Convergence rates of self-repelling diffusions on Riemannian manifolds
Francis Lörler
TL;DR
Using a spectral decomposition of the interaction kernel V, the paper shows the self-repelling diffusion on a compact manifold is a second-order lift of an Ornstein-Uhlenbeck process. This viewpoint links non-reversible degenerate dynamics to a reversible diffusion in an enlarged state space, enabling explicit upper and lower bounds on convergence to equilibrium via flow Poincaré inequalities. In the periodic torus setting, the bounds improve previous results and can match the upper bound order in some regimes. The work advances hypocoercivity methodology by providing parameter-sensitive, non-asymptotic relaxation rates for self-interacting diffusions.
Abstract
We study a class of self-repelling diffusions on compact Riemannian manifolds whose drift is the gradient of a potential accumulated along their trajectory. When the interaction potential admits a suitable spectral decomposition, the dynamics and its environment are equivalent to a finite-dimensional degenerate diffusion. We show that this diffusion is a second-order lift of an Ornstein-Uhlenbeck process whose invariant law corresponds to the Gaussian invariant measure of the environment, and immediately obtain a general upper bound on the rate of convergence to stationarity using the framework of second-order lifts. Furthermore, using a flow Poincaré inequality, we develop lower bounds on the convergence rate. We show that, in the periodic case, these lower bounds improve upon those of Benaïm and Gauthier (Probab. Theory Relat. Fields, 2016), and even match the order of the upper bound in some cases.