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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 Soc.*, **149**, 1970) never superseded the standard Ray-Knight approach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: 28, 283-289, LNM 191 (1971)

**WALSH, John B.**

Two footnotes to a theorem of Ray (Markov processes)

Ray's theorem (*Ann. of Math.*, **70**, 1959) is the construction of a good semigroup (and process) from a Ray resolvent. The first ``footnote'' gives the construction of a second semigroup with nice properties from the left instead of the right side. The second ``footnote'' studies the filtration of a Ray process

Comment: See Meyer-Walsh,*Invent. Math.*, **14**, 1971

Keywords: Ray compactification

Nature: Original

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VI: 16, 168-172, LNM 258 (1972)

**MEYER, Paul-André**; **WALSH, John B.**

Un résultat sur les résolvantes de Ray (Markov processes)

This is a complement to the authors' paper on Ray processes in*Invent. Math.,* **14**, 1971: a lemma is proved on the existence of many martingales which are continuous whenever the process is continuous (a wrong reference for it was given in the paper). Then it is shown that the mapping $x\rightarrow P_x$ is continuous in the weak topology of measures, when the path space is given the topology of convergence in measure. Note that a correction is mentioned on the errata page of vol. VII

Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng*Ann. Inst. Henri Poincaré* **20**, 1984

Keywords: Ray compactification, Weak convergence of measures

Nature: Original

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VI: 21, 215-232, LNM 258 (1972)

**WALSH, John B.**

Transition functions of Markov processes (Markov processes)

Assume that a cadlag process satisfies the strong Markov property with respect to some family of kernels $P_t$ (not necessarily a semigroup). It is shown that these kernels can be modified into a true strong Markov transition function with a few additional properties. A similar problem is solved for a left continuous, moderate Markov process. The technique involves a Ray compactification which is eliminated at the end, and a useful lemma shows how to construct supermedian functions which separate points

Comment: The problem discussed here has great theoretical importance, but little practical importance except for time reversal. The construction of a nice transition function for a Markov process has been also discussed by Kuznetsov ()

Keywords: Transition functions, Strong Markov property, Moderate Markov property, Ray compactification

Nature: Original

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VII: 02, 25-32, LNM 321 (1973)

**MEYER, Paul-André**

Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)

This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification

Comment: All this material has become classical. See for instance Dellacherie-Meyer,*Probabilités et Potentiel,* vol. D, chapter XV, 31--35

Keywords: Lévy systems, Ray compactification

Nature: Original

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VIII: 15, 289-289, LNM 381 (1974)

**MEYER, Paul-André**

Une note sur la compactification de Ray (Markov processes)

This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification

Comment: This is now considered a standard fact

Keywords: Ray compactification, Right processes

Nature: Original

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X: 14, 216-234, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams*Diffusions, Markov Processes and Martingales,* vol. 1 (second edition), Wiley 1994. See also 1024

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

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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: 28, 283-289, LNM 191 (1971)

Two footnotes to a theorem of Ray (Markov processes)

Ray's theorem (

Comment: See Meyer-Walsh,

Keywords: Ray compactification

Nature: Original

Retrieve article from Numdam

VI: 16, 168-172, LNM 258 (1972)

Un résultat sur les résolvantes de Ray (Markov processes)

This is a complement to the authors' paper on Ray processes in

Comment: The idea of using the topology of convergence in measure on a path space turned out to be a fruitful idea; see Meyer and Zheng

Keywords: Ray compactification, Weak convergence of measures

Nature: Original

Retrieve article from Numdam

VI: 21, 215-232, LNM 258 (1972)

Transition functions of Markov processes (Markov processes)

Assume that a cadlag process satisfies the strong Markov property with respect to some family of kernels $P_t$ (not necessarily a semigroup). It is shown that these kernels can be modified into a true strong Markov transition function with a few additional properties. A similar problem is solved for a left continuous, moderate Markov process. The technique involves a Ray compactification which is eliminated at the end, and a useful lemma shows how to construct supermedian functions which separate points

Comment: The problem discussed here has great theoretical importance, but little practical importance except for time reversal. The construction of a nice transition function for a Markov process has been also discussed by Kuznetsov ()

Keywords: Transition functions, Strong Markov property, Moderate Markov property, Ray compactification

Nature: Original

Retrieve article from Numdam

VII: 02, 25-32, LNM 321 (1973)

Une mise au point sur les systèmes de Lévy. Remarques sur l'exposé de A. Benveniste (Markov processes)

This is an addition to the preceding paper 701, extending the theory to right processes by means of a Ray compactification

Comment: All this material has become classical. See for instance Dellacherie-Meyer,

Keywords: Lévy systems, Ray compactification

Nature: Original

Retrieve article from Numdam

VIII: 15, 289-289, LNM 381 (1974)

Une note sur la compactification de Ray (Markov processes)

This short note shows that (contrary to the belief of Meyer and Walsh in a preceding paper) the state space of a Ray process is universally measurable in its Ray compactification

Comment: This is now considered a standard fact

Keywords: Ray compactification, Right processes

Nature: Original

Retrieve article from Numdam

X: 14, 216-234, LNM 511 (1976)

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

Retrieve article from Numdam