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 in which g denotes the Lie algebra corresponding to G. After identifying the target Fokker-Planck equation, we especially consider the unitary group G= U(d) and show that the equation on U(d) 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). Finally, we restrict our concern to G= SU(2) to provide explicit calculation and present the nonlinear stability of an incoherent state for the Fokker-Planck equation depending on the relation between parameters.