Stochastic averaging for SDEs with Hopf Drift and polynomial diffusion coefficients
محورهای موضوعی : Probability theory and stochastic processes
M. Alvand
1
(Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Iran)
کلید واژه: Stochastic Differential Equation, stochastic averaging, system of complex bilinear equations, stochastic ow, Cholesky decomposition,
چکیده مقاله :
It is known that a stochastic differential equation (SDE) induces two probabilisticobjects, namely a difusion process and a stochastic flow. While the diffusion process isdetermined by the infinitesimal mean and variance given by the coefficients of the SDE,this is not the case for the stochastic flow induced by the SDE. In order to characterize thestochastic flow uniquely the infinitesimal covariance given by the coefficients of the SDE isneeded in addition. The SDEs we consider here are obtained by a weak perturbation of a rigidrotation by random fields which are white in time. In order to obtain information about thestochastic flow induced by this kind of multiscale SDEs we use averaging for the infinitesimal covariance. The main result here is an explicit determination of the coefficients of the averagedSDE for the case that the diffusion coefficients of the initial SDE are polynomial. To do thiswe develop a complex version of Cholesky decomposition algorithm.
[1] N. Abourashchi, A. Yu Veretennikov. On stochastic averaging and mixing, Theory Stoch. Process. 16, (1), (2010) 111-129.
[2] M. Alvand, Constructing an SDE from its two-point generator, Stoch. Dyn. DOI: 10.1142/S0219493715500252
[3] L. Arnold, Random Dynamical Systems, Springer-Verlag, 1998.
[4] P. H. Baxendale, Stochastic averaging and asymptotic behaviour of the stochastic Duffing - Van der Pol equation, Stochastic Process. Appl. 113, No. 2 (2004) 235-272.
[5] ———, Brownian motion in the diffeomorphisms group, Compositio Math. 53, No.1 (1984) 19-50.
[6] P. Bernard, Stochastic averaging, Nonlinear Stochastic Dynamics, (2002) 29-42.
[7] M. I. Freıdlin, The factorization of nonnegative definite matrices, Teor. Verojatnost. i Primenen. 13 (1968) 375-378.
[8] Z. L. Huang and W. Q. Zhu, Stochastic averaging of quasi-generalized Hamiltonian systems, Int. J. Nonlinear Mech. No.44 (2009) 71-80.
[9] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.
[10] J. A. Sanders, F. Verhulst and J. Murdack, Averaging Methods in Nonlinear Dynamical Systems, 2nd edition, Springer, 2007.
[11] R. B. Sowers, Averaging of stochastic flows: Twist maps and escape from resonance, Stochastic Process. Appl. No. 119, (2009) 3549-3582.
[12] S. Wiggins, An Introduction to Applied Nonlonear Dynamical Systems and Chaos, second edition, SpringerVerlag, 2009.
[13] W.Q. Zhu, Z.L. Huang and Y. Suzuki, Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems, Int. J. Nonlinear Mech. No.37 (2002) 419-437.
