Invariance principle in the singular perturbations limit (Q2321078)

The author examines the invariance principle within a general singularly perturbed differential equation \[ \frac{dx}{dt} = \frac{1}{\epsilon}F(x) + G(x), \ \ x \in \mathbb{R}^{n}, \tag{1} \] where \(\epsilon\) is a small positive parameter, the functions \(F,G\) are continuous and satisfy some additional conditions (in particular, it is assumed that solutions of (1) are uniquely determined by their initial conditions, and exist on the entire time line). The goal of the paper is to study the limit of the dynamics generated by equation (1) as \(\epsilon \to +0\). A key point is that the limit of trajectories of (1) may not be described as a trajectory in \(\mathbb{R}^{n}\), but by a trajectory in the space of probability measures on \(\mathbb{R}^{n}\). In the present paper, the limit dynamics of (1) is presented by the evolution of a Young measure, whose values are invariant measures of the fast equation \[ \frac{dx}{ds} = F(x), \ \ t = \epsilon s. \] Using this approach, the author establishes an invariance principle for the limit dynamics, and examines the relations, at times subtle, with the singularly perturbed system itself.