Abstract
The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.
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REFERENCES
Abraham, R. and Marsden, J. (1978). Foundations of Mechanics, 2nd ed., Benjamin, New York.
Albeverio, S. and Fei, S. M. (1995). A remark on symmetry of stochastic dynamical systems and their conserved quantities, J. Phys. A, 28, 6363–6371.
Arnold, L. (1973). Stochastic Differential Equations: Theory and Applications, Wiley, New York.
Bismut, J. M. (1981). Méchanique Aléatoire, Lecture Notes in Math., 866, Springer, Berlin.
Bluman, G. W. and Cole, J. (1974). Similarity Method for Differential Equations, Springer, Berlin.
Bluman, G. W. and Kumei, S. (1989). Symmetries and Differential Equations, Springer, Berlin.
Eisenhert, L. F. (1961). Continuous Groups of Transformations, Dover, New York.
Greenspan, D. (1984). Conservative numerical methods for x = ∫(x), J. Comput. Phys., 56, 28–41.
Hojman, S. (1992). A new conservation law constructed without using Lagrangians and Hamiltonians, J. Phys. A, 25, L291–L295.
Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland/Kodansha, Amsterdam/Tokyo.
Itoh, Y. (1993). Stochastic model of an integrable non-linear system, J. Phys. Soc. Japan, 62, 1826–1828.
Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer, Berlin.
Kobayashi, S. and Nomizu, K. (1969). Foundations of Differential Geometry, Wiley, New York.
Maeda, S. (1987). The similarity method for difference equations, IMA J. Appl. Math., 38, 129–134.
Maruyama, K. and Itoh, Y. (1991). Stochastic differential equations for the Wright-Fisher model, Proc. Inst. Statist. Math., 39, 47–52 (in Japanese).
Maruyama, K. and Itoh, Y. (1997). An application of isotropic diffusion on a sphere to Fisher-Wright model (submitted).
Mimura, F. and Nono, T. (1995). A method for deriving new conservation laws, Bulletin of Kyushu Institute of Technology: Mathematics and Natural Sciences, 42, 1–17.
Misawa, T. (1994a). Conserved quantities and symmetry for stochastic dynamical systems, Phys. Lett. A, 195, 185–189.
Misawa, T. (1994b). New conserved quantities derived from symmetry for stochastic dynamical systems, J. Phys. A, 27, L777–L782.
Nakamura, Y. (1994). Stochastic Lax representation and random collision models, J. Phys. Soc. Japan, 63, 827–829.
Samuelson, P. A. (1972). The general saddlepoint property of optimal-control motions, J. Econom. Theory, 5, 102–120.
Sató, R. and Ramachandran, R. V. (1990). Conservation Laws and Symmetry: Application to Economics and Finance, Kluwer, Boston.
Thieullen, M. and Zambrini, J.-C. (1997). Probability and quantum symmetries I: The theorem of Noether in Schrödinger's Euclidean quantum mechanics, Ann. Inst. II. Poincaré, Phys. Théor., 67, 297–338.
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Misawa, T. Conserved Quantities and Symmetries Related to Stochastic Dynamical Systems. Annals of the Institute of Statistical Mathematics 51, 779–802 (1999). https://doi.org/10.1023/A:1004095516648
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DOI: https://doi.org/10.1023/A:1004095516648