Reflection couplings and contraction rates for diffusions

Abstract

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.

To be published in PTRF. The final publication is available at Springer via http://dx.doi.org/10.1007/s00440-015-0673-1

References in corpus (4)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• Separation cut-offs for birth and death chains
• A new Poisson-type deviation inequality for Markov jump processes with positive Wasserstein curvature
• Error bounds for Metropolis-Hastings algorithms applied to perturbations of Gaussian measures in high dimensions

Cited by in corpus (23)

• Geometric integrators and the Hamiltonian Monte Carlo method
• Propagation of chaos: a review of models, methods and applications. II. Applications
• Coupling and Convergence for Hamiltonian Monte Carlo
• Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes
• Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics
• Long-time behaviors of mean-field interacting particle systems related to McKean-Vlasov equations
• On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions
• Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo
• Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models
• Slow-Fast Systems with Fractional Environment and Dynamics
• Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions
• Couplings for Andersen Dynamics
• Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling
• Time-uniform log-Sobolev inequalities and applications to propagation of chaos
• Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions
• Entropic curvature and convergence to equilibrium for mean-field dynamics on discrete spaces
• Propagation of chaos: a review of models, methods and applications. I. Models and methods
• Simple Nonlinear Models with Rigorous Extreme Events and Heavy Tails
• Numerical computations of geometric ergodicity for stochastic dynamics
• Well-posedness and approximation of reflected McKean-Vlasov SDEs with applications
• Logarithmic Sobolev inequalities for non-equilibrium steady states
• Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method
• Multimodal sampling via Schrödinger-Föllmer samplers with temperatures