Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space II

Viorel Barbu, Giuseppe Da Prato, and Luciano Tubaro

Full-text: Open access

Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 3 (2011), 699-724.

Dates
First available in Project Euclid: 23 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1308834856

Digital Object Identifier
doi:10.1214/10-AIHP381

Mathematical Reviews number (MathSciNet)
MR2841072

Zentralblatt MATH identifier
1230.60081

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 15A63: Quadratic and bilinear forms, inner products [See mainly 11Exx] 31C25: Dirichlet spaces

Keywords
Neumann problem Ornstein–Uhlenbeck operator Kolmogorov operator Reflection problem Infinite-dimensional analysis

Citation

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. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 3, 699--724. doi:10.1214/10-AIHP381. https://projecteuclid.org/euclid.aihp/1308834856


Export citation

References

  • [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.
  • [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.
  • [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.
  • [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.
  • [5] V. I. Bogachev. Gaussian Measures. Mathematical Surveys and Monographs 62. Amer. Math. Soc., Providence, RI, 1998.
  • [6] E. Cepà. Problème de Skorohod multivoque. Ann. Probab. 26 (1998) 500–532.
  • [7] G. Da Prato. Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Basel, 2004.
  • [8] G. Da Prato and A. Lunardi. Elliptic operators with unbounded drift-coefficients and Neumann boundary condition. J. Differential Equations 198 (2004) 35–52.
  • [9] G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes 229. Cambridge Univ. Press, 1996.
  • [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.
  • [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.
  • [12] P. Malliavin. Stochastic Analysis. Springer, Berlin, 1997.
  • [13] P. A. Meyer and W. A. Zheng. Tightness criteria for laws of semimartingales. Ann. Inst. H. Poincaré 20 (1984) 353–372.
  • [14] D. Nualart and E. Pardoux. White noise driven by quasilinear SPDE’s with reflection. Probab. Theory Related Fields 93 (1992) 77–89.
  • [15] A. V. Skorohod. Integration in Hilbert Space. Springer, New York, 1974.
  • [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.