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

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

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

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topS.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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- [2] Ya.I. Belopolskaya and Yu.L. Dalecky, Stochastic Processes and Differential Geometry, Kluwer Academic Publishers, Dordrecht, 1989.
- [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
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- [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] 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] 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] Yu.E. Gliklikh, Stochastic and Global Analysis in Problems of Mathematical Physics, KomKniga, Moscow, 2005 (in Russian).
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- [13] X. He, A probabilistic method for Navier-Stokes vorticies, J. Appl. Probab. 38 (2001), 1059-1066. Zbl1005.76018
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- [15] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20-63. Zbl0070.38603
- [16] E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Reviews 150 (4) (1966), 1079-1085.
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- [18] E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1985. Zbl0563.60001
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