Spatial discretization of dynamical systems


Monday, September 24th, 2018
10:00-11:00, WASTE4THINK Room at DeustoTech

Peter E. Kloeden

Institute of Mathematics, University of Tubingen, Germany

Abstract:
Random ordinary differential equations (RODEs) are pathwise ordinary differen-tial equations that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been very much overshadowed by stochastic ordinary differential equations (SODEs). The stochastic process could be a fractional Brownian motion, but when it is a diffusuion process there is a close connection between RODEs and SODEs through the Doss-Sussmann transformation and its generalisations, which relate a RODE and an SODE with the same (transformed) solutions. RODEs play an important role in the theory of random dynamical systems and random attractors.
Classical numerical schemes such as Runge-Kutta schemes can be used for RODEs but do not achieve their usual high order since the vector field does not inherit enough smoothness in time from the driving process. It will be shown how, nevertheless, Taylor expansions of the solutions of RODES can be obtained when the stochastic process has H¨older continuous sample paths and then used to de-rive pathwise convergent numerical schemes of arbitrarily high order. An appli-cation in biology with a RODE and SODE will also be discussed.
References:

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[3] Xiaoying Han and P. E. Kloeden: Random Ordinary Differential Equations and their Numerical Solution, Springer Natur Singapore, 2017.
[4] A. Jentzen and P. E. Kloeden: Pathwise Taylor schemes for random ordinary differential equation. BIT. 49 (2009), 113–140.
[5] P.E. Kloeden and A. Jentzen: Pathwise convergent higher order numerical schemes for random ordinary differential equations. Proc. Roy. Soc. London A. 463 (2007), 2929–2944.