Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

Abstract

We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

Mixing time guarantees for unadjusted Hamiltonian Monte Carlo

“…singular interactions [44,32,70,19,74,69]. An elementary coupling approach to uniform-in-time propagation of chaos has also been developed for various μ * -preserving McKean-Vlasov processes [25]; for various extensions, see [24,33,73,63], and in particular, for first steps towards MCMC for μ * , see [63, §4.2]. Nonlinear overdamped/kinetic Langevin diffusions have also been studied in the context of non-convex learning [43,47].…”

Coupling by reflection for controlled diffusion processes: turnpike property and large time behavior of Hamilton Jacobi Bellman equations

“…However, we stress here that with the sole exception of some multiplicative factors in Theorem 1.4(iv), all constants appearing in the exponential estimates at Theorem and 1. [15], it seems plausible that some progress in this direction could be made: we plan to explore this research line in the near future. We now introduce some standard basic notation and eventually proceed to the presentation of the main results.…”

“…• An important difference between the constant λ appearing at (13) and λ ∞ appearing at (15) is that the latter does not depend on g ′ , whereas the first does.…”

Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

“…Using this coupling, the distance between the two processes is controlled by the solution of a SDE on [0, +∞) which is sticky at 0. Based on this idea, the question of the long-time behaviour of nonlinear McKean-Vlasov processes and of the uniform in time propagation of chaos has been recently revisited in [128].…”