Linear Response Theory for Nonlinear Stochastic Differential Equations with α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-Stable Lévy Noises

We consider a nonlinear stochastic differential equation driven by an α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-stable Lévy process (1<α<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<\alpha <2$$\end{document}). We first prove existence and uniqueness of the invariant measure by the Bogoliubov-Krylov argument. Then we obtain some regularity results for the probability density of its invariant measure by establishing the a priori estimate of the corresponding stationary Fokker-Planck equation. Finally, by the a priori estimate of the Kolmogorov backward equation and the perturbation property of the Markov semigroup, we derive the response function and generalize the famous linear response theory in nonequilibrium statistical mechanics to non-Gaussian stochastic dynamic systems.