The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive Monte-Carlo methods. We study here an advanced version of familiar Markov-chain Monte-Carlo (MCMC) algorithms that sample from target distributions defined as change of measures from Gaussian laws on general Hilbert spaces. Such a model structure arises in several contexts: we focus here at the important class of statistical models driven by diffusion paths whence the Wiener process constitutes the reference Gaussian law. Particular emphasis is given on advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in the state space (in contrast with local-move Random-walk-type algorithms) with analytical and experimental results. We illustrate its computational advantages in various diffusion processes and observation regimes; examples include stochastic volatility and latent survival models. In contrast with their standard MCMC counterparts, the advanced versions have mesh-free mixing times, as these will not deteriorate upon refinement of the approximation of the inherently infinitedimensional diffusion paths by finite-dimensional ones used in practice when applying the algorithms on a computer.