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 It has been superseded by the theory of $\Gamma_2$ due to Bakry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
1253Keywords: Littlewood-Paley theory,
Riesz transforms,
Brownian motion,
Inequalities,
Harmonic functions,
Singular integrals,
Carré du champ,
Infinitesimal generators,
Semigroup theoryNature: Original
Retrieve article from Numdam
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
1010Keywords: Littlewood-Paley theory,
Carré du champ,
Infinitesimal generators,
Semigroup theoryNature: Correction Retrieve article from Numdam
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 generatorsNature: Original Retrieve article from Numdam
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
2917Keywords: Semimartingales with jumps,
Lévy processes,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
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
2916Keywords: Semimartingales with jumps,
Lévy processes,
Subordination,
Infinitesimal generatorsNature: Original Retrieve article from Numdam
