Approximation of a diffusion induced by a dynamical system on a kinetic equation (Q2722496)

Let \((\Omega,{\mathcal F},\mathbb{P},T)\) be an ergodic dynamical system, and \(f\in L^2(\mathbb{P};\mathbb{R}^d)\). For \(\varepsilon> 0\), \(t\geq 0\), \(x\in \mathbb{R}^d\), consider \(X_{t,x}^\varepsilon(\omega):= x-\varepsilon_x \sum_{k=0}^{[t/\varepsilon^2]} f(T^k\omega)\). In the spirit of an article by \textit{C. Bardos, F. Golse} and \textit{J.-F. Colonna} [Physica D 104, 32-60 (1997; Zbl 0902.35088)], it is proved that: NEWLINENEWLINENEWLINEIf \(T\) is automorphic and \(\Omega\) is the \(n\)-torus and if \(f\) is smooth and centred, then the sequence \(\mathbb{E} [(\frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} f\circ T^k)^{\otimes 2}]\) converges to some positive matrix \(D\), and for any \(\varphi\in {\mathcal C}_b(\mathbb{R}^d,\mathbb{R})\) we have as \(\varepsilon\searrow 0\), uniformly on the compacts of \(\mathbb{R}_+\times \mathbb{R}^d\), convergence of \(\mathbb{E} [\varphi (X_{t,x}^\varepsilon)]\) to \(\mathbb{E} [\varphi(x- B_t)]\), where \(B\) is centred Brownian with covariance \(D\). NEWLINENEWLINENEWLINEMoreover \(\varphi(X_{t,x}^\varepsilon)\) converges weakly in \(L^\infty (\mathbb{R}_+\times \mathbb{R}^d\times \Omega)\) towards \(\mathbb{E} [\varphi(x-B_t)]\). NEWLINENEWLINENEWLINEThe same results hold for general \(\Omega\) and \(T\) if \(f\) is supposed \(T\)-homologous to the sequence of increases of some martingale.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].