On Lie-point symmetries for Ito stochastic differential equations
Tóm tắt
In the deterministic realm, both differential equations and symmetry generators are geometrical objects, and behave properly under changes of coordinates; actually this property is essential to make symmetry analysis independent of the choice of coordinates and applicable. When trying to extend symmetry analysis to stochastic (Ito) differential equations, we are faced with a problem inherent to their very nature: they are not geometrical object, and they behave in their own way (synthesized by the Ito formula) under changes of coordinates. Thus it is not obvious that symmetries are preserved under a change of coordinates. We will study when this is the case, and when it is not; the conclusion is that this is always the case for so called simple symmetries. We will also note that Kozlov theory relating symmetry and integrability for stochastic differential equations is confirmed by our considerations and results, as symmetries of the type relevant in it are indeed of the type preserved under coordinate changes.
Tài liệu tham khảo
D.V. Alexseevsky, A.M. Vinogradov and V.V. Lychagin, Basic Ideas and Concepts of Differential Geometry, Springer 1991
L. Arnold, Random dynamical systems, Springer 1988
L. Arnold and P. Imkeller, “Normal forms for stochastic differential equations”, Prob. Th. Rel. Fields 110 (1998), 559–588
S.S. Chern, W.H. Chen and K.S. Lam, Lectures on Differential Geometry, World Scientific, 2000
G. Cicogna and G. Gaeta, Symmetry and perturbation theory in nonlinear dynamics, Springer 1999
L.C. Evans, An introduction to stochastic differential equations, A.M.S. 2013
D. Freedman, Brownian motion and diffusion, Springer 1983
G. Gaeta, “Lie-point symmetries and stochastic differential equations II”, J. Phys. A 33 (2000), 4883–4902
G. Gaeta, “Symmetry of stochastic non-variational differential equations”, Phys. Rep. 686 (2017), 1–62
G. Gaeta and C.Lunini, “Symmetry and integrability for stochastic differential equations”, forthcoming paper
G. Gaeta and N. Rodríguez-Quintero, “Lie-point symmetries and stochastic differential equations”, J. Phys. A 32 (1999), 8485–8505
G. Gaeta and F. Spadaro, “Random Lie-point symmetries of stochastic differential equations”, J. Math. Phys. 58 (2017), 053503
N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North Holland 1981
N.G. van Kampen, Stochastic processes in Physics and Chemistry, North Holland 1992
R. Kozlov, “Symmetry of systems of stochastic differential equations with diffusion matrices of full rank”, J. Phys. A 43 (2010), 245201
R. Kozlov, “The group classification of a scalar stochastic differential equations”, J. Phys. A 43 (2010), 055202
R. Kozlov, “On maximal Lie point symmetry groups admitted by scalar stochastic differential equations”, J. Phys. A 44 (2011), 205202
M. Nakahara, Geometry, Topology and Physics, IOP, 1990
Ch. Nash and S. Sen, Topology and Geometry for Physicists, Academic Press, 1983; reprinted by Dover, 2011
B. Oksendal, Stochastic differential equations (4th edition), Springer 1985
P.J. Olver, Application of Lie groups to differential equations, Springer 1986
P.J. Olver,Equivalence, Invariants and Symmetry, Cambridge University Press 1995
L.S. Schulman, Techniques and applications of path integration, Wiley 1981; reprinted by Dover 2005
B. Srihirun, S.V. Meleshko and E. Schulz, “On the definition of an admitted Lie group for stochastic differential equations with multi-Brownian motion”, J. Phys. A 39 (2006), 13951–13966
H. Stephani, Differential equations. Their solution using symmetries, Cambridge University Press 1989
D.W. Stroock, Markov processes from K.Ito’s perspective, Princeton UP 2003
G. Unal, “Symmetries of Ito and Stratonovich dynamical systems and their conserved quantities”, Non-lin. Dyn. 32 (2003), 417–426