Probability Theory
A Bismut type formula for the Hessian of heat semigroups
Comptes Rendus. Mathématique, Volume 336 (2003) no. 8, pp. 661-666.

We obtain an intrinsic version of a Bismut type formula for the Hessian of heat semigroups, resp. harmonic functions, by computing second order directional derivatives of families of martingales, along with filtering of redundant noise. As applications we provide a Hessian estimate in the general case as well as a slightly improved one in the radially symmetric situation.

On écrit une formule de Bismut intrinsèque pour la hessienne d'un semigroupe de la chaleur ou d'une fonction harmonique sur une variété, en calculant des dérivées secondes directionnelles de familles de martingales, et en filtrant ensuite le bruit superflu. Cela nous permet d'obtenir des estimées de la hessienne dans un cadre très général. Avec des hypothèses de symétrie radiale de la variété, on améliore encore ces estimées.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00123-7
Arnaudon, Marc 1; Plank, Holger 2; Thalmaier, Anton 3

1 Département de mathématiques, Université de Poitiers, Téléport 2, BP 30179, 86962 Futuroscope Chasseneuil cedex, France
2 Universität Regensburg, NWF I, Mathematik, 93040 Regensburg, Germany
3 Université d'Evry, laboratoire d'analyse et probabilité, département de mathématiques, bd François Mitterrand, 91025 Evry cedex, France
@article{CRMATH_2003__336_8_661_0,
     author = {Arnaudon, Marc and Plank, Holger and Thalmaier, Anton},
     title = {A {Bismut} type formula for the {Hessian} of heat semigroups},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {661--666},
     publisher = {Elsevier},
     volume = {336},
     number = {8},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00123-7},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00123-7/}
}
TY  - JOUR
AU  - Arnaudon, Marc
AU  - Plank, Holger
AU  - Thalmaier, Anton
TI  - A Bismut type formula for the Hessian of heat semigroups
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 661
EP  - 666
VL  - 336
IS  - 8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S1631-073X(03)00123-7/
DO  - 10.1016/S1631-073X(03)00123-7
LA  - en
ID  - CRMATH_2003__336_8_661_0
ER  - 
%0 Journal Article
%A Arnaudon, Marc
%A Plank, Holger
%A Thalmaier, Anton
%T A Bismut type formula for the Hessian of heat semigroups
%J Comptes Rendus. Mathématique
%D 2003
%P 661-666
%V 336
%N 8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00123-7/
%R 10.1016/S1631-073X(03)00123-7
%G en
%F CRMATH_2003__336_8_661_0
Arnaudon, Marc; Plank, Holger; Thalmaier, Anton. A Bismut type formula for the Hessian of heat semigroups. Comptes Rendus. Mathématique, Volume 336 (2003) no. 8, pp. 661-666. doi : 10.1016/S1631-073X(03)00123-7. http://www.numdam.org/articles/10.1016/S1631-073X(03)00123-7/

[1] Arnaudon, M.; Thalmaier, A. Bismut type differentiation of semigroups, Probab. Theory and Math. Statist., Vilnius, 1998, VSP/TEV, Utrecht and Vilnius, 1999, pp. 23-32

[2] Arnaudon, M.; Thalmaier, A. Horizontal martingales in vector bundles, Séminaire de Probabilités, XXXVI, Lecture Notes in Math., 1801, Springer, Berlin, 2003, pp. 419-456

[3] Elworthy, K.D.; Le Jan, Y.; Li, X.-M. On the Geometry of Diffusion Operators and Stochastic Flows, Lecture Notes in Math., 1720, Springer, Berlin, 1999

[4] Elworthy, K.D.; Li, X.-M. Formulae for the derivatives of heat semigroups, J. Funct. Anal., Volume 125 (1994), pp. 252-286

[5] Malliavin, P. Stochastic Analysis, Grundlehren Math. Wiss., 313, Springer-Verlag, Berlin, 1997

[6] H. Plank, Stochastic representation of the gradient and Hessian of diffusion semigroups on Riemannian manifolds, Ph.D. Thesis, Universität Regensburg, 2002

[7] Thalmaier, A.; Wang, F.-Y. Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal., Volume 155 (1998), pp. 109-124

Cited by Sources: