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

2025/09/23 by Yong Xu, Yang, Xiaoyu, Xu, Yong
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #60F10 #60G15 #60H10 #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR) #Stochastic processes and financial applications #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2509.18675

openalex publication_date 2025/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

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.

Citations

• Averaging principle for rough slow-fast systems of level 3
• Averaging principle for semilinear slow-fast rough partial differential equations
• Moderate deviations for rough differential equations
• Pathwise central limit theorem and moderate deviations via rough paths for SPDEs with multiplicative noise
• Large deviation principle for slow-fast system with mixed fractional Brownian motion
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• Diffusion approximation for fully coupled stochastic differential equations
• Large deviation principles for stochastic dynamical systems with a fractional Brownian noise
• Averaging Principles for Mixed Fast-Slow Systems Driven by Fractional Brownian Motion
• Typical dynamics and fluctuation analysis of slow-fast systems driven by\n fractional Brownian motion
• Large deviation for two-time-scale stochastic Burgers equation
• Large deviations and averaging for systems of slow--fast stochastic reaction--diffusion equations
• Large deviation principle of Freidlin-Wentzell type for pinned diffusion\n processes
• Large Deviation Principle for Enhanced Gaussian Processes
• A Variation Embedding Theorem and Applications

Related

• Moderate deviations for rough differential equations
• Large deviation principle for slow-fast system with mixed fractional Brownian motion
• Precise Laplace approximation for mixed rough differential equation
• Averaging principle for slow-fast systems of rough differential equations via controlled paths
• Large deviation principle for slow-fast systems with infinite-dimensional mixed fractional Brownian motion