no. 3
Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 699-724.

Dans cet article nous étudions l'existence et la régularité des solutions d'un problème de Neumann associé à un opérateur de Ornstein-Uhlenbeck défini sur un domaine convexe K, borné et régulier dans un espace de Hilbert H. Le problème est lié à un problème de réflexion associé à une équation différentielle stochastique dans le domaine K.

This work is concerned with the existence and regularity of solutions to the Neumann problem associated with a Ornstein-Uhlenbeck operator on a bounded and smooth convex set K of a Hilbert space H. This problem is related to the reflection problem associated with a stochastic differential equation in K.

DOI : https://doi.org/10.1214/10-AIHP381
Classification : 60J60,  47D07,  15A63,  31C25
Mots clés : Neumann problem, Ornstein-Uhlenbeck operator, Kolmogorov operator, reflection problem, infinite-dimensional analysis 
@article{AIHPB_2011__47_3_699_0,
     author = {Barbu, Viorel and Da Prato, Giuseppe and Tubaro, Luciano},
     title = {Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a {Hilbert} space {II}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {699--724},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {3},
     year = {2011},
     doi = {10.1214/10-AIHP381},
     zbl = {1230.60081},
     mrnumber = {2841072},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP381/}
}
TY  - JOUR
AU  - Barbu, Viorel
AU  - Da Prato, Giuseppe
AU  - Tubaro, Luciano
TI  - Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
DA  - 2011///
SP  - 699
EP  - 724
VL  - 47
IS  - 3
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/10-AIHP381/
UR  - https://zbmath.org/?q=an%3A1230.60081
UR  - https://www.ams.org/mathscinet-getitem?mr=2841072
UR  - https://doi.org/10.1214/10-AIHP381
DO  - 10.1214/10-AIHP381
LA  - en
ID  - AIHPB_2011__47_3_699_0
ER  - 
Barbu, Viorel; Da Prato, Giuseppe; Tubaro, Luciano. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 699-724. doi : 10.1214/10-AIHP381. http://www.numdam.org/articles/10.1214/10-AIHP381/

[1] L. Ambrosio, G. Savaré and L. Zambotti. Existence and stability for Fokker-Planck equations with log-concave reference measure. Probab. Theory Related Fields. 145 (2009) 517-564. | MR 2529438

[2] V. Barbu and G. Da Prato. The Neumann problem on unbounded domains of ℝd and stochastic variational inequalities. Comm. PDE 30 (2005) 1217-1248. | MR 2180301 | Zbl 1130.35137

[3] V. Barbu and G. Da Prato. The generator of the transition semigroup corresponding to a stochastic variational inequality. Comm. PDE 33 (2008) 1318-1338. | MR 2450160 | Zbl 1155.60034

[4] V. Barbu, G. Da Prato and L. Tubaro. Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space. Ann. Probab. 37 (2009) 1427-1458. | MR 2546750 | Zbl 1205.60141

[5] V. I. Bogachev. Gaussian Measures. Mathematical Surveys and Monographs 62. Amer. Math. Soc., Providence, RI, 1998. | MR 1642391 | Zbl 0913.60035

[6] E. Cepà. Problème de Skorohod multivoque. Ann. Probab. 26 (1998) 500-532. | MR 1626174 | Zbl 0937.34046

[7] G. Da Prato. Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Basel, 2004. | MR 2111320 | Zbl 1066.60061

[8] G. Da Prato and A. Lunardi. Elliptic operators with unbounded drift-coefficients and Neumann boundary condition. J. Differential Equations 198 (2004) 35-52. | MR 2037749 | Zbl 1046.35025

[9] G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes 229. Cambridge Univ. Press, 1996. | MR 1417491 | Zbl 0849.60052

[10] G. Da Prato and J. Zabczyk. Second Order Partial Differential Equations in Hilbert Spaces. London Mathematical Society Lecture Notes 293. Cambridge Univ. Press, 2002. | MR 1985790 | Zbl 1012.35001

[11] A. Hertle. Gaussian surface measures and the Radon transform on separable Banach spaces. In Measure Theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979) 513-531. Lecture Notes in Math. 794. Springer, Berlin, 1980. | MR 577995 | Zbl 0431.28011

[12] P. Malliavin. Stochastic Analysis. Springer, Berlin, 1997. | MR 1450093 | Zbl 0878.60001

[13] P. A. Meyer and W. A. Zheng. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré 20 (1984) 353-372. | Numdam | MR 771895 | Zbl 0551.60046

[14] D. Nualart and E. Pardoux. White noise driven by quasilinear SPDE's with reflection. Probab. Theory Related Fields 93 (1992) 77-89. | MR 1172940 | Zbl 0767.60055

[15] A. V. Skorohod. Integration in Hilbert Space. Springer, New York, 1974. | MR 466482

[16] L. Zambotti. Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection. Probab. Theory Related Fields 123 (2002) 579-600. | MR 1921014 | Zbl 1009.60047

Cité par Sources :