On differential equations and inclusions with mean derivatives on a compact manifold

S.V. Azarina; Yu.E. Gliklikh

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2007)

  • Volume: 27, Issue: 2, page 385-397
  • ISSN: 1509-9407

Abstract

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We introduce and investigate a new sort of stochastic differential inclusions on manifolds, given in terms of mean derivatives of a stochastic process, introduced by Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. For inclusions with forward mean derivatives on manifolds we prove some results on the existence of solutions.

How to cite

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S.V. Azarina, and Yu.E. Gliklikh. "On differential equations and inclusions with mean derivatives on a compact manifold." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 27.2 (2007): 385-397. <http://eudml.org/doc/271188>.

@article{S2007,
abstract = {We introduce and investigate a new sort of stochastic differential inclusions on manifolds, given in terms of mean derivatives of a stochastic process, introduced by Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. For inclusions with forward mean derivatives on manifolds we prove some results on the existence of solutions.},
author = {S.V. Azarina, Yu.E. Gliklikh},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {mean derivatives; differential inclusions; stochastic processes on manifolds; stochastic process},
language = {eng},
number = {2},
pages = {385-397},
title = {On differential equations and inclusions with mean derivatives on a compact manifold},
url = {http://eudml.org/doc/271188},
volume = {27},
year = {2007},
}

TY - JOUR
AU - S.V. Azarina
AU - Yu.E. Gliklikh
TI - On differential equations and inclusions with mean derivatives on a compact manifold
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2007
VL - 27
IS - 2
SP - 385
EP - 397
AB - We introduce and investigate a new sort of stochastic differential inclusions on manifolds, given in terms of mean derivatives of a stochastic process, introduced by Nelson for the needs of the so called stochastic mechanics. This class of stochastic inclusions is ideologically the closest one to ordinary differential inclusions. For inclusions with forward mean derivatives on manifolds we prove some results on the existence of solutions.
LA - eng
KW - mean derivatives; differential inclusions; stochastic processes on manifolds; stochastic process
UR - http://eudml.org/doc/271188
ER -

References

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  1. [1] S.V. Azarina and Yu.E. Gliklikh, Differential inclusions with mean derivatrives, Dynamic Syst. Appl. 16 (2007), 49-71. Zbl1130.34006
  2. [2] Ya.I. Belopolskaya and Yu.L. Dalecky, Stochastic Processes and Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1989. 
  3. [3] Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis and V.V. Obukhovskii, Introduction to the Theory of Multivalued Maps and Differential Inclusions, KomKniga, Moscow, 2005 (in Russian). Zbl1126.34001
  4. [4] K.D. Elworthy, Stochastic Differential Equations on Manifolds, Lect. Notes of London Math. Soc., vol. 70, Cambridge University Press, Cambridge, 1982. Zbl0514.58001
  5. [5] I.I. Gihman and A.V. Skorohod, Theory of Stochastic Processes, Vol. 3, Springer-Verlag, New York, 1979. Zbl0404.60061
  6. [6] Yu.E. Gliklikh, Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, Kluwer, Dordrecht, 1996. Zbl0865.60042
  7. [7] Yu.E. Gliklikh, Global Analysis in Mathematical Physics. Geometric and Stochastic Methods, Springer-Verlag, New York, 1997. 
  8. [8] Yu.E. Gliklikh, Stochastic equations in mean derivatives and their applications. I., Transactions of Russian Acacdemy of Natural Sciences, Series MMMIC. 1 (4) (1997), 26-52 (in Russian). Zbl0915.60065
  9. [9] Yu.E. Gliklikh, Stochastic equations in mean derivatives and their applications. II, Transactions of Russian Acacdemy of Natural Sciences, Series MMMIC 4 (4) (2000), 17-36 (in Russian). 
  10. [10] Yu.E. Gliklikh, Deterministic viscous hydrodynamics via stochastic processes on groups of diffeomorphisms, Probabilistic Metohds in Fluids (I.M. Davis et al., eds), World Scientific, Singapore, 2003, 179-190. Zbl1068.76018
  11. [11] Yu.E. Gliklikh, Stochastic and Global Analysis in Problems of Mathematical Physics, KomKniga, Moscow, 2005 (in Russian). 
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  13. [13] X. He, A probabilistic method for Navier-Stokes vorticies, J. Appl. Probab. 38 (2001), 1059-1066. Zbl1005.76018
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