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arXiv · 1907.04825

Superdiffusive limits for deterministic fast-slow dynamical systems

Ilya Chevyrev·Peter K. Friz·Alexey Korepanov·Ian Melbourne
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Abstract

We consider deterministic fast-slow dynamical systems on $\mathbb{R}^m\times Y$ of the form \[ \begin{cases} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a(x_k^{(n)}) + n^{-1/α} b(x_k^{(n)}) v(y_k)\;,\quad y_{k+1} = f(y_k)\;, \end{cases} \] where $α\in(1,2)$. Under certain assumptions we prove convergence of the $m$-dimensional process $X_n(t)= x_{\lfloor nt \rfloor}^{(n)}$ to the solution of the stochastic differential equation \[ \mathop{}\!\mathrm{d} X = a(X)\mathop{}\!\mathrm{d} t + b(X) \diamond \mathop{}\!\mathrm{d} L_α \; , \] where $L_α$ is an $α$-stable Lévy process and $\diamond$ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps $f$ of Pomeau-Manneville type.

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Ilya Chevyrev, Peter K. Friz, Alexey Korepanov, Ian Melbourne. 2020-07-11. Superdiffusive limits for deterministic fast-slow dynamical systems. https://doi.org/10.1007/s00440-020-00988-5

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