On characterizations of Metropolis type algorithms in continuous time

International audienceIn the continuous time framework, a new definition is proposed for the Metropolis algorithm $(\wi X_t)_{t\geq0}$ associated to an a priori given exploratory Markov process $( X_t)_{t\geq0}$ and to a tarjet probability distribution $\pi$. It should be the minimizer for the relative entropy of the trajectorial law of $(\wi X_t)_{t\in[0,T]}$ with respect to the law of $( X_t)_{t\in[0,T]}$, when both processes start with $\pi$ as initial law and when $\pi$ is assumed to be reversible for $(\wi X_t)_{t\geq0}$. This definition doesn't depend on the time horizon $T>0$ and the corresponding minimizing process is not difficult to describe. Even if this procedure can be made general, the details were only worked out in situation of finite jump processes and of compact manifold-valued diffusion processes (a sketch is also given for Markov processes admitting both a diffusive part and a jump part). The proofs rely on an alternative approach to general Girsanov transformations in the spirit of Kunita. The case of $\varphi$-relative entropies is also investigated, in particular to make a link with a previous work of Billera and Diaconis on the traditional Metropolis algorithm in the discrete time setting