Markovian embedding of generalized Langevin equations with a nonlinear friction kernel and configuration-dependent mass

We consider a generalized Langevin equation (GLE) in which the deterministic force, the mass and the friction kernel are configuration-dependent, i.e. general nonlinear functions of the reaction coordinate. We introduce a projection operator that allows for a self-consistent Markovian embedding of such GLEs. Self -consistency means that trajectories generated by the Markovian embedding are described by a GLE with the same configuration-dependent deterministic force, mass and friction kernel. Using the projection operator, we derive a closed-form relation between the parameters of the Markovian embedding Langevin equations and the parameters of the GLE. This is accomplished by applying the projection operator formalism to the system of Markovian embedding stochastic equations.