Superdiffusive limits for deterministic fast-slow dynamical systems

Probab. Theory Related Fields 178 (2020) 735-770.

Ilya Chevyrev, Peter Friz, Alexey Korepanov and Ian Melbourne

Abstract

We consider deterministic fast-slow dynamical systems on R^m x Y of the form

x_k+1⁽ⁿ⁾ = x_k⁽ⁿ⁾ + n^-1a(x_k⁽ⁿ⁾) + n^-1/αb(x_k⁽ⁿ⁾)v(y_k),     y_k+1 = f(y_k).

where α ∈ (1,2). Under certain assumptions we prove convergence of the m-dimensional process X_n(t)= x_[nt]⁽ⁿ⁾ to the solution of the stochastic differential equation

dX = a(X)dt + b(X) ◇ dL_α

where L_α is an α-stable Lévy process and ◇ 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.