Spectral analysis of a semiclassical random walk associated to a general confining potential

We consider a semiclassical random walk with respect to a probability measure associated to a potential with a finite number of critical points. We recover the spectral results from [1] on the corresponding operator in a more general setting and with improved accuracy. In particular we do not make any assumption on the distribution of the critical points of the potential, in the spirit of [15]. Our approach consists in adapting the ideas from [15] to the recent gaussian quasimodes framework which appears to be more robust than the usual methods, especially when dealing with non local operators.