Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 12 (2007) > Paper 37 open journal systems 


Dynamical properties and characterization of gradient drift diffusions

Sébastien Darses, Boston University
Ivan Nourdin, University Paris VI


Abstract
We study the dynamical properties of the Brownian diffusions having σ Id as diffusion coefficient matrix and b=∇U as drift vector. We characterize this class through the equality D+2=D-2, where D+ (resp. D-) denotes the forward (resp. backward) stochastic derivative of Nelson's type. Our proof is based on a remarkable identity for D+2-D-2 and on the use of the martingale problem.


Full text: PDF

Pages: 390-400

Published on: October 21, 2007
Bibliography
  1. Cresson, J.; Darses, S. Plongement stochastique des systèmes lagrangiens. (French) [Stochastic embedding of Lagrangian systems] C. R. Math. Acad. Sci. Paris 342 (2006), no. 5, 333-336. Math. Review MR2201959 (2006i:37115)
  2. Darses, S.; Nourdin, I. Stochastic derivatives for fractional diffusions Ann. Probab. 35 (2007), no. 5, 1998-2020. Math. Review number not yet available.
  3. Föllmer, H. Time reversal on Wiener space. Stochastic processes---mathematics and physics (Bielefeld, 1984), 119-129, Lecture Notes in Math., 1158, Springer, Berlin, 1986. Math. Review MR0838561 (88a:60140)
  4. Karatzas, I.; Shreve, S. E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8 Math. Review MR1121940 (92h:60127)
  5. Kolmogoroff, A. Zur Umkehrbarkeit der statistischen Naturgesetze. (German) Math. Ann. 113 (1937), no. 1, 766--772. Math. Review MR1513121
  6. Millet, A.; Nualart, D.; Sanz, M. Integration by parts and time reversal for diffusion processes. Ann. Probab. 17 (1989), no. 1, 208--238. Math. Review MR0972782 (90m:60088)
  7. Nelson, E. Dynamical theory of Brownian motion. Princeton University Press. Second edition. Math. Review number not available.
  8. Pardoux, É. Grossissement d'une filtration et retournement du temps d'une diffusion. (French) [Enlargement of a filtration and time reversal of a diffusion] Séminaire de Probabilités, XX, 1984/85, 48--55, Lecture Notes in Math., 1204, Springer, Berlin, 1986. Math. Review MR0942014 (89e:60150)
  9. Rogers, L. C. G.; Williams, D. Diffusions, Markov processes, and martingales. Vol. 2. Itô calculus. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1987. xiv+475 pp. ISBN: 0-471-91482-7 Math. Review MR0921238 (89k:60117)
  10. Roelly, S.; Thieullen, M. Duality formula for the bridges of a Brownian diffusion: application to gradient drifts. Stochastic Process. Appl. 115 (2005), no. 10, 1677--1700. Math. Review MR2165339 (2006k:60069)
  11. Royer, G. Une initiation aux inégalités de Sobolev logarithmiques. (French) [An introduction to logarithmic Sobolev inequalities] Cours Spécialisés [Specialized Courses], 5. Société Mathématique de France, Paris, 1999. viii+114 pp. ISBN: 2-85629-075-2 Math. Review MR1704288 (2000k:60002)
  12. Thieullen, M. Second order stochastic differential equations and non-Gaussian reciprocal diffusions. Probab. Theory Related Fields 97 (1993), no. 1-2, 231--257. Math. Review MR1240725 (94k:60081)
  13. Zheng, W. A.; Meyer, P.-A. Quelques résultats de "mécanique stochastique". (French) [Some results in "stochastic mechanics"] Seminar on probability, XVIII, 223--244, Lecture Notes in Math., 1059, Springer, Berlin, 1984. Math. Review MR0770964 (86c:81026)
















Research
Support Tool
Capture Cite
View Metadata
Supplementary
Files and / or
CORRECTIONS
Printer Friendly
Context
Author Address
Action
Email Author
Email Others


Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X