Stochastic Lohe Matrix Model on the Lie Group and Mean-Field Limit

We propose a Lohe matrix model in a random environment where each oscillator can be regarded as an element of a general matrix Lie group G. In order to make the stochastic system stays on G for all time, we introduce suitable noise terms so that the underlying manifold G is positively invariant under the stochastic system. Then, we formally derive the Fokker-Planck type equation defined on G×g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\times \mathfrak {g}$$\end{document} in which g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {g}$$\end{document} denotes the Lie algebra corresponding to G. After identifying the target Fokker-Planck equation, we especially consider the unitary group G=U(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\mathbf {U}(d)$$\end{document} and show that the equation on U(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {U}(d)$$\end{document} admits a global unique solution and that it can be rigorously derived using a stochastic mean-field limit procedure with a convergence rate of order O(1/N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(1/\sqrt{N})$$\end{document}. Finally, we restrict our concern to G=SU(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\mathbf {SU}(2)$$\end{document} to provide explicit calculation and present the nonlinear stability of an incoherent state for the Fokker-Planck equation depending on the relation between parameters.