We consider contractivity for diffusion semigroups w.r.t. Kantorovich ($L^1$
Wasserstein) distances based on appropriately chosen concave functions. These
distances are inbetween total variation and usual Wasserstein distances. It is
shown that by appropriate explicit choices of the underlying distance,
contractivity with rates of close to optimal order can be obtained in several
fundamental classes of examples where contractivity w.r.t. standard Wasserstein
distances fails. Applications include overdamped Langevin diffusions with
locally non-convex potentials, products of these processes, and systems of
weakly interacting diffusions, both of mean-field and nearest neighbour type.Comment: To be published in PTRF. The final publication is available at
Springer via http://dx.doi.org/10.1007/s00440-015-0673-
We consider R d -valued diffusion processes of type dXt = b(Xt)dt + dBt.Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich (L 1 Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattingly's extension of Harris' Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.MSC 2010 subject classifications: Primary 60J60, 60H10.
We introduce a new probabilistic approach to quantify convergence to equilibrium for (kinetic) Langevin processes. In contrast to previous analytic approaches that focus on the associated kinetic Fokker-Planck equation, our approach is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. It yields contractions in a particular Wasserstein distance, and it provides rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. In particular, we are able to recover kinetic behavior in terms of explicit lower bounds for the contraction rate. For example, for a rescaled double-well potential with local minima at distance a, we obtain a lower bound for the contraction rate of order Ω(a −1 ) provided the friction coefficient is of order Θ(a −1 ). * Financial support from DAAD and French government through the PROCOPE program, and from the German Science foundation through the Hausdorff Center for Mathematics is gratefully acknowledged.MSC 2010 subject classifications: Primary 60J60, 60H10, 35Q84, 35B40
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