Stochastic differential inclusions of Langevin type on Riemannian manifolds

Yuri E. Gliklikh; Andrei V. Obukhovskiĭ

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2001)

  • Volume: 21, Issue: 2, page 173-190
  • ISSN: 1509-9407

Abstract

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We introduce and investigate a set-valued analogue of classical Langevin equation on a Riemannian manifold that may arise as a description of some physical processes (e.g., the motion of the physical Brownian particle) on non-linear configuration space under discontinuous forces or forces with control. Several existence theorems are proved.

How to cite

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Yuri E. Gliklikh, and Andrei V. Obukhovskiĭ. "Stochastic differential inclusions of Langevin type on Riemannian manifolds." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 21.2 (2001): 173-190. <http://eudml.org/doc/271454>.

@article{YuriE2001,
abstract = {We introduce and investigate a set-valued analogue of classical Langevin equation on a Riemannian manifold that may arise as a description of some physical processes (e.g., the motion of the physical Brownian particle) on non-linear configuration space under discontinuous forces or forces with control. Several existence theorems are proved.},
author = {Yuri E. Gliklikh, Andrei V. Obukhovskiĭ},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {stochastic differential inclusions; Langevin equation; Riemannian manifolds; stochastic differential equation; Riemannian manifold},
language = {eng},
number = {2},
pages = {173-190},
title = {Stochastic differential inclusions of Langevin type on Riemannian manifolds},
url = {http://eudml.org/doc/271454},
volume = {21},
year = {2001},
}

TY - JOUR
AU - Yuri E. Gliklikh
AU - Andrei V. Obukhovskiĭ
TI - Stochastic differential inclusions of Langevin type on Riemannian manifolds
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2001
VL - 21
IS - 2
SP - 173
EP - 190
AB - We introduce and investigate a set-valued analogue of classical Langevin equation on a Riemannian manifold that may arise as a description of some physical processes (e.g., the motion of the physical Brownian particle) on non-linear configuration space under discontinuous forces or forces with control. Several existence theorems are proved.
LA - eng
KW - stochastic differential inclusions; Langevin equation; Riemannian manifolds; stochastic differential equation; Riemannian manifold
UR - http://eudml.org/doc/271454
ER -

References

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  1. [1] P. Billingsley, Convergence of Probability Measures, New York et al., Wiley 1969. Zbl0944.60003
  2. [2] R.L. Bishop and R.J. Crittenden, Geometry of Manifolds, New York-London, Academic Press 1964. Zbl0132.16003
  3. [3] Yu.G. Borisovich and Yu.E. Gliklikh, On Lefschetz number for a certain class of set-valued maps, 7-th Summer Mathematical School., Kiev (1970), 283-294 (in Russian). 
  4. [4] E.D. Conway, Stochastic equations with discontinuous drift, Trans. Amer. Math. Soc. 157 (1) (1971), 235-245. Zbl0276.60058
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  7. [7] Yu.E. Gliklikh, Fixed points of multivalued mappings with nonconvex images and the rotation of multivalued vector fields, Sbornik Trudov Aspirantov Matematicheskogo Fakul'teta, Voronezh University (1972), 30-38 (in Russian). 
  8. [8] Yu.E. Gliklikh and I.V. Fedorenko, On the geometrization of a certain class of mechanical systems with random perturbations of the force, Voronezh University, Deposited in VINITI, October 21, 1980, N 4481 (in Russian). 
  9. [9] Yu.E. Gliklikh and I.V. Fedorenko, Equations of geometric mechanics with random force fields, Priblizhennye metody issledovaniya differentsial'nykh uravneni i ikh prilozheniya, Kubyshev 1981, 64-72 (in Russian). 
  10. [10] Yu.E. Gliklikh, Riemannian parallel translation in non-linear mechanics, Lect. Notes Math. 1108 (1984), 128-151. 
  11. [11] Yu.E. Gliklikh, Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics, Dordrecht, Kluwer 1996, xvi+189. Zbl0865.60042
  12. [12] Yu.E. Gliklikh, Global Analysis in Mathematical Physics, Geometric and Stochastic Methods, New York, Springer-Verlag 1997, xv+213. 
  13. [13] A.N. Kolmogorov and S.V. Fomin, Elements of theory of functions and functional analysis, Moscow, Nauka 1968. Zbl0235.46001
  14. [14] W. Kryszewski, Homotopy properties of set-valued mappings, Toruń, Toruń University 1997, 243. Zbl1250.54022
  15. [15] J. Motyl, On the Solution of Stochastic Differential Inclusion, J. Math. Anal. and Appl. 192 (1995), 117-132. Zbl0826.60053
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