Averaging of stochastic flows: twist maps and escape from resonance
Let \((\Omega,{\mathfrak I},P)\) be a probability space, \(W\) standard Brownian motion, \(\theta_0\), \(x\in \mathbb{R}\) and \(\varepsilon\in (0,1)\). In the present paper the \(\mathbb{R}^2\)-valued stochastic differential equation \[ d\theta^\varepsilon_t= {1\over\varepsilon}\omega(X^\varepsilon_t)\, dt,\quad dX^\varepsilon_t= \sigma(\theta^\varepsilon_t)\,dW_t,\quad (\theta^\varepsilon_t, X^\varepsilon_t)= (\theta_0, x_0), \] where \(\omega\in C^\infty(\mathbb{R})\) and \(\sigma\in C^\infty(S^1)= \{\varphi\in CJ^\infty(\mathbb{R}): (\varphi(\theta)= \varphi(\theta+ 1),\theta\in \mathbb{R}\}\) is considered. The \(\widehat\sigma\) is defined by \[ \widehat\sigma= \Biggl\{\int^1_0 \sigma^2(\theta)\,d\theta\Biggr\}^{1/2} \] and for each \(f\in C^2(\mathbb{R})\), \(L_1 f=\widehat\sigma^2\ddot f\). For all \(f\in C^1(\mathbb{R}^2)\) and \((\theta, x)\in \mathbb{R}^2\) the vector fields \(V_0\), \(V_1\) on \(\mathbb{R}^2\) are defined as \[ (V_0f)(\theta,x)=\omega(x){\partial f\over\partial\theta} (\theta,x),\quad (V_1 f)(\theta, x)= \sigma(\theta){\partial f\over\partial x}(\theta, X). \] For each \(\varepsilon\in E(0,1)\), let \(\{\phi^\varepsilon_t: t\geq 0\}\) be the \(\text{Diff}(\mathbb{R}^2)\)-valued stochastic process (the \(\text{Diff}(\mathbb{R}^2)\) denote the group of \(C^\infty\) diffeomorphisms of \(\mathbb{R}^2\)) such that \[ d\phi^\varepsilon_t= {1\over\varepsilon} V_0(\phi^\varepsilon_t)\,dt+ V_1(\phi^\varepsilon_t)\circ dW_t,\quad t> 0,\qquad \phi^\varepsilon_0= \text{id} \] (id denote the identity map, and \(\circ\) Stratonovich integration). Let \(\Phi^\varepsilon_t= \pi_2\circ\phi^\varepsilon_t\), where \(\pi_2(p)= x\) for all \(p= (\theta, x)\in\mathbb{R}^2\). The basic result of this paper is that: \(p_0= (\theta_0, x_0)\) the law of \(\{\Phi^\varepsilon_t(p_0): t\geq 0\}\) converges as \(\varepsilon\to 0\) to an \(\mathbb{R}\)-valued Markov process with generator \(L_1\) aid initial distribution \(\delta_{\pi_2(p_0)}\).
scientific article; zbMATH DE number 5614628
scientific article