Superdiffusive limits beyond the Marcus regime for deterministic fast-slow systems

Abstract

We consider deterministic fast-slow dynamical systems of the form \[ x k + 1 ( n ) = x k ( n ) + n − 1 A ( x k ( n ) ) + n − 1 / α B ( x k ( n ) ) v ( y k ) , y k + 1 = T y k , x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} A(x_k^{(n)}) + n^{-1/\alpha } B(x_k^{(n)}) v(y_k), \quad y_{k+1} = Ty_k, \] where α ∈ ( 1 , 2 ) \alpha \in (1,2) and x k ( n ) ∈ R m x_k^{(n)}\in \mathbb {R}^m . Here, T T is a slowly mixing nonuniformly hyperbolic dynamical system and the process W n ( t ) = n − 1 / α ∑ k = 1 [ n t ] v ( y k ) W_n(t)=n^{-1/\alpha }\sum _{k=1}^{[nt]}v(y_k) converges weakly to a d d -dimensional α \alpha -stable Lévy process L α L_\alpha . We are interested in convergence of the m m -dimensional process X n ( t ) = x [ n t ] ( n ) X_n(t)= x_{[nt]}^{(n)} to the solution of a stochastic differential equation (SDE) \[ d X = A ( X ) d t + B ( X ) d L α . dX = A(X)\,dt + B(X)\, dL_\alpha . \] 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 B is exact or if the excursions for W n W_n converge to straight lines as n → ∞ 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 ≥ 2 m\ge 2 .