Sharp spectral gap of adaptive Langevin dynamics

We consider a degenerated Fokker-Planck type differential operator associated to an adaptive Langevin dynamic. We prove Eyring-Kramers formulas for the bottom of the spectrum of this operator in the low temperature regime. The main ingredients are resolvent estimates obtained via hypocoercive techniques and the construction of sharp Gaussian quasimodes through an adaptation of the WKB method.