Large and moderate deviation for rough slow-fast system with level 3 geometric rough path

This work is to give the large deviation principle (LDP) and moderate deviation principle (MDP) for a slow-fast system driven by the mixed fractional Brownian motion (FBM) (1/4,1/3) via the level-3 geometric rough path (RP).

Firstly, we provide a different way to lift the translation of mixed FBM in the Cameron-Martin direction to the geometric RP.

The LDP turns to the weak convergence of the controlled system by averaging the fast one.

Lacking the invariant measure for the controlled fast one, a "replaced process" whose limiting measure could be decoupled from the control term and owes the exponential ergodicity is constructed.

Here, the stability under level-3 RPs is established under more elaborate estimates.

Besides, different from LDP we give the MDP, that requires more intricate bounded estimates of the deviation component.

The MDP for level-2 RP system is recovered as a special case.