The diffusive behaviour of simple random-walk proposals of many Markov Chain
Monte Carlo (MCMC) algorithms results in slow exploration of the state space
making inefficient the convergence to a target distribution. Hamiltonian/Hybrid
Monte Carlo (HMC), by introducing fictious momentum variables, adopts
Hamiltonian dynamics, rather than a probability distribution, to propose future
states in the Markov chain. Splitting schemes are numerical integrators for
Hamiltonian problems that may advantageously replace the St\"ormer-Verlet
method within HMC methodology. In this paper a family of stable methods for
univariate and multivariate Gaussian distributions, taken as guide-problems for
more realistic situations, is proposed. Differently from similar methods
proposed in the recent literature, the considered schemes are featured by null
expectation of the random variable representing the energy error. The
effectiveness of the novel procedures is shown for bivariate and multivariate
test cases taken from the literature