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X: 10, 125-183, LNM 511 (1976)

**MEYER, Paul-André**

Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book*Topics in Harmonic Analysis related to the Littlewood-Paley theory,* Princeton 1970. The classical Littlewood-Paley theory shows that the $L^p$ norm ($1<p<\infty$) of a function $f$ on $**R**^n$ is equivalent to that of several kinds of non-linear functionals of $f$ called Littlewood-Paley functions, which are square roots of quadratic expressions involving the harmonic extension of $f$ to the half-space $**R**^n\times **R**_+$, and its derivatives. Using these equivalences, it is easy to prove that the Riesz transforms are bounded in~$L^p$. The classical theory is given a probabilistic interpretation, the L-P functions appearing as conditional expectations of functionals of a Brownian motion on the half-space, given its final position on the limit hyperplane, and then the L-P inequalities follow from the Burkholder inequalities of martingale theory. The original L-P theory concerned the unit disk; Stein had extended it to $**R**^n$ and had started extending it to symmetric semigroups. Here a new tool is introduced, the squared-field operator (carré du champ) introduced by J.P.~Roth (*CRAS Paris,* **278A**, 1974, p.1103) in potential theory and by Kunita (*Nagoya M. J.*, **36**, 1969) in probability. This paper consists of 4 talks, and in the last one theorems 1' and 3 are false

Comment: This paper was rediscovered by Varopoulos (*J. Funct. Anal.*, **38**, 1980), and was then rewritten by Meyer in 1510 in a simpler form. Its main application has been to the Ornstein-Uhlenbeck semigroup in 1816. It has been superseded by the theory of $\Gamma_2$ due to Bakry 1910, see also Bakry-Émery 1912, and Meyer 1908 reporting on Cowling's extension of Stein's work. An erratum is given in 1253

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

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XII: 53, 741-741, LNM 649 (1978)

**MEYER, Paul-André**

Correction à ``Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)

This is an erratum to 1010

Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Correction

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XV: 11, 167-188, LNM 850 (1981)

**BOULEAU, Nicolas**

Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)

The main result of the paper of Kunita (*Nagoya Math. J.*, **36**, 1969) showed that the domain of the extended generator $A$ of a right Markov semigroup is an algebra if and only if the angle brackets of all martingales are absolutely continuous with respect to the measure $dt$. See also 1010. Such semigroups are called here ``semigroups of Lebesgue type''. Kunita's result is sharpened here: it is proved in particular that if some non-affine convex function $f$ operates on the domain, then the semigroup is of Lebesgue type (Kunita's result corresponds to $f(x)=x^2$) and if the second derivative of $f$ is not absolutely continuous, then the semigroup has no diffusion part (i.e., all martingales are purely discontinuous). The second part of the paper is devoted to the behaviour of the extended domain under an absolutely continuous change of probability (arising from a multiplicative functional)

Keywords: Semigroup theory, Carré du champ, Infinitesimal generators

Nature: Original

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XXIX: 16, 166-180, LNM 1613 (1995)

**APPLEBAUM, David**

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (*Stochastics Stochastics Rep.* **56**, 1996). The same question is addressed by Cohen in the next article 2917

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

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XXIX: 17, 181-193, LNM 1613 (1995)

**COHEN, Serge**

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (*Stochastics Stochastics Rep.* **56**, 1996). Here this language is used to study the Markov property of solutions to SDE's with jumps. In particular,two definitions of a Lévy process in a Riemannian manifold are compared: One as the solution to a SDE driven by some Euclidean Lévy process, the other by subordinating some Riemannian Brownian motion. It is shown that in general the former is not of the second kind

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

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Démonstration probabiliste de certaines inégalités de Littlewood-Paley (4 talks) (Applications of martingale theory, Markov processes)

This long paper consists of four talks, suggested by E.M.~Stein's book

Comment: This paper was rediscovered by Varopoulos (

Keywords: Littlewood-Paley theory, Riesz transforms, Brownian motion, Inequalities, Harmonic functions, Singular integrals, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Original

Retrieve article from Numdam

XII: 53, 741-741, LNM 649 (1978)

Correction à ``Inégalités de Littlewood-Paley'' (Applications of martingale theory, Markov processes)

This is an erratum to 1010

Keywords: Littlewood-Paley theory, Carré du champ, Infinitesimal generators, Semigroup theory

Nature: Correction

Retrieve article from Numdam

XV: 11, 167-188, LNM 850 (1981)

Propriétés d'invariance du domaine du générateur infinitésimal étendu d'un processus de Markov (Markov processes)

The main result of the paper of Kunita (

Keywords: Semigroup theory, Carré du champ, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XXIX: 16, 166-180, LNM 1613 (1995)

A horizontal Lévy process on the bundle of orthonormal frames over a complete Riemannian manifold (Stochastic differential geometry, Markov processes)

This is an attempt to define a manifold-valued Lévy process by solving a SDE driven by a Euclidean Lévy process; but the author shows that the so-obtained processes are not Markovian in general.

Comment: The existence and uniqueness statements are a particular case of general theorems due to Cohen (

Keywords: Semimartingales with jumps, Lévy processes, Infinitesimal generators

Nature: Original

Retrieve article from Numdam

XXIX: 17, 181-193, LNM 1613 (1995)

Some Markov properties of stochastic differential equations with jumps (Stochastic differential geometry, Markov processes)

The Schwartz-Meyer theory of second-order calculus for manifold-valued continuous semimartingales (see 1505 and 1655) was extended by Cohen to càdlàg semimartingales (

Comment: The first definition is independently introduced by David Applebaum 2916

Keywords: Semimartingales with jumps, Lévy processes, Subordination, Infinitesimal generators

Nature: Original

Retrieve article from Numdam