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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 Soc.*, **149**, 1970) never superseded the standard Ray-Knight approach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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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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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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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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam