Geodesic
On differential equations and inclusions with mean derivatives on a compact manifold
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 27 (2007) no. 2, pp. 385-397.

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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.
Keywords: mean derivatives, differential inclusions, stochastic processes on manifolds 
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Azarina, S.; Gliklikh, Yu. On differential equations and inclusions with mean derivatives on a compact manifold. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 27 (2007) no. 2, pp. 385-397. http://geodesic.mathdoc.fr/item/DMDICO_2007_27_2_a8/

[1] S.V. Azarina and Yu.E. Gliklikh, Differential inclusions with mean derivatrives, Dynamic Syst. Appl. 16 (2007), 49-71.

[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).

[4] K.D. Elworthy, Stochastic Differential Equations on Manifolds, Lect. Notes of London Math. Soc., vol. 70, Cambridge University Press, Cambridge, 1982.

[5] I.I. Gihman and A.V. Skorohod, Theory of Stochastic Processes, Vol. 3, Springer-Verlag, New York, 1979.

[6] Yu.E. Gliklikh, Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, Kluwer, Dordrecht, 1996.

[7] Yu.E. Gliklikh, Global Analysis in Mathematical Physics. Geometric and Stochastic Methods, Springer-Verlag, New York, 1997.

[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).

[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.

[11] Yu.E. Gliklikh, Stochastic and Global Analysis in Problems of Mathematical Physics, KomKniga, Moscow, 2005 (in Russian).

[12] Yu.E. Gliklikh and L.A. Morozova, On Itô stochastic differential equations on infinite products of Riemannian manifolds, Transactions of Russian Academy of Natural Scences. Series MMMIC 2 (1) (1998), 71-79 (in Russian).

[13] X. He, A probabilistic method for Navier-Stokes vorticies, J. Appl. Probab. 38 (2001), 1059-1066.

[14] K. Itô, Stochastic Differential Equations in a Differentiable Manifold (2), Mem. Coll. Sci. Univ. Kyoto., Ser. A 28 (10 (1953), 81-85.

[15] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. 63 (1956), 20-63.

[16] E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Reviews 150 (4) (1966), 1079-1085.

[17] E. Nelson, Dynamical Theory of Brownian Motion, Princeton University Press, Princeton, 1967.

[18] E. Nelson, Quantum Fluctuations, Princeton University Press, Princeton, 1985.

[19] D. Pollard, Convergence of Stochastic Processes, Springer-Verlag, Berlin, 1984.

[20] A.N. Shiryaev, Probability, Springer-Verlag, New York, 1984.

[21] Y. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965.