Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions

In this paper we solve problems of eigenvalues of stochastic Hamiltonian systems with boundary conditions and construct the corresponding eigenfunctions. This is a sort of forward-backward stochastic differential equations (FBSDE) parameterized by . The problem is to find non-trivial solutions while the trivial solution 0 exists. We show that, as the classical cases, the phenomenon of statistic periodicity and related stochastic oscillations appear. A method of dual transformation of stochastic Hamiltonian systems is introduced and applied, as a main tool, in the construction of eigenfunctions. This eigenvalue problem is also formulated in a standard way in functional analysis.