Superdiffusive limits for deterministic fast-slow dynamical systems

TL;DR

This paper proves that certain deterministic fast-slow systems with intermittent dynamics converge to a stochastic differential equation driven by an alpha-stable Levy process, extending the understanding of superdiffusive limits in dynamical systems.

Contribution

It establishes the convergence of fast-slow deterministic systems to alpha-stable Levy-driven SDEs and verifies assumptions for Pomeau-Manneville type maps.

Findings

Convergence to alpha-stable Levy-driven SDEs for specific dynamical systems

Verification of assumptions for intermittent Pomeau-Manneville maps

Extension of superdiffusive limit theory to deterministic systems

Abstract

We consider deterministic fast-slow dynamical systems on $\mathbb{R}^m\times Y$ of the form \[ \begin{cases} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a(x_k^{(n)}) + n^{-1/\alpha} b(x_k^{(n)}) v(y_k)\;,\quad y_{k+1} = f(y_k)\;, \end{cases} \] 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 \[ \mathop{}\!\mathrm{d} X = a(X)\mathop{}\!\mathrm{d} t + b(X) \diamond \mathop{}\!\mathrm{d} L_\alpha \; , \] where $L_\alpha$ is an $\alpha$-stable L\'evy 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.