Superdiffusive limits for deterministic fast-slow dynamical systems

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 $\alpha \in (1,2)$. 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 $L_{\alpha }$ is an $\alpha$-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.

36 pages, 3 figures. Minor revision. To appear in Probability Theory and Related Fields