Balancing of partially-observed stochastic differential equations

We study Balanced Truncation for stochastic differential equations. In doing so, we adopt ideas from large deviations theory and discuss notions of controllability and obervability for dissipative Hamiltonian systems with degenerate noise term, also known as Langevin equations. For partially-observed Langevin equations, we illustrate model reduction by balanced truncation with an example from molecular dynamics and discuss aspects of structure-preservation.