Abstract
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 on 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Albeverio and S. Fei, “Remark on symmetry of stochastic dynamical systems and their conserved quantities,” J. Phys. A, 28, 6363–6371 (1995).
J.-M. Bismut, Mecanique Aleatoire, Lect. Notes Math., Vol. 866, Springer, Berlin (1981).
M. Blaszak, Z. Domański, A. Sergyeyev, and B. Szablikowski, “Integrable quantum Stäckel systems,” Phys. Lett. A, 377, No. 38, 2564–2572 (2013).
A. V. Bolsinov and V. S. Matveev, “Geometrical interpretation of Benenti systems,” J. Geom. Phys., 44, No. 4, 489–506 (2003).
A. N. Borodin and M. I. Freidlin, “Fast oscillating random perturbations of dynamical systems with conservation laws,” Ann. Inst. H. Poincaré Probab. Stat., 31, No. 3, 485–525 (1995).
C. Duval and G. Valent, “Quantum integrability of quadratic Killing tensors,” J. Math. Phys., 46, No. 5, 053516 (2005).
A. T. Fomenko and A. V. Bolsinov, Integrable Hamiltonian Systems: Geometry, Topology, Classification, Chapman & Hall/CRC, Boca Raton (2004).
M. Freidlin and M. Weber, “Random perturbations of dynamical systems and diffusion processes with conservation laws,” Probab. Theory Relat. Fields, 128, No. 3, 441–466 (2004).
A. R. Galmarino, “Representation of an isotropic diffusion as a skew product,” Z. Wahrsch. Verw. Gebiete, 1, No. 4, 359–378 (1963).
M. Gitterman, The Noisy Oscillator: The First Hundred Years, from Einstein until Now, World Scientific, New York (2005).
K. Grove, H. Karcher, and E. A. Ruh, “Group actions and curvature,” Invent. Math., 23, 31–48 (1974).
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Math. Lib., Vol. 24, North-Holland (1981).
B. Jovanovic, “Symmetries and integrability,” Publ. Inst. Math. (Beograd), 84 (98), 1–36 (2008).
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, Cambridge (1997).
J.-A. Lázaro-Camíand J.-P. Ortega, “Reduction, reconstruction, and skew-product decomposition of symmetric stochastic differential equations,” Stoch. Dyn., 9, No. 1, 1–46 (2009).
Xue-Mei Li, “An averaging principle for a completely integrable stochastic Hamiltonian system,” Nonlinearity, 21, No. 4, 803–822 (2008).
M. Liao, “A decomposition of Markov processes via group action,” J. Theor. Probab., 22, No. 1, 164–185 (2009).
J. Liouville, “Note sur l’intégration des équations différentielles de la dynamique,” J. Math. Pures Appl., 20 137–138 (1855).
L. Markus and A. Weerasinghe, “Stochastic oscillators,” J. Differ. Equ., 71, No. 2, 288–314 (1988).
V. S. Matveev, “Quantum integrability of the Beltrami–Laplace operator for geodesically equivalent metrics,” Russ. Math. Dokl., 61, No. 2 216–219 (2000).
T. Misawa, “Conserved quantities and symmetries related to stochastic dynamical systems,” Ann. Inst. Stat. Math., 51, No. 4, 779–802 (1999).
B. Øksendal, Stochastic Differential Equations, Springer, Berlin (2003).
E. J. Pauwels and L. C. G. Rogers, “Skew-product decompositions of Brownian motions,” Contemp. Math., 73, 237–262 (1988).
M. Taylor, Pseudodifferential Operators, Springer, New York (1996).
N. T. Zung, “Torus actions and integrable systems,” in: Topological Methods in the Theory of Integrable Systems, Cambridge Sci. Publ., Cambridge (2006), pp. 289–328.
N. T. Zung, “A general approach to the problem of action-angle variables,” in preparation (see also the earlier version: “Action-angle variables on Dirac manifolds,” arXiv:1204.3865).
N. T. Zung and N. T. Thien, “Physics-like second-order models of financial assets prices,” in preparation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Anatoly Timofeevich Fomenko on the occasion of his 70th birthday
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 20, No. 3, pp. 213–249, 2015.
Rights and permissions
About this article
Cite this article
Zung, N.T., Thien, N.T. Reduction and Integrability of Stochastic Dynamical Systems. J Math Sci 225, 681–706 (2017). https://doi.org/10.1007/s10958-017-3486-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-017-3486-1