Superdiffusive limits beyond the Marcus regime for deterministic fast-slow systems
Ilya Chevyrev, Alexey Korepanov, Ian Melbourne
TL;DR
This work develops a general theory to derive the superdiffusive limits of deterministic fast–slow systems driven by α‑stable noise, even when the noise exhibits nonlinear excursions that preclude Marcus limits. By introducing decorated càdlàg path spaces and a decorated SDE framework, the authors capture excursion geometry and establish continuous dependence of solutions on decorated drivers, enabling convergence of the slow component X_n to the solution of dX = A(X) dt + B(X) dL_α^P. They prove that, in typical nonexact cases (e.g., billiards with flat cusps), the limiting dynamics are governed by decorated Lévy processes L_α^P rather than Marcus SDEs, and that Marcus limits only arise when excursions are linear. The approach combines Young/rough‑path techniques with Gibbs–Markov/Young tower dynamics to obtain both W_n convergence to decorated Lévy processes and tightness in p‑variation, culminating in a robust convergence of X_n to a decorated SDE solution. This framework unifies Marcus and Itô interpretations in a single, path‑space enriched setting and broadens the applicability of stochastic homogenisation to deterministic systems with complex excursion structures, with notable implications for dispersing billiards and other nonuniformly hyperbolic systems.
Abstract
We consider deterministic fast-slow dynamical systems of the form \[ x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} A(x_k^{(n)}) + n^{-1/α} B(x_k^{(n)}) v(y_k), \quad y_{k+1} = Ty_k, \] where $α\in(1,2)$ and $x_k^{(n)}\in{\mathbb R}^m$. Here, $T$ is a slowly mixing nonuniformly hyperbolic dynamical system and the process $W_n(t)=n^{-1/α}\sum_{k=1}^{[nt]}v(y_k)$ converges weakly to a $d$-dimensional $α$-stable Lévy process $L_α$. We are interested in convergence of the $m$-dimensional process $X_n(t)=x_{[nt]}^{(n)}$ to the solution of a stochastic differential equation (SDE) \[ dX = A(X)\,dt + B(X)\, dL_α. \] In the simplest cases considered in previous work, the limiting SDE has the Marcus interpretation. In particular, the SDE is Marcus if the noise coefficient $B$ is exact or if the excursions for $W_n$ converge to straight lines as $n\to\infty$. Outside these simplest situations, it turns out that typically the Marcus interpretation fails. We develop a general theory that does not rely on exactness or linearity of excursions. To achieve this, it is necessary to consider suitable spaces of ``decorated'' càdlàg paths and to interpret the limiting decorated SDE. In this way, we are able to cover more complicated examples such as billiards with flat cusps where the limiting SDE is typically non-Marcus for $m\ge2$.