Taylor-like approximations of center manifolds for rough differential equations
Alexandra Blessing, Dennis Rudik
TL;DR
This work develops Taylor-like, polynomial approximations of local random center manifolds for rough differential equations driven by geometric rough paths. By coupling rough-path analysis with deterministic center-manifold theory, the authors propose an ansatz h^c(\xi,\mathbf{W}) ≈ \phi(x) with \phi(x)=\sum_{i=2}^q \alpha_i(\mathbf{W}) x^i, where the coefficients \alpha_i solve auxiliary rough differential equations driven by the same rough path; a pathwise truncation, discretization, and fixed-point framework justify the approximation and yield an error of order O(\|\xi\|^{q+1}). The method handles nonlinear multiplicative and non-Markovian noise, generalizing existing results that rely on linear noise or flow transforms, and is illustrated through SDEs with both linear and nonlinear multiplicative noise. The approach provides a practical, pathwise tool for center-manifold reduction in rough dynamics and paves the way for rough PDE extensions and numerical computation of the approximation terms.
Abstract
The dynamics of rough differential equations (RDEs) has recently received a lot of interest. For example, the existence of local random center manifolds for RDEs has been established. In this work, we present an approximation for local random center manifolds for RDEs driven by geometric rough paths. To this aim, we combine tools from rough path and deterministic center manifold theory to derive Taylor-like approximations of local random center manifolds. The coefficients of this approximation are stationary solutions of RDEs driven by the same geometric rough path as the original equation. We illustrate our approach for stochastic differential equations (SDEs) with linear and nonlinear multiplicative noise.