| Title:
|
Stochastic parallel transport and connections of $H^2M$ (English) |
| Author:
|
Catuogno, Pedro |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
35 |
| Issue:
|
4 |
| Year:
|
1999 |
| Pages:
|
305-315 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we prove that there is a bijective correspondence between connections of $H^2M$, the principal bundle of the second order frames of $M$, and stochastic parallel transport in the tangent space of $M$. We construct in a direct geometric way a prolongation of connections without torsion of $M$ to connections of $H^2M$. We interpret such prolongation in terms of stochastic calculus. (English) |
| Keyword:
|
second order geometry |
| Keyword:
|
stochastic calculus |
| Keyword:
|
connections |
| Keyword:
|
parallel transport |
| MSC:
|
53B15 |
| MSC:
|
53C05 |
| MSC:
|
58J65 |
| idZBL:
|
Zbl 1049.58035 |
| idMR:
|
MR1744518 |
| . |
| Date available:
|
2008-06-06T22:24:41Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107705 |
| . |
| Reference:
|
[1] Bismut J. M.: Mécanique Aléatorie.Lecture Notes in Mathematics 866, Springer 1981. MR 0629977 |
| Reference:
|
[2] Catuogno P. J.: Second Order Connections and Stochastic Calculus.Relatorio de Pesquisa. Unicamp, 1995. |
| Reference:
|
[3] Cordero L. A., Dobson C. T. J., de León M.: Differential Geometry of Frame Bundles.Kluwer Acad. Pub. 1988. |
| Reference:
|
[4] Emery M.: Stochastic Calculus in Manifolds.Springer-Verlag, 1989. Zbl 0697.60060, MR 1030543 |
| Reference:
|
[5] Emery M.: On Two Transfer Principles in Stochastic Differential Geometry.Séminaire de Probabilités XXIV. Lecture Notes in Mathematics 1426, Springer 1990. Zbl 0704.60066, MR 1071558 |
| Reference:
|
[6] Gancarzewicz J.: Connections of Order r.Ann. Pol. Math. 34, 70–83, 1977. Zbl 0347.53008, MR 0440471 |
| Reference:
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[7] Ikeda N., Watanabe S.: Stochastic Differential Equations and Diffusion Processes.North-Holland 1981. Zbl 0495.60005, MR 1011252 |
| Reference:
|
[8] Itô K.: Stochastic Parallel Displacement.Probabilistic Methods in Differential Equations. Lecture Notes in Mathematics 451, Springer 1975. Zbl 0308.60027, MR 0394905 |
| Reference:
|
[9] Janyška J., Kolář I.: On the Connections Naturally Induced on the Second Order Frame Bundle.Archivum Mathematicum (Brno) 22, 21–28, 1986. Zbl 0628.53034, MR 0868117 |
| Reference:
|
[10] Kobayashi S.: Canonical Forms on Frame Bundles of Higher Order Contact.Proc. Symp. Pure Math. 3, 186–193, 1961. MR 0126810 |
| Reference:
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[11] Kobayashi S., Nomizu K.: Foundations of Differential Geometry.Interscience, 1 (1963), 2 (1968). Zbl 0119.37502, MR 0152974 |
| Reference:
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[12] Kolář I.: On some Operations with Connections.Math. Nachr. 69, 297–306, 1975. Zbl 0318.53034, MR 0391157 |
| Reference:
|
[13] Meyer P. A. : Géométrie Différentielle Stochastique.Séminaire de Probabilités XV. Lecture Notes in Mathematics 851, Springer 1981. |
| Reference:
|
[14] Meyer P. A. : Géométrie Différentielle Stochastique (bis).Séminaire de Probabilités XVI. Lecture Notes in Mathematics 921, Springer 1982. Zbl 0539.58039, MR 0658725 |
| Reference:
|
[15] Schwartz L.: Géométrie Différentielle du 2$^{\text{e}}$ordre, Semimartingales et Équations Différentielle Stochastiques sur une Variété Différentielle.Séminaire de Probabilités XVI. Lecture Notes in Mathematics 921, Springer 1982. MR 0658722 |
| Reference:
|
[16] Shigekawa I.: On Stochastic Horizontal Lifts.Z. Wahrscheinlichkeitsheorie verw. Geviete 59, 211–221, 1982. Zbl 0487.60056, MR 0650613 |
| . |
