Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 2, pp. 407-438.

We study points of density 1/2 of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density 1/2 is formulated in terms of the pointwise behaviour of the Ornstein-Uhlembeck semigroup.

Dans cet article nous étudions la structure de l’ensemble des points avec densité 1/2 pour les ensemble de périmètre fini dans un espace gaussien infini-dimensionnel. Nous démontrons que, comme dans le cas de dimension finie, la mesure de surface est concentrée sur cet ensemble de points. Ici, la définition de points avec densité 1/2 est donnée en utilisant le comportement ponctuel du semigroupe de Ornstein-Uhlembeck.

DOI: 10.5802/afst.1297
Ambrosio, Luigi 1; Figalli, Alessio 2

1 Scuola Normale Superiore, p.za dei Cavalieri 7, I-56126 Pisa, Italy.
2 The University of Texas at Austin, Department of Mathematics, 1 University Station C1200, Austin TX 78712, USA
@article{AFST_2011_6_20_2_407_0,
     author = {Ambrosio, Luigi and Figalli, Alessio},
     title = {Surface measures and convergence of the {Ornstein-Uhlenbeck} semigroup in {Wiener} spaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {407--438},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 20},
     number = {2},
     year = {2011},
     doi = {10.5802/afst.1297},
     zbl = {1228.60063},
     mrnumber = {2847889},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1297/}
}
TY  - JOUR
AU  - Ambrosio, Luigi
AU  - Figalli, Alessio
TI  - Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2011
SP  - 407
EP  - 438
VL  - 20
IS  - 2
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1297/
DO  - 10.5802/afst.1297
LA  - en
ID  - AFST_2011_6_20_2_407_0
ER  - 
%0 Journal Article
%A Ambrosio, Luigi
%A Figalli, Alessio
%T Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2011
%P 407-438
%V 20
%N 2
%I Université Paul Sabatier, Institut de mathématiques
%C Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1297/
%R 10.5802/afst.1297
%G en
%F AFST_2011_6_20_2_407_0
Ambrosio, Luigi; Figalli, Alessio. Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 2, pp. 407-438. doi : 10.5802/afst.1297. http://www.numdam.org/articles/10.5802/afst.1297/

[1] Airault (H.) and Malliavin (P.).— Intégration géométrique sur l’espace de Wiener, Bull. des Sciences Math., 112, p. 25-74 (1988). | MR | Zbl

[2] Ambrosio (L.), Fusco (N.) and Pallara (D.).— “Functions of bounded variation and free discontinuity problems", Oxford Mathematical Monographs (2000). | MR | Zbl

[3] Ambrosio (L.), Maniglia (S.), Miranda Jr. (M.) and Pallara (D.).— BV functions in abstract Wiener spaces, J. Funct. Anal., 258, p. 785-813 (2010). | MR | Zbl

[4] Ambrosio (L.), Da Prato (G.) and Pallara (D.).— BV functions in a Hilbert space with respect to a Gaussian measure, Preprint, (2009), http://cvgmt.sns.it/cgi/get.cgi/papers/ambdappal/

[5] Ambrosio (L.), Miranda (M.) and Pallara (D.).— Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability, Discrete Contin. Dyn. Syst. Series A, 28, p. 591-606 (2010). | MR | Zbl

[6] Besicovitch (A.P.).— On the existence of subsets of finite measure of sets of infinite measure, Indag. Math., 14, p. 339-344 (1952). | MR | Zbl

[7] Bogachev (V.I.).— “Gaussian Measures", American Mathematical Society (1998). | MR | Zbl

[8] De Giorgi (E.).— Definizione ed espressione analitica del perimetro di un insieme, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat., 8, p. 390-393 (1953). | MR | Zbl

[9] De Giorgi (E.).— Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl., 4, p. 191-213 (1954). | MR | Zbl

[10] Federer (H.).— A note on the Gauss-Green theorem, Proc. Amer. Math. Soc., 9, p. 447-451 (1958). | MR | Zbl

[11] Federer (H.).— “Geometric measure theory", Springer (1969). | MR | Zbl

[12] Feyel (D.) and De la Pradelle (A.).— Hausdorff measures on the Wiener space, Potential Anal., 1, p. 177-189 (1992). | MR | Zbl

[13] Figalli (A.), Maggi (F.) and Pratelli (A.).— A mass transportation approach to quantitative isoperimetric inequalities, Invent. Math., 182, p. 167-211 (2010). | MR | Zbl

[14] Fukushima (M.).— On semimartingale characterization of functionals of symmetric Markov processes, Electron J. Probab., 4, p. 1-32 (1999). | Zbl

[15] Fukushima (M.).— BV functions and distorted Ornstein-Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal., 174, p. 227-249 (2000). | MR | Zbl

[16] Fukushima (M.) and M. Hino (M.).— On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal., 183, p. 245-268 (2001). | MR | Zbl

[17] Hino (M.).— Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space, J. Funct. Anal., 258, p. 1656-1681 (2010). | MR | Zbl

[18] Ledoux (M.).— Isoperimetry and Gaussian analysis, in ”Lectures on Probability Theory and Statistics“, Saint Flour, 1994 Lecture Notes in Mathematics, 1648, Springer (1996). | MR | Zbl

[19] Ledoux (M.).— Semigroup proof of the isoperimetric inequaliy in Euclidean and Gaussian spaces, Bull. Sci. Math., 118, p. 485-510 (1994). | MR | Zbl

[20] Preiss (D.).— Gaussian measures and the density theorem, Comment. Math. Univ. Carolin., 22, p. 181-193 (1981). | MR | Zbl

[21] Stein (E.M.).— “Topics in Harmonic Analysis related to the Littlewood-Paley theory", Annals of Mathematics Studies 63, Princeton University Press (1970). | MR | Zbl

[22] Zambotti (L.).— Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection, Probab. Theory Relat. Fields, 123, p. 579-600 (2002). | MR | Zbl

[23] Ziemer (W.P.).— “Weakly differentiable functions”, Graduate Texts in Mathematics, Springer (1989). | MR | Zbl

Cited by Sources: