XV: 47, 671-672, LNM 850 (1981)
BAKRY, Dominique
Une remarque sur les semi-martingales à deux indices (
Several parameter processes)
Let $({\cal F}^1_s)$ and $({\cal F}^2_t)$ be two filtrations whose conditional expectations commute. Let $(A_t)$ be a bounded increasing process adapted to $({\cal F}^2_t)$. It had been proved under stringent absolute continuity conditions on $A$ that the process $X_{st}=E[A_t\,|\,{\cal F}^1_s]$ was a semimartingale (a stochastic integrator). A counterexample is given here to show that this is not true in general
Keywords: Two-parameter semimartingalesNature: Original
Retrieve article from Numdam
XVI: 08, 134-137, LNM 920 (1982)
BAKRY, Dominique
Remarques sur le processus d'Ornstein-Uhlenbeck en dimension infinie (
Malliavin's calculus,
Several parameter processes)
A process taking values in a space of sample paths can be considered as a two parameter process. Considering in this way the Ornstein-Uhlenbeck process (
1606) raises a few natural questions, like the commutation of conditional expectations relative to the two filtrations---which is shown to hold true
Keywords: Ornstein-Uhlenbeck processNature: Original Retrieve article from Numdam
XVI: 09, 138-150, LNM 920 (1982)
BAKRY, Dominique;
MEYER, Paul-André
Sur les inégalités de Sobolev logarithmiques (two parts) (
Applications of martingale theory)
These two papers are variations on a paper of G.F. Feissner (
Trans. Amer Math. Soc.,
210, 1965). Let $\mu$ be a Gaussian measure, $P_t$ be the corresponding Ornstein-Uhlenbeck semigroup. Nelson's hypercontractivity theorem states (roughly) that $P_t$ is bounded from $L^p(\mu)$ to some $L^q(\mu)$ with $q\ge p$. In another celebrated paper, Gross showed this to be equivalent to a logarithmic Sobolev inequality, meaning that if a function $f$ is in $L^2$ as well as $Af$, where $A$ is the Ornstein-Uhlenbeck generator, then $f$ belongs to the Orlicz space $L^2Log_+L$. The starting point of Feissner was to translate this again as a result on the ``Riesz potentials'' of the semi-group (defined whenever $f\in L^2$ has integral $0$) $$R^{\alpha}={1\over \Gamma(\alpha)}\int_0^\infty t^{\alpha-1}P_t\,dt\;.$$ Note that $R^{\alpha}R^{\beta}=R^{\alpha+\beta}$. Then the theorem of Gross implies that $R^{1/2}$ is bounded from $L^2$ to $L^2Log_+L$. This suggests the following question: which are in general the smoothing properties of $R^\alpha$? (Feissner in fact considers a slightly different family of potentials).\par The complete result then is the following : for $\alpha$ complex, with real part $\ge0$, $R^\alpha$ is bounded from $L^pLog^r_+L$ to $L^pLog^{r+p\alpha}_+L$. The method uses complex interpolation between two cases: a generalization to Orlicz spaces of a result of Stein, when $\alpha$ is purely imaginary, and the case already known where $\alpha$ has real part $1/2$. The first of these two results, proved by martingale theory, is of a quite general nature
Keywords: Logarithmic Sobolev inequalities,
Hypercontractivity,
Gaussian measures,
Riesz potentialsNature: Original Retrieve article from Numdam
XVI: 31, 355-369, LNM 920 (1982)
BAKRY, Dominique
Semimartingales à deux indices Retrieve article from Numdam
XVII: 42, 498-501, LNM 986 (1983)
BAKRY, Dominique
Une remarque sur les processus gaussiens définissant des mesures $L^2$ Retrieve article from Numdam
XVIII: 18, 197-218, LNM 1059 (1984)
BAKRY, Dominique
Étude probabiliste des transformées de Riesz et de l'espace $H^1$ sur les sphères Retrieve article from Numdam
XIX: 09, 130-174, LNM 1123 (1985)
BAKRY, Dominique
Transformation de Riesz pour les semi-groupes symétriques (two parts) Retrieve article from Numdam
XIX: 10, 175-175, LNM 1123 (1985)
BAKRY, Dominique
Une remarque sur les inégalités de Littlewood-Paley sous l'hypothèse $\Gamma_2\ge0$ Retrieve article from Numdam
XIX: 12, 177-206, LNM 1123 (1985)
BAKRY, Dominique;
ÉMERY, Michel
Diffusions hypercontractives Retrieve article from Numdam
XX: 40, 614-614, LNM 1204 (1986)
BAKRY, Dominique
Correction au Séminaire XIX Retrieve article from Numdam
XXI: 08, 137-172, LNM 1247 (1987)
BAKRY, Dominique
Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée Retrieve article from Numdam
XXII: 01, 1-50, LNM 1321 (1988)
BAKRY, Dominique
La propriété de sous-harmonicité des diffusions dans les variétés Retrieve article from Numdam
XXIII: 01, 1-20, LNM 1372 (1989)
BAKRY, Dominique
Sur l'interpolation complexe des semigroupes de diffusion Retrieve article from Numdam
XXV: 20, 234-261, LNM 1485 (1991)
BAKRY, Dominique
Inégalités de Sobolev faibles : un critère $\Gamma_2$ Retrieve article from Numdam
XXVI: 17, 170-188, LNM 1526 (1992)
BAKRY, Dominique;
MICHEL, Dominique
Sur les inégalités FKG Retrieve article from Numdam
XXX: 12, 178-206, LNM 1626 (1996)
BAKRY, Dominique;
ECHERBAULT, Mireille
Sur les inégalités GKS Retrieve article from Numdam
XXXVII: 02, 60-80, LNM 1832 (2003)
BAKRY, Dominique;
MAZET, Olivier
Characterization of Markov semigroups on $\bf R$ associated to some families of orthogonal polynomials
XLVII: 10, 157-185, LNM 2137 (2015)
BAKRY, Dominique;
ZRIBI, Olfa
$h$-Transforms and Orthogonal PolynomialsNature: Original
