Properties and limitations of geometric tempering for gradient flow dynamics

ArXi:2604.20301v1 Announce Type: cross We consider the problem of sampling from a probability distribution $\pi$. It is well known that this can be written as an optimisation problem over the space of probability distributions in which we aim to minimise the Kullback--Leibler divergence from $\pi$. We consider the effect of replacing $\pi$ with a sequence of moving targets $(\pi_t)_{t\ge0}$ defined via geometric tempering on the Wasserstein and Fisher--Rao gradient flows. We show that convergence occurs exponentially in continuous time, providing novel bounds in both cases.