Superdiffusive limit and recurrence for locally perturbed infinite-horizon Lorentz processes

Let $\{q_n\mid n\geq 0\}$ be a random trajectory in $\mathbb{R}^2$. Its superdiffusively scaled flow is the piecewise linear continuous process $L_N$ defined by

$$L_N\left(\frac{j}{N}\right)=\frac{q_j}{\sqrt{N\log N}}.$$

Consider the superdiffusively scaled flow $\{L_N(t)\mid t\geq 0\}$ for a locally perturbed infinite-horizon Lorentz process.

Infinite-horizon Lorentz-process conjecture. As $N\to\infty$,

$$L_N(t)\Rightarrow W_{\sigma^2}(t)$$

in $C[0,\infty)$, where $W_{\sigma^2}(t)$ is Brownian motion with non-degenerate covariance matrix $\sigma^2$, and this limiting covariance matrix coincides with that of the unperturbed periodic Lorentz process. Moreover, the locally perturbed planar Lorentz process with infinite horizon is almost surely recurrent.

The statement predicts both the superdiffusive invariance principle and recurrence for the locally perturbed infinite-horizon model, extending the corresponding results for the unperturbed process. The supplied text does not state whether these assertions have been proved, so their status remains open.