Geodesic
Reduction and integrability of stochastic dynamical systems
Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 3, pp. 213-249.

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This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction, and integrability. In particular, we show that an SDS that is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS's, and extend the results about the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS's. We also show how integrable SDS's are related to compatible families of integrable Riemannian metrics on manifolds. 
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Nguyen Tien Zung; Nguyen Thanh Thien. Reduction and integrability of stochastic dynamical systems. Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 3, pp. 213-249. http://geodesic.mathdoc.fr/item/FPM_2015_20_3_a10/

[1] Albeverio S., Fei S., “Remark on symmetry of stochastic dynamical systems and their conserved quantities”, J. Phys. A, 28 (1995), 6363–6371 | DOI | MR | Zbl

[2] Bismut J. M., Mecanique Aleatoire, Lect. Notes Math., 866, Springer, Berlin, 1981 | DOI | MR | Zbl

[3] Błaszak M, Domański Z., Sergyeyev A., Szablikowski B., “Integrable quantum Stäckel systems”, Phys. Lett. A, 377:38 (2013), 2564–2572 | DOI | MR

[4] Bolsinov A. V., Matveev V. S., “Geometrical interpretation of Benenti systems”, J. Geom. Phys., 44:4 (2003), 489–506 | DOI | MR | Zbl

[5] Borodin A. N., Freidlin M. I., “Fast oscillating random perturbations of dynamical systems with conservation laws”, Ann. Inst. H. Poincaré Probab. Statist., 31:3 (1995), 485–525 | MR | Zbl

[6] Duval C., Valent G., “Quantum integrability of quadratic Killing tensors”, J. Math. Phys., 46:5 (2005), 053516 | DOI | MR | Zbl

[7] Fomenko A. T., Bolsinov A. V., Integrable Hamiltonian Systems: Geometry, Topology, Classification, Chapman Hall/CRC, Boca Raton, 2004 | MR | Zbl

[8] Freidlin M., Weber M., “Random perturbations of dynamical systems and diffusion processes with conservation laws”, Probab. Theory Related Fields, 128:3 (2004), 441–466 | DOI | MR | Zbl

[9] Galmarino A. R., “Representation of an isotropic diffusion as a skew product”, Z. Wahrsch. Verw. Gebiete, 1:4 (1963), 359–378 | DOI | MR | Zbl

[10] Gitterman M., The Noisy Oscillator: The First Hundred Years, from Einstein Until Now, World Scientific, New York, 2005 | MR | Zbl

[11] Grove K., Karcher H., Ruh E. A., “Group actions and curvature”, Invent. Math., 23 (1974), 31–48 | DOI | MR | Zbl

[12] Ikeda N., Watanabe S., Stochastic Differential Equations and Diffusion Processes, North-Holland Math. Lib., 24, North-Holland, 1981 | MR | Zbl

[13] Jovanovic B., “Symmetries and integrability”, Publ. Inst. Math. (Beograd), 84(98) (2008), 1–36 | DOI | MR | Zbl

[14] Kunita H., Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[15] Lázaro-Camí J.-A., Ortega J.-P., “Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations”, Stoch. Dyn., 9:1 (2009), 1–46 | DOI | MR | Zbl

[16] Li Xue-Mei, “An averaging principle for a completely integrable stochastic Hamiltonian system”, Nonlinearity, 21:4 (2008), 803–822 | DOI | MR | Zbl

[17] Liao M., “A decomposition of Markov processes via group action”, J. Theor. Probab., 22:1 (2009), 164–185 | DOI | MR | Zbl

[18] Liouville J., “Note sur l'intégration des équations différentielles de la dynamique”, J. Math. Pures Appl., 20 (1855), 137–138

[19] Markus L., Weerasinghe A., “Stochastic oscillators”, J. Differ. Equ., 71:2 (1988), 288–314 | DOI | MR | Zbl

[20] Matveev V. S., “Quantum integrability of the Beltrami–Laplace operator for geodesically equivalent metrics”, Russ. Math. Dokl., 61:2 (2000), 216–219 | MR | Zbl

[21] Misawa T., “Conserved quantities and symmetries related to stochastic dynamical systems”, Ann. Inst. Stat. Math., 51:4 (1999), 779–802 | DOI | MR | Zbl

[22] Øksendal B., Stochastic Differential Equations, Springer, Berlin, 2003 | MR

[23] Pauwels E. J., Rogers L. C. G., “Skew-product decompositions of Brownian motions”, Contemp. Math., 73 (1988), 237–262 | DOI | MR | Zbl

[24] Taylor M., Pseudodifferential Operators, Springer, New York, 1996 | MR

[25] Zung N. T., “Torus actions and integrable systems”, Topological Methods in the Theory of Integrable Systems, Cambridge Sci. Publ., Cambridge, 2006, 289–328 | MR | Zbl

[26] Zung N. T., A general approach to the problem of action-angle variables, In preparation

[27] Zung N. T., Thien N. T., Physics-like second-order models of financial assets prices, In preparation