On averaging principle for diffusion processes with null-recurrent fast component.
The asymptotic behaviour as \(\varepsilon \downarrow 0\) of the solution \((X_ \varepsilon ,Y_ \varepsilon )\) to a system of stochastic differential equations with fast and slow components \[ dX_ \varepsilon (t) = \frac 1{\varepsilon }\varphi ( X_ \varepsilon (t),Y_ \varepsilon (t)) \,dw(t), \qquad dY_ \varepsilon (t) = b(X_ \varepsilon (t), Y_ \varepsilon (t))\,dt + \sigma (X_ \varepsilon (t),Y_ \varepsilon (t)) \,dw(t) \] is studied. It is supposed that \(w\) is a \(k\)-dimensional Brownian motion and \(\varphi : \mathbb R\times \mathbb R^ {d}\to \mathbb R^ {1\times k}\), \(b: \mathbb R\times \mathbb R^ {d}\to \mathbb R^ {d}\), \(\sigma :\mathbb R\times \mathbb R^ {d}\to \mathbb R^ {d\times k}\) are Lipschitz continuous functions satisfying suitable regularity and smoothness hypotheses. If the diffusion processes \(X^ {y}\), \(y\in \mathbb R^ {d}\), defined by the stochastic differential equation \(dX^ {y} = \varphi (X^ {y},y)\,dw\) are positive recurrent, then the standard averaging theory would imply that the process \(Y_ \varepsilon \) converges weakly to a solution of a stochastic differential equation, the coefficients of which may be computed from \(b\) and \(\sigma \) by an averaging procedure. However, under the assumptions that \(X^ {y}\) is a null-recurrent one-dimensional Markov process for all \(y\), nonetheless, it is proven that the pair \((\varepsilon X_ \varepsilon ,Y_ \varepsilon )\) converges as \(\varepsilon \downarrow 0\) to a diffusion process whose generator is identified.
scientific article; zbMATH DE number 2119497
scientific article