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Nonlinear effects within invariance principles
Maximilian Engel, Peter K. Friz, Tal Orenshtein · 2024-02-21 · via math.PR updates on arXiv.org

The combination of functional limit theorems with the pathwise analysis of deterministic and stochastic differential equations has proven to be a powerful approach to the analysis of fast-slow systems. In a multivariate setting, this requires rough path ideas, as already suggested in the seminal work [Melbourne-Stuart, Nonlinearity, 24, 2011]. This initiated a program pursued by numerous authors and which has required substantial results on invariance principles (also known as functional central limit theorems) in rough path topologies. We take a unified point of view and provide simple and exact formulas, of the Green-Kubo type, that characterize the relevant Brownian rough path limits and discuss how they naturally apply in different settings.

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