Rough Path Theory to approximate Random Dynamical Systems
| Autor: | A. Gu, Kening Lu, Hongjun Gao, M. J. Garrido Atienza, Björn Schmalfuß |
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| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Physics
Rough path Differential equation Probability (math.PR) Dynamical Systems (math.DS) 01 natural sciences Random dynamical systems Omega 010305 fluids & plasmas Modeling and Simulation 0103 physical sciences FOS: Mathematics Mathematics - Dynamical Systems Analysis Brownian motion Mathematics - Probability Mathematical physics |
| Popis: | We consider the rough differential equation $dY=f(Y)d\bm \om$ where $\bm \om=(\omega,\bbomega)$ is a rough path defined by a Brownian motion $\omega$ on $\RR^m$. Under the usual regularity assumption on $f$, namely $f\in C^3_b (\RR^d, \RR^{d\times m})$, the rough differential equation has a unique solution that defines a random dynamical system $\phi_0$. On the other hand, we also consider an ordinary random differential equation $dY_\delta=f(Y_\delta)d\omega_\de$, where $\omega_\de$ is a random process with stationary increments and continuously differentiable paths that approximates $\omega$. The latter differential equation generates a random dynamical system $\phi_\delta$ as well. We show the convergence of the random dynamical system $\phi_\delta$ to $\phi_0$ for $\delta\to 0$ in H\"older norm. Comment: 23 pages |
| Databáze: | OpenAIRE |
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