Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions and Markov chain

In this paper, we study the eigenvalue problem of stochastic Hamiltonian system driven by Brownian motion and Markov chain with boundary conditions and time-dependent coefficients. For any dimensional case, the existence of the first eigenvalue is proven and the corresponding eigenfunctions are constructed by virtue of dual transformation and generalized Riccati equation system. Furthermore, we have more finely characterized the existence of all eigenvalues and constructed the related eigenfunctions for one-dimensional Hamiltonian system. Moreover, the increasing order of these eigenvalues have also been given.