Adaptive Langevin dynamics allow to sample the Boltzmann–Gibbs distribution at a prescribed temperature for underdamped Langevin dynamics with unknown fluctuation magnitudes. The latter situation appears for instance when sampling a posteriori distributions of parameters in Bayesian inference, in cases when there are many data points and one uses a mini-batching strategy to approximate the gradient of the likelihood function. The idea of Adaptive Langevin is to consider the friction in underdamped Langevin dynamics as a dynamical variable, updated according to some Nose-Hoover feedback. We show here using techniques from hypocoercivity that the law of Adaptive Langevin dynamics converges exponential fast to equilibrium, with a rate which can be quantified in terms of the key parameters of the dynamics (mass of the extra variable and magnitude of the fluctuation in the Langevin dynamics). This allows us in particular to obtain a Central Limit Theorem on time averages along realizations of the dynamics.
Depending on time, I will discuss how hypocoercive techniques can be used for other dynamics such as Langevin dynamics with non-quadratic kinetic energies, nonequilibrium Langevin dynamics, spectral discretization of Langevin dynamics and Temperature accelerated molecular dynamics.