Asymptotics of fixed point distributions for inexact Monte Carlo algorithms

We introduce a simple general method for finding the equilibrium distribution for a class of widely used inexact Markov chain Monte Carlo algorithms. The explicit error due to the noncommutativity of the updating operators when numerically integrating Hamilton's equations can be derived using the Baker-Campbell-Hausdorff formula. This error is manifest in the conservation of a shadow Hamiltonian that lies close to the desired Hamiltonian. The fixed point distribution of inexact hybrid algorithms may then be derived taking into account that the fixed point of the momentum heat bath and that of the molecular dynamics do not coincide exactly. We perform this derivation for various inexact algorithms used for lattice QCD calculations.