Averaging Principle and Uniform Large Deviations for Time-Dependent Slow-Fast Systems via Young-Rough Integration

The main aim of this work is to investigate the limiting behavior and asymptotic properties of time-dependent slow-fast stochastic dynamical systems. The slow component is driven by a stochastic process with Hölder-continuous paths of regularity greater than 1/2, while the fast component is driven by a Brownian rough path. We introduce a novel Young-rough integration framework tailored to such systems and establish well-posedness results, providing the analytical foundation for further asymptotic analysis. Building on this framework, we employ Khasminskii's time-discretization method to establish the strong convergence of the slow component to the solution of the corresponding averaged equation in the L^p-sense. Furthermore, when the driving noise in the slow equation is specified as a fractional Brownian motion, we apply the weak convergence approach to establish a uniform large deviation principle for the slow component, uniformly over compact sets of initial conditions, in the Hölder topology.