, but inducing possibly a coarser topology. It is assumed that all potentials $Uf$ with $f\in C_k$ extend to continuous functions on $F$, and that points of $F$ are separated by continuous functions on $F$ whose restriction to $E$ is supermedian. Then it is shown how to extend the resolvent to $F$ and imitate the construction of a Ray semigroup and a strong Markov process. This was an attempt to compactify the space using only supermedian functions, not $p$-supermedian for all $p>0$. An application to Markov chains is given

Comment: This method of compactification suggested by Chung's boundary theory for Markov chains (similarly Doob,*Trans. Amer. Math. Soc.*, **149**, 1970) never superseded the standard Ray-Knight approach

Keywords: Resolvents, Ray compactification, Martin boundary, Boundary theory

Nature: Original

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III: 09, 144-151, LNM 88 (1969)

**MEYER, Paul-André**

Un résultat de théorie du potentiel (Potential theory, Markov processes)

Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded

Comment: See Meyer,*Processus de Markov,* Lecture Notes in M. **26**

Keywords: Green potentials, Dual semigroups

Nature: Original

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IV: 07, 73-75, LNM 124 (1970)

**DELLACHERIE, Claude**

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

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IV: 15, 170-194, LNM 124 (1970)

**MOKOBODZKI, Gabriel**

Densité relative de deux potentiels comparables (Potential theory)

The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$

Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in*Séminaire Bourbaki,* **422**, November 1972. See also Chapter XII of Dellacherie-Meyer, *Probabilités et potentiel,* containing new proofs due to Feyel

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 16, 195-207, LNM 124 (1970)

**MOKOBODZKI, Gabriel**

Quelques propriétés remarquables des opérateurs presque positifs (Potential theory)

A sequel to the preceding paper 415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent

Comment: Same as 415

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 17, 208-215, LNM 124 (1970)

**REVUZ, Daniel**

Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (Potential theory, Markov processes)

Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see 210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure

Comment: For additional work of Revuz on recurrence, see*Ann. Inst. Fourier,* **21**, 1971

Keywords: Recurrent potential theory

Nature: Original

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V: 06, 76-76, LNM 191 (1971)

**CHUNG, Kai Lai**

A simple proof of Doob's convergence theorem (Potential theory)

Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set

Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set

Keywords: Excessive functions, Semi-polar sets

Nature: New exposition of known results

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V: 19, 196-208, LNM 191 (1971)

**MEYER, Paul-André**

Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)

Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here

Comment: See Mokobodzki's work on cones of potentials,*Séminaire Bourbaki,* May 1970

Keywords: Minimal excessive functions, Martin boundary, Integral representations

Nature: Exposition

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V: 21, 211-212, LNM 191 (1971)

**MEYER, Paul-André**

Deux petits résultats de théorie du potentiel (Potential theory)

Excessive functions are characterized by their domination property over potentials. The strong ordering relation between two functions is carried over to their réduites

Comment: See Dellacherie-Meyer*Probability and Potentials,* Chapter XII, \S2

Keywords: Excessive functions, Réduite, Strong ordering

Nature: Original

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V: 23, 237-250, LNM 191 (1971)

**MEYER, Paul-André**

Travaux de H. Rost en théorie du balayage (Potential theory, Ergodic theory)

The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (*Zeit. für W-theorie,* **15**, 1970; *Ann. Inst. Fourier,* **21**, 1971): it provides a solution to Skorohod's imbedding problem for measures on discrete time Markov processes. Here it is also used to prove Brunel's Lemma in pointwise ergodic theory

Comment: Extension to continuous time in Meyer 612. See also 806, 1012. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Brunel's lemma, Skorohod imbedding

Nature: Exposition, Original additions

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V: 24, 251-269, LNM 191 (1971)

**MEYER, Paul-André**

Solutions de l'équation de Poisson dans le cas récurrent (Potential theory, Markov processes)

The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used

Comment: The paper was motivated by Métivier (*Ann. Math. Stat.*, **40**, 1969) and is completely superseded by one of Revuz (*Ann. Inst. Fourier,* **21**, 1971)

Keywords: Recurrent potential theory, Filling scheme, Harris recurrence, Poisson equation

Nature: Original

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V: 25, 270-274, LNM 191 (1971)

**MEYER, Paul-André**

Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)

Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (*Ann. Inst. Fourier,* **20-1**, 1970) showed it was quite general

Comment: Shih's paper is the origin of the general definition of ``right processes''

Keywords: Excessive functions, Réduite

Nature: Exposition

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V: 26, 275-277, LNM 191 (1971)

**REVUZ, Daniel**

Remarque sur les potentiels de mesure (Markov processes, Potential theory)

The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,*Markov Processes and Potential Theory,* 1968) uses the assumption that excessive functions are lower semicontinuous. This assumption is weakened

Comment: To be asked

Keywords: Polar sets, Semi-polar sets, Excessive functions

Nature: Original

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V: 29, 290-310, LNM 191 (1971)

**WALSH, John B.**

Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)

It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called*essential topology,* used in the paper of Chung and Walsh 522 in the same volume

Comment: See Doob*Bull. Amer. Math. Soc.*, **72**, 1966. An important application in given by Walsh 623 in the next volume. See the paper 1025 of Benveniste. For the use of a different topology see Ito *J. Math. Soc. Japan,* **20**, 1968

Keywords: Essential topology

Nature: Original

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V: 30, 311-341, LNM 191 (1971)

**WATANABE, Takesi**

On balayées of excessive measures and functions with respect to resolvents (Potential theory)

A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process

Comment: See Kunita and T. Watanabe,*Ill. J. Math.*, **9**, 1965. For the modern theory of balayage of measures (using Kuznetsov's processes) see Getoor, *Excessive Measures,* 1990, Chapter 4

Keywords: Excessive measures, Balayage

Nature: Original

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V: 32, 347-361, LNM 191 (1971)

**WEIL, Michel**

Quasi-processus et énergie (Markov processes, Potential theory)

The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil 418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy

Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons*Sem. Stoch. Proc.*, 1987

Keywords: Hunt quasi-processes, Energy

Nature: Original

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VI: 12, 130-150, LNM 258 (1972)

**MEYER, Paul-André**

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (*Invent. Math.,* **14**, 1971) the construction is extended to continuous time Markov processes. In the transient case, the results are translated in potential-theoretic language, and proved using techniques due to Mokobodzki. Then the general case follows from this result applied to a space-time extension of the semi-group

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

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VI: 15, 164-167, LNM 258 (1972)

**MEYER, Paul-André**

Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)

The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set

Keywords: Réduite, Fine topology, Absolute continuity hypothesis

Nature: Original

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VI: 17, 173-176, LNM 258 (1972)

**MOKOBODZKI, Gabriel**

Pseudo-quotient de deux mesures par rapport à un cône de potentiels. Application à la dualité (Potential theory)

The last four pages of this paper have been omitted by mistake, and appear in the following volume as 729. The general results concerning the axiomatically defined cones of potentials (see for instance the author's exposition in*Séminaire Bourbaki,* 1969-70, **377**) are quickly reviewed first, and then applied to the following problem concerning the potential kernel $V$ of a resolvent: given a pair of measures $\lambda\le\mu$ in the sense of balayage, then we have $\lambda V\le \mu V$ in the ordinary sense. The corresponding density (dominated by $1$) does not depend on the resolvent, but only on the potential cones of excessive functions and potentials associated with it, and a way to compute it is indicated

Keywords: Cones of potentials

Nature: Original

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VI: 20, 202-214, LNM 258 (1972)

**REVUZ, Daniel**

Le principe semi-complet du maximum (Potential theory)

The problem studied here (and not completely solved) consists in finding potential theoretic characterizations for the recurrent potential operators constructed in the basic paper of Neveu,*Ann. Inst. Fourier,* **22-2**, 1972. It is shown that these operators satisfy suitable maximum principles (as usual, slightly stronger in the discrete case than in the continuous case). The converse is delicate and some earlier work of Kondo (*Osaka J. Math.*, **4**, 1967) and Oshima (same journal, **6**, 1969) is discussed in this new set-up

Comment: This topic is discussed again in Revuz' book*Markov Chains,* North-Holland

Keywords: Recurrent potential theory, Maximum principles, Recurrent Markov chains

Nature: Original

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VII: 16, 155-171, LNM 321 (1973)

**MEYER, Paul-André**; **TRAKI, Mohammed**

Réduites et jeux de hasard (Potential theory)

This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage*How to Gamble if you Must * in the language of standard (countably additive) measure theory, using the methods of descriptive set theory (analytic sets, section theorems, etc). The attempt is successful, since all general theorems can be proved in this set-up. More recent results in the same line, due to Strauch and Sudderth, are extended too. An appendix includes useful comments by Mokobodzki on the case of a gambling house consisting of a single kernel (discrete potential theory)

Comment: This material is reworked in Dellacherie-Meyer,*Probabilités et Potentiel,* Vol. C, Chapter X

Keywords: Balayage, Gambling house, Réduite, Optimal strategy

Nature: Original

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VII: 27, 291-300, LNM 321 (1973)

**TAYLOR, John C.**

On the existence of resolvents (Potential theory)

Since the basic results of Hunt, a kernel satisfying the complete maximum principle is expected to be the potential kernel of a sub-Markov resolvent. This is not always the case, however, and one should also express that, so to speak, ``potentials vanish at the boundary''. Such a condition is given here on an abstract space, which supersedes an earlier result of the author (*Invent. Math.* **17**, 1972) and a result of Hirsch (*Ann. Inst. Fourier,* **22-1**, 1972)

Comment: The definitive paper of Taylor on this subject appeared in*Ann. Prob.*, **3**, 1975

Keywords: Complete maximum principle, Resolvents

Nature: Original

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VII: 29, 319-321, LNM 321 (1973)

**MOKOBODZKI, Gabriel**

Pseudo-quotient de deux mesures, application à la dualité (Potential theory)

Contains the four last pages of 617 omitted from Volume VI

Nature: Correction

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VIII: 06, 27-36, LNM 381 (1974)

**DINGES, Hermann**

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

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VIII: 11, 150-154, LNM 381 (1974)

**HEATH, David C.**

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $**R**^n$ is given. A theorem of Mokobodzki on réduites is used as an important technical tool

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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IX: 31, 515-517, LNM 465 (1975)

**HEATH, David C.**

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XI: 01, 1-20, LNM 581 (1977)

**AVANISSIAN, Vazgain**

Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (Potential theory, Miscellanea)

This paper studies two classes of functions in (an open set of) $**R**^n$, $n\ge1$: 1) Harmonic functions of infinite order (see Avanissian and Fernique, *Ann. Inst. Fourier,* **18-2**, 1968), which are $C^\infty$ functions satisfying a growth condition on their iterated laplacians, and are shown to be real analytic. 2) Infinitely differentiable functions (or distributions) similar to completely monotonic functions on the line, i.e., whose iterated laplacians are alternatively positive and negative (they were introduced by Lelong). Among the results is the fact that the second class is included in the first

Keywords: Harmonic functions, Real analytic functions, Completely monotonic functions

Nature: Original

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XI: 03, 27-33, LNM 581 (1977)

**CHUNG, Kai Lai**

Pedagogic notes on the barrier theorem (Potential theory)

Let $D$ a bounded open set in $**R**^n$, and let $z$ be a boundary point. Then a barrier at $z$ is a superharmonic function in $D$, strictly positive and with a limit equal to $0$ at $z$. The barrier theorem asserts that if there is a barrier at $z$, then $z$ is regular. An elegant proof of this is given using Brownian motion. Then it is shown that the expectation of $S$, the hitting time of $D^c$, is bounded, upper semi-continuous in $R^n$ and continuous in $D$, and is a barrier at every regular point

Comment: An error is corrected in 1247

Keywords: Classical potential theory, Barrier, Regular points

Nature: New proof of known results

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XI: 12, 132-195, LNM 581 (1977)

**MEYER, Paul-André**

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(**R**^n)$ is $BMO$, using methods adapted from the probabilistic Littlewood-Paley theory (of which this is a kind of limiting case). Some details of the proof are interesting in their own right

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XII: 26, 378-397, LNM 649 (1978)

**BROSSARD, Jean**

Comportement non-tangentiel et comportement brownien des fonctions harmoniques dans un demi-espace. Démonstration probabiliste d'un théorème de Calderon et Stein (Potential theory, Real analysis)

Given a harmonic function $u$ in a half space, Stein (*Acta Math.* 106, 1961) shows that the boundary points $x$ such that 1) $u$ has a non-tangential limit at $x$, 2) $u$ is ``non tangentially bounded'' near $x$, 3) $\nabla u$ is locally $L^2$ in the non-tangential cones at $x$, are the sames, except for sets of measure $0$. This result is given here a probabilistic proof using conditional Brownian motion

Keywords: Harmonic functions in a half-space, Non-tangential limits

Nature: Original

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XII: 47, 739-739, LNM 649 (1978)

**CHUNG, Kai Lai**

Correction to "Pedagogic Notes on the Barrier Theorem" (Potential theory)

Corrects an error in 1103

Keywords: Classical potential theory, Barrier, Regular points

Nature: Correction

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XIV: 40, 357-391, LNM 784 (1980)

**FALKNER, Neil**

On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (Brownian motion, Potential theory)

The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $**R**^n$ with the initial measure $\mu$, and $T$ is a *non-randomized * stopping time. The conditions given are sufficient in all cases, necessary if $\mu$ does not charge polar sets

Comment: A general survey on the Skorohod embedding problem is Ob\lój,*Probab. Surv.* **1**, 2004

Keywords: Skorohod imbedding

Nature: Original

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XIV: 44, 418-436, LNM 784 (1980)

**RAO, Murali**

A note on Revuz measure (Markov processes, Potential theory)

The problem is to weaken the hypotheses of Chung (*Ann. Inst. Fourier,* **23**, 1973) implying the representation of the equilibrium potential of a compact set as a Green potential. To this order, Revuz measure techniques are used, and interesting auxiliary results are proved concerning the Revuz measures of natural additive functionals of a Hunt process

Keywords: Revuz measures, Additive functionals, Hunt processes, Equilibrium potentials

Nature: Original

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Sur l'harmonicité des fonctions séparément harmoniques (Potential theory)

This paper proves a harmonic version of Hartogs' theorem: separately harmonic functions are jointly harmonic (without any boundedness assumption) using a complex extension procedure. The talk is an extract from the author's original work in

Comment: This talk was justified by the current interest of the seminar in doubly excessive functions, see Cairoli 102 in the same volume

Keywords: Doubly harmonic functions

Nature: Exposition

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I: 02, 18-33, LNM 39 (1967)

Semi-groupes de transition et fonctions excessives (Markov processes, Potential theory)

A study of product kernels, product semi-groups and product Markov processes

Comment: This paper was the first step in R.~Cairoli's study of two-parameter processes

Keywords: Product semigroups

Nature: Original

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I: 03, 34-51, LNM 39 (1967)

Noyaux de convolution singuliers opérant sur les fonctions höldériennes et noyaux de convolution régularisants (Potential theory)

The Poisson equation $ėlta\,(Uf)=-f$, where $U$ is the Newtonian potential is proved to be true in the strictest sense when $f$ is a Hölder function (while it is not for mere continuous functions). This involves an exposition of singular integral kernels on Hölder spaces

Comment: This talk was a by-product of the extensive work of Courrège, Bony and Priouret on Feller semi-groups on manifolds with boundary (

Keywords: Newtonian potential

Nature: Exposition

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I: 07, 163-165, LNM 39 (1967)

Sur un théorème de Deny (Potential theory, Measure theory)

In the potential theory of a resolvent which satisfies the absolute continuity hypothesis, every sequence of excessive functions contains a subsequence which converges except on a set of potential zero. It is also proved that a sequence which converges weakly in $L^1$ but not strongly must oscillate around its limit

Comment: a version of this result in classical potential theory was proved by Deny,

Keywords: A.e. convergence, Subsequences

Nature: Original

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I: 09, 177-189, LNM 39 (1967)

Résolvantes en dualité (Markov processes, Potential theory)

Given two sub-Markov resolvents in duality, whose kernels are absolutely continuous with respect to a given measure, it is shown how to choose their densities to get true Green kernels, excessive in one variable and coexcessive in the other one. It is shown also that coexcessive functions are exactly the densities of excessive measures

Comment: These results, now classical, are due to Kunita-T. Watanabe,

Keywords: Green potentials, Dual semigroups

Nature: Exposition

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II: 10, 171-174, LNM 51 (1968)

Les résolvantes fortement fellériennes d'après Mokobodzki (Potential theory)

On a compact space, a submarkov kernel $N$ has the strong Feller property if it maps Borel bounded functions into continuous functions, and the stronger Feller property if the mapping $x\rightarrow \epsilon_x N$ is continuous in the norm topology of measures. It is proved that the product of two strong Feller kernels is stronger Feller, and as a consequence if the kernels of are resolvent are strong Feller they are automatically stronger Feller

Comment: This follows from a result on weakly compact operators on continuous functions due to Grothendieck (

Keywords: Resolvents, Strong Feller properties

Nature: Exposition

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II: 11, 175-199, LNM 51 (1968)

Compactifications associées à une résolvante (Potential theory)

Let $E$ be a locally compact space, $(U_p)$ be a submarkovian resolvent, with a potential kernel $U=U_0$ which maps $C_k$ (the continuous functions with compact support) into continuous bounded functions. Let $F$ be a compact space containing $E$ as a dense subset, but inducing possibly a coarser topology. It is assumed that all potentials $Uf$ with $f\in C_k$ extend to continuous functions on $F$, and that points of $F$ are separated by continuous functions on $F$ whose restriction to $E$ is supermedian. Then it is shown how to extend the resolvent to $F$ and imitate the construction of a Ray semigroup and a strong Markov process. This was an attempt to compactify the space using only supermedian functions, not $p$-supermedian for all $p>0$. An application to Markov chains is given

Comment: This method of compactification suggested by Chung's boundary theory for Markov chains (similarly Doob,

Keywords: Resolvents, Ray compactification, Martin boundary, Boundary theory

Nature: Original

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III: 09, 144-151, LNM 88 (1969)

Un résultat de théorie du potentiel (Potential theory, Markov processes)

Under strong duality hypotheses, it is shown that a measure which does not charge polar sets is equivalent to a measure whose Green potential is bounded

Comment: See Meyer,

Keywords: Green potentials, Dual semigroups

Nature: Original

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IV: 07, 73-75, LNM 124 (1970)

Potentiels de Green et fonctionnelles additives (Markov processes, Potential theory)

Under duality hypotheses, the problem is to associate an additive functional with a Green potential, which may assume the value $+\infty$ on a polar set: the corresponding a.f. may explode at time $0$

Comment: Such additive functionals appear very naturally in the theory of Dirichlet spaces

Keywords: Green potentials, Additive functionals

Nature: Original

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IV: 15, 170-194, LNM 124 (1970)

Densité relative de deux potentiels comparables (Potential theory)

The main problem considered here is the following: given a transient resolvent $(V_{\lambda})$ on a measurable space, a finite potential $Vg$, an excessive function $u$ dominated by $Vg$ in the strong sense (i.e., $Vg-u$ is excessive), show that $u=Vf$ for some $f\leq g$, and compute $f$ by some ``derivation'' procedure, like $\lim_{\lambda\rightarrow\infty} \lambda(I-\lambda V_{\lambda})\,u$

Comment: The main theorem and the technical tools of its proof have been landmarks in the potential theory of a resolvent, though in the case of the resolvent of a good Markov process there is a simple probabilistic proof of the main result. Another exposition can be found in

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 16, 195-207, LNM 124 (1970)

Quelques propriétés remarquables des opérateurs presque positifs (Potential theory)

A sequel to the preceding paper 415. Almost positive operators are candidates to the role of derivation operators relative to a resolvent

Comment: Same as 415

Keywords: Resolvents, Strong ordering, Lebesgue derivation theorem

Nature: Original

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IV: 17, 208-215, LNM 124 (1970)

Application d'un théorème de Mokobodzki aux opérateurs potentiels dans le cas récurrent (Potential theory, Markov processes)

Mokododzki's theorem asserts that if the kernels of a resolvent are strong Feller, i.e., map bounded functions into continuous functions, then they must satisfy a norm continuity property (see 210). This is used to show the existence for``normal'' recurrent processes of a nice potential operator, defined for suitable functions of zero integral with respect to the invariant measure

Comment: For additional work of Revuz on recurrence, see

Keywords: Recurrent potential theory

Nature: Original

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V: 06, 76-76, LNM 191 (1971)

A simple proof of Doob's convergence theorem (Potential theory)

Doob's theorem is a version of the main convergence theorem of potential theory: the limit of a decreasing sequence of excessive functions differs of its regularized version on a semi-polar set

Comment: It is also shown that a function $f$ satisfying $f\ge P_Kf$ for all compact sets $K$ differs from its regularized function on a semi-polar set

Keywords: Excessive functions, Semi-polar sets

Nature: New exposition of known results

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V: 19, 196-208, LNM 191 (1971)

Représentation intégrale des fonctions excessives. Résultats de Mokobodzki (Markov processes, Potential theory)

Main result: the convex cone of excessive functions for a resolvent which satisfies the absolute continuity hypothesis is the union of convex compact metrizable ``hats''\ in a suitable topology, and therefore has the integral representation property. The original proof of Mokobodzki, self-contained and unpublished, is given here

Comment: See Mokobodzki's work on cones of potentials,

Keywords: Minimal excessive functions, Martin boundary, Integral representations

Nature: Exposition

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V: 21, 211-212, LNM 191 (1971)

Deux petits résultats de théorie du potentiel (Potential theory)

Excessive functions are characterized by their domination property over potentials. The strong ordering relation between two functions is carried over to their réduites

Comment: See Dellacherie-Meyer

Keywords: Excessive functions, Réduite, Strong ordering

Nature: Original

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V: 23, 237-250, LNM 191 (1971)

Travaux de H. Rost en théorie du balayage (Potential theory, Ergodic theory)

The ``filling scheme'' is a technique used in ergodic theory to prove Hopf's maximal Lemma and the Chacon-Ornstein theorem, studied in detail by H.~Rost (

Comment: Extension to continuous time in Meyer 612. See also 806, 1012. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Brunel's lemma, Skorohod imbedding

Nature: Exposition, Original additions

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V: 24, 251-269, LNM 191 (1971)

Solutions de l'équation de Poisson dans le cas récurrent (Potential theory, Markov processes)

The problem is to solve the Poisson equation for measures, $\mu-\mu P=\theta$ for given $\theta$, in the case of a recurrent transition kernel $P$. Here a ``filling scheme'' technique is used

Comment: The paper was motivated by Métivier (

Keywords: Recurrent potential theory, Filling scheme, Harris recurrence, Poisson equation

Nature: Original

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V: 25, 270-274, LNM 191 (1971)

Balayage pour les processus de Markov continus à droite, d'après C.T. Shih (Markov processes, Potential theory)

Hunt's fundamental theorem on balayage gives the probabilistic interpretation of the réduite of an excessive function set on a set. It was proved originally within the special class of Hunt's processes, then extended to ``standard'' processes. Using a method of compactification, Shih (

Comment: Shih's paper is the origin of the general definition of ``right processes''

Keywords: Excessive functions, Réduite

Nature: Exposition

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V: 26, 275-277, LNM 191 (1971)

Remarque sur les potentiels de mesure (Markov processes, Potential theory)

The standard proof of the equivalence between semi-polar sets being polar and a very precise domination principle (Blumenthal-Getoor,

Comment: To be asked

Keywords: Polar sets, Semi-polar sets, Excessive functions

Nature: Original

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V: 29, 290-310, LNM 191 (1971)

Some topologies connected with Lebesgue measure (Markov processes, General theory of processes, Potential theory)

It is a recurrent theme in the theory of stochastic processes that time sets of measure $0$ should be ignored. Thus topologies on the line which ignore sets of measure $0$ are useful. The main topic here is the so-called

Comment: See Doob

Keywords: Essential topology

Nature: Original

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V: 30, 311-341, LNM 191 (1971)

On balayées of excessive measures and functions with respect to resolvents (Potential theory)

A general study of balayage of excessive measures as dual to réduite of excessive functions, first for a single kernel, then for a resolvent on a measurable space, and finally for a standard process

Comment: See Kunita and T. Watanabe,

Keywords: Excessive measures, Balayage

Nature: Original

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V: 32, 347-361, LNM 191 (1971)

Quasi-processus et énergie (Markov processes, Potential theory)

The energy of an excessive function $f$ with respect to an excessive measure $\xi$ has a simple proba\-bi\-listic interpretation if $\xi$ is is the potential of a measure $\mu$ and $f$ is the potential of an additive functional $(A_t)$, as ${1\over2}E_\mu[A_\infty^2]$. If $\xi$ is not a potential, still it can be associated with it a quasi-process (see Weil 418) with a birthtime $b$ and a death time $d$, and the formal expression ${1\over2}E[(A_d-A_b)^2]$ is given a precise meaning and represents the energy

Comment: This subject has been renewed by the introduction of Kuznetsov's measures. See Fitzsimmons

Keywords: Hunt quasi-processes, Energy

Nature: Original

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VI: 12, 130-150, LNM 258 (1972)

Le schéma de remplissage en temps continu, d'après H. Rost (Ergodic theory, Potential theory)

The work of H. Rost on the so-called discrete filling scheme was presented to the Seminar as 523. Here following Rost himself (

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Filling scheme, Balayage of measures, Skorohod imbedding

Nature: Exposition, Original additions

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VI: 15, 164-167, LNM 258 (1972)

Une note sur le théorème du balayage de Hunt (Markov processes, Potential theory)

The theorem involved is the characterization of the réduite $P_A u$ of an excessive function $u$ on a set $A$ as equal (except on a well-defined semipolar set) to the infimum of the excessive functions that dominate $u$ on $A$. This theorem is slightly improved under the absolute continuity hypothesis. The proof rests on the following property of the fine topology: every (nearly Borel) finely closed set is contained in a fine $G_\delta$ which differs from it by a polar set

Keywords: Réduite, Fine topology, Absolute continuity hypothesis

Nature: Original

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VI: 17, 173-176, LNM 258 (1972)

Pseudo-quotient de deux mesures par rapport à un cône de potentiels. Application à la dualité (Potential theory)

The last four pages of this paper have been omitted by mistake, and appear in the following volume as 729. The general results concerning the axiomatically defined cones of potentials (see for instance the author's exposition in

Keywords: Cones of potentials

Nature: Original

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VI: 20, 202-214, LNM 258 (1972)

Le principe semi-complet du maximum (Potential theory)

The problem studied here (and not completely solved) consists in finding potential theoretic characterizations for the recurrent potential operators constructed in the basic paper of Neveu,

Comment: This topic is discussed again in Revuz' book

Keywords: Recurrent potential theory, Maximum principles, Recurrent Markov chains

Nature: Original

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VII: 16, 155-171, LNM 321 (1973)

Réduites et jeux de hasard (Potential theory)

This paper arose from an attempt (by the second author) to rewrite the results of Dubins-Savage

Comment: This material is reworked in Dellacherie-Meyer,

Keywords: Balayage, Gambling house, Réduite, Optimal strategy

Nature: Original

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VII: 27, 291-300, LNM 321 (1973)

On the existence of resolvents (Potential theory)

Since the basic results of Hunt, a kernel satisfying the complete maximum principle is expected to be the potential kernel of a sub-Markov resolvent. This is not always the case, however, and one should also express that, so to speak, ``potentials vanish at the boundary''. Such a condition is given here on an abstract space, which supersedes an earlier result of the author (

Comment: The definitive paper of Taylor on this subject appeared in

Keywords: Complete maximum principle, Resolvents

Nature: Original

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VII: 29, 319-321, LNM 321 (1973)

Pseudo-quotient de deux mesures, application à la dualité (Potential theory)

Contains the four last pages of 617 omitted from Volume VI

Nature: Correction

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VIII: 06, 27-36, LNM 381 (1974)

Stopping sequences (Markov processes, Potential theory)

Given a discrete time Markov process $(X_n)$ with transition kernel $P$, a stopping sequence with initial distribution $\mu$ is a family $(\mu_n)$ of measures such that $\mu\ge\mu_0$ and $\mu_{k-1}P\ge\mu_k$. The stopping sequence associated with a stopping time $T$ is the sequence of distributions of $X_{T}, k< T<\infty$ under the law $P_\mu$. Every stopping sequence arises in this way from some randomized stopping time $T$, and the distribution of $X_T, T<\infty$ is independent of $T$ and called the final distribution. Then several constructions of stopping sequences are described, including Rost's ``filling scheme'', and several operations on stopping sequences, aiming at the construction of ``short'' stopping times in the Skorohod imbedding problem, without assuming transience of the process

Comment: This is a development of the research of H.~Rost on the ``filling scheme'', for which see 523, 524, 612. This article contains announcements of further results

Keywords: Discrete time Markov processes, Skorohod imbedding, Filling scheme

Nature: Original

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VIII: 11, 150-154, LNM 381 (1974)

Skorohod stopping via Potential Theory (Potential theory, Markov processes)

The original construction of the Skorohod imbedding of a measure into Brownian motion is translated into a language meaningful for general Markov processes, and the extension to Brownian motion in $

Comment: This paper is best read in connection with 931. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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IX: 31, 515-517, LNM 465 (1975)

Skorohod stopping in discrete time (Markov processes, Potential theory)

Using ideas of Mokobodzki, it is shown how the imbedding of a measure $\mu_1$ in the discrete Markov process with initial measure $\mu_0$ can be achieved by a random mixture of hitting times

Comment: This is a potential theoretic version of the original construction of Skorohod. This paper is better read in conjunction with Heath 811. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XI: 01, 1-20, LNM 581 (1977)

Fonctions harmoniques d'ordre infini et l'harmonicité réelle liée à l'opérateur laplacien itéré (Potential theory, Miscellanea)

This paper studies two classes of functions in (an open set of) $

Keywords: Harmonic functions, Real analytic functions, Completely monotonic functions

Nature: Original

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XI: 03, 27-33, LNM 581 (1977)

Pedagogic notes on the barrier theorem (Potential theory)

Let $D$ a bounded open set in $

Comment: An error is corrected in 1247

Keywords: Classical potential theory, Barrier, Regular points

Nature: New proof of known results

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XI: 12, 132-195, LNM 581 (1977)

Le dual de $H^1({\bf R}^\nu)$~: démonstrations probabilistes (Potential theory, Applications of martingale theory)

This is a self-contained exposition and proof of the celebrated (Fefferman-Stein) result that the dual of $H^1(

Comment: Though the proof is complete, it misses an essential point in the Fefferman-Stein theorem, namely, it depends on the Cauchy (Poisson) semigroup while the original result the convolution with quite general smooth functions in its definition of $H^1$. Similar methods were used by Bakry in the case of spheres, see 1818. The reasoning around (3.1) p.178 needs to be corrected

Keywords: Harmonic functions, Hardy spaces, Poisson kernel, Carleson measures, $BMO$, Riesz transforms

Nature: Exposition, Original additions

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XII: 26, 378-397, LNM 649 (1978)

Comportement non-tangentiel et comportement brownien des fonctions harmoniques dans un demi-espace. Démonstration probabiliste d'un théorème de Calderon et Stein (Potential theory, Real analysis)

Given a harmonic function $u$ in a half space, Stein (

Keywords: Harmonic functions in a half-space, Non-tangential limits

Nature: Original

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XII: 47, 739-739, LNM 649 (1978)

Correction to "Pedagogic Notes on the Barrier Theorem" (Potential theory)

Corrects an error in 1103

Keywords: Classical potential theory, Barrier, Regular points

Nature: Correction

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XIV: 40, 357-391, LNM 784 (1980)

On Skorohod embedding in $n$-dimensional Brownian motion by means of natural stopping times (Brownian motion, Potential theory)

The problem discussed here is the Skorohod representation of a measure $\nu$ as the distribution of $B_T$, where $(B_t)$ is Brownian motion in $

Comment: A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

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XIV: 44, 418-436, LNM 784 (1980)

A note on Revuz measure (Markov processes, Potential theory)

The problem is to weaken the hypotheses of Chung (

Keywords: Revuz measures, Additive functionals, Hunt processes, Equilibrium potentials

Nature: Original

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