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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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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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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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VI: 22, 233-242, LNM 258 (1972)

**WALSH, John B.**

The perfection of multiplicative functionals (Markov processes)

In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure

Keywords: Multiplicative functionals, Perfection, Essential topology

Nature: Original

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X: 02, 19-23, LNM 511 (1976)

**CHACON, Rafael V.**; **WALSH, John B.**

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. 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: 18, 327-339, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

Prolongement de processus holomorphes. Cas ``carré intégrable'' (Several parameter processes)

This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in*Acta Math.* **4**, 1975. These processes have stochastic integral representations along (increasing) paths, with a common kernel called their derivative. Under an integrability restriction, a process holomorphic in a region of the plane is shown to be extendable as a holomorphic process to a larger region of a canonical shape (intersection of a rectangle and a disk centered at the origin)

Comment: See also 1119

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 19, 340-348, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

Some examples of holomorphic processes (Several parameter processes)

This is a sequel to the preceding paper 1118. It also extends the definition to processes defined on a random domain

Comment: See the author's paper in*Ann. Prob.* **5**, 1971 for additional results

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

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XI: 20, 349-355, LNM 581 (1977)

**CAIROLI, Renzo**; **WALSH, John B.**

On changing time (Several parameter processes)

The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet

Keywords: Brownian sheet

Nature: Original

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XI: 33, 490-492, LNM 581 (1977)

**WALSH, John B.**

A property of conformal martingales (Martingale theory)

Almost every path of a (complex) conformal martingale on the open time interval $]0,\infty[$ has the following behaviour at time $0$: either it has a limit in the Riemann sphere, or it is everywhere dense

Comment: See also 1408

Keywords: Conformal martingales

Nature: Original

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XV: 21, 290-306, LNM 850 (1981)

**CHACON, Rafael V.**; **LE JAN, Yves**; **WALSH, John B.**

Spatial trajectories (Markov processes, General theory of processes)

It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated

Comment: See Chacon-Jamison,*Israel J. of M.*, **33**, 1979

Keywords: Spatial trajectories

Nature: Original

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XVI: 16, 212-212, LNM 920 (1982)

**WALSH, John B.**

A non-reversible semi-martingale (Stochastic calculus)

A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale

Keywords: Semimartingales, Time reversal

Nature: Original

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XXVII: 17, 173-176, LNM 1557 (1993)

**WALSH, John B.**; **YOR, Marc**

Some remarks on $A(t,B_t)$

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XXXIX: 04, 117-118, LNM 1874 (2006)

**WALSH, John B.**

A lost scroll

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

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

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

VI: 22, 233-242, LNM 258 (1972)

The perfection of multiplicative functionals (Markov processes)

In the definition of multiplicative functionals the problem arose from the beginning whether the exceptional null set in the relation $M_{s+t}=M_s\,M_t\circ\theta_s$ was allowed to depend on $s$ or not---in the latter case the functional is said to be perfect. C.~Doléans showed by a detailed analysis (see 203) that every functional has a perfect modification, see also Dellacherie 304. Here a perfect version is constructed directly as $\lim_{s\rightarrow 0} M_{t-s}\circ\theta_s$, the limit being taken in the essential topology of the line, which ignores sets of zero Lebesgue measure

Keywords: Multiplicative functionals, Perfection, Essential topology

Nature: Original

Retrieve article from Numdam

X: 02, 19-23, LNM 511 (1976)

One-dimensional potential imbedding (Brownian motion)

The problem is to find a Skorohod imbedding of a given measure into one-dimensional Brownian motion using non-randomized stopping times. One-dimensional potential theory is used as a tool

Comment: The construction is related to that of Dubins (see 516). In this volume 1012 also constructs non-randomized Skorohod imbeddings. A general survey on the Skorohod embedding problem is Ob\lój,

Keywords: Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XI: 18, 327-339, LNM 581 (1977)

Prolongement de processus holomorphes. Cas ``carré intégrable'' (Several parameter processes)

This paper concerns a class of two-parameter (real) processes adapted to the filtration of the Brownian sheet, and called holomorphic in the seminal paper of the authors in

Comment: See also 1119

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

Retrieve article from Numdam

XI: 19, 340-348, LNM 581 (1977)

Some examples of holomorphic processes (Several parameter processes)

This is a sequel to the preceding paper 1118. It also extends the definition to processes defined on a random domain

Comment: See the author's paper in

Keywords: Holomorphic processes, Brownian sheet

Nature: Original

Retrieve article from Numdam

XI: 20, 349-355, LNM 581 (1977)

On changing time (Several parameter processes)

The analogue of the well-known result that any continuous martingale can be time changed into a Brownian motion using its own quadratic variation process is answered negatively for two-parameter martingales (even strong ones) in the filtration of the Brownian sheet

Keywords: Brownian sheet

Nature: Original

Retrieve article from Numdam

XI: 33, 490-492, LNM 581 (1977)

A property of conformal martingales (Martingale theory)

Almost every path of a (complex) conformal martingale on the open time interval $]0,\infty[$ has the following behaviour at time $0$: either it has a limit in the Riemann sphere, or it is everywhere dense

Comment: See also 1408

Keywords: Conformal martingales

Nature: Original

Retrieve article from Numdam

XV: 21, 290-306, LNM 850 (1981)

Spatial trajectories (Markov processes, General theory of processes)

It is well known that Markov processes with the same excessive functions are the same up to a strictly increasing continuous time-change. It is therefore natural to study spatial trajectories, i.e., trajectories up to a strictly increasing continuous time changes, and in particular to provide the space of all spatial trajectories with a reasonable $\sigma$-field so that it may carry measures. It is shown here that the space of right-continuous spatial trajectories with left-hand limits is a Blackwell space. The class of intrinsic stopping times defined on this space is also investigated

Comment: See Chacon-Jamison,

Keywords: Spatial trajectories

Nature: Original

Retrieve article from Numdam

XVI: 16, 212-212, LNM 920 (1982)

A non-reversible semi-martingale (Stochastic calculus)

A simple example is given of a continuous semimartingale (a Brownian motion which stops at time $1$ and starts moving again at time $T>1$, $T$ encoding all the information up to time $1$) whose reversed process is not a semimartingale

Keywords: Semimartingales, Time reversal

Nature: Original

Retrieve article from Numdam

XXVII: 17, 173-176, LNM 1557 (1993)

Some remarks on $A(t,B_t)$

Retrieve article from Numdam

XXXIX: 04, 117-118, LNM 1874 (2006)

A lost scroll