Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes

By Andreas Eberle, Arnaud Guillin, Raphael Zimmer. Transactions of the American Mathematical Society, 2016

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We consider $\mathbb{R}^d$-valued diffusion processes of type \begin{align*} dX_t\ =\ b(X_t)dt\, +\, dB_t. \end{align*} 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.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/04bbecdbc09de20072c5583e30a9ad8f843b753a

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Citations in the corpus

Couplings and quantitative contraction rates for Langevin dynamics (2017, methodology): “By designing a special semi-metric ρ on R2d that is based on a concave function of the distance and on a Lyapunov function, we can then (similarly as in [16]) make use of the random fluctuations and of a drift condition in order to derive average contractivity for the coupling distance ρ((Xt, Vt), (”
Quantitative contraction rates for Markov chains on general state spaces (2018, methodology): “in [23, 24, 3, 19] to study properties of Markov chains and diffusion processes.”
Large and Moderate Deviation Principles for McKean-Vlasov SDEs with Jumps (2020, background): “One can refer to [13, 25, 30, 48, 49] and the references therein for the existence and uniqueness of solutions to MVSDEs, [2, 20, 32, 47, 46, 54] for propagation of chaos, [21, 22, 23, 24, 28, 38, 41, 42, 43, 52] for exponential ergodicity and functional inequalities, [1, 19, 27, 40, 53] and the ref”