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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 Stat.*, **40**, 1969) and is completely superseded by one of Revuz (*Ann.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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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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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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam

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

Retrieve article from Numdam