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We consider deterministic fast-slow dynamical systems of the form \[x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} A(x_k^{(n)}) + n^{-1/\alpha} B(x_k^{(n)})v(y_k), \quad y_{k+1} = Ty_k, \] where $\alpha\in(1,2)$ and$x_k^{(n)}\in{\mathbb R}^m$. Here, $T$ is a slowly mixing nonuniformlyhyperbolic dynamical system and the process$W_n(t)=n^{-1/\alpha}\sum_{k=1}^{[nt]}v(y_k)$ converges weakly to a$d$-dimensional $\alpha$-stable L\'evy process $L_\alpha$.  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) \[ dX = A(X)\,dt + B(X)\, dL_\alpha. \] In the simplest cases consideredin previous work, the limiting SDE has the Marcus interpretation. Inparticular, the SDE is Marcus if the noise coefficient $B$ is exact or if theexcursions for $W_n$ converge to straight lines as $n\to\infty$.  Outside these simplest situations, it turns out that typically the Marcusinterpretation fails. We develop a general theory that does not rely onexactness or linearity of excursions. To achieve this, it is necessary toconsider suitable spaces of ``decorated'' c\`adl\`ag paths and to interpret thelimiting decorated SDE. In this way, we are able to cover more complicatedexamples such as billiards with flat cusps where the limiting SDE is typicallynon-Marcus for $m\ge2$.

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