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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 Soc.*, **149**, 1970) never superseded the standard Ray-Knight approach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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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 The original proof of Mokobodzki, self-contained and unpublished, is given here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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IX: 13, 305-317, LNM 465 (1975)

**FÖLLMER, Hans**

Phase transition and Martin boundary (Miscellanea)

To be completed

Comment: To be completed

Keywords: Random fields, Martin boundary

Nature: Original

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

IX: 13, 305-317, LNM 465 (1975)

Phase transition and Martin boundary (Miscellanea)

To be completed

Comment: To be completed

Keywords: Random fields, Martin boundary

Nature: Original

Retrieve article from Numdam