Stochastic averaging for SDEs with Hopf Drift and polynomial diffusion coefficients

Journal of Linear and Topological Algebra

مقاله 2، دوره 04، شماره 02، مرداد 2015، صفحه 101-114   

نوع مقاله: Research Paper

نویسنده

M. Alvand

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

چکیده

It is known that a stochastic differential equation (SDE) induces two probabilistic objects, namely a difusion process and a stochastic flow. While the diffusion process is determined 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 the stochastic flow uniquely the infinitesimal covariance given by the coefficients of the SDE is needed in addition. The SDEs we consider here are obtained by a weak perturbation of a rigid rotation by random fields which are white in time. In order to obtain information about the stochastic 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 averaged SDE for the case that the diffusion coefficients of the initial SDE are polynomial. To do this we develop a complex version of Cholesky decomposition algorithm.

کلیدواژه‌ها

Stochastic Differential Equation؛ stochastic ow؛ stochastic averaging؛ Cholesky decomposition؛ system of complex bilinear equations

مراجع

[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.