Dissipative Linear Stochastic Hamiltonian Systems
Provides theoretical foundations for analyzing open dynamical systems with random forces, but is incremental for the stochastic systems community.
This paper studies linear stochastic Hamiltonian systems, deriving stability conditions, invariant measure structure, and dissipation relations via deformed Hamiltonians, with applications to feedback connections.
This paper is concerned with stochastic Hamiltonian systems which model a class of open dynamical systems subject to random external forces. Their dynamics are governed by Ito stochastic differential equations whose structure is specified by a Hamiltonian, viscous damping parameters and system-environment coupling functions. We consider energy balance relations for such systems with an emphasis on linear stochastic Hamiltonian (LSH) systems with quadratic Hamiltonians and linear coupling. For LSH systems, we also discuss stability conditions, the structure of the invariant measure and its relation with stochastic versions of the virial theorem. Using Lyapunov functions, organised as deformed Hamiltonians, dissipation relations are also considered for LSH systems driven by statistically uncertain external forces. An application of these results to feedback connections of LSH systems is outlined.