A note on transportation cost inequalities for diffusions with reflections

Abstract

We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusion processes with suitable drift and diffusion coefficients. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.

Convergence rate to equilibrium in Wasserstein distance for reflected jump-diffusions

“…Therefore, d(X 2 (t) − X 1 (t)) = [g(X 2 (t)) − g(X 1 (t))] dt − dℓ 1 (t) ≤ [g(X 2 (t)) − g(X 1 (t))] dt ≤ G [X 2 (t) − X 1 (t)] dt. This proves the first line in (19). Letting t ↑ τ k+1 in X 1 (t) ≤ X 2 (t), we get: y 1 := X 1 (τ k+1 −) ≤ y 2 := X 2 (τ k+1 −), By Assumption 2 and Assumption 4 for jump measures, there is a coupling (z 1 , z 2 ) of normalized jump measures:…”

“…Combining this observation with the second result in (19), we complete the proof of induction step and with it the proof of Lemma 1. To prove (18) and the first line in (19), for t ∈ (τ k , τ k+1 ], use induction over k = 0, 1, . .…”

“…Assume we proved (18) and (19). Use induction over k to show the statement of the lemma for τ k < t ≤ τ k+1 , assuming this for t = τ k .…”

“…We note that the interplay between Wasserstein distance and Lyapunov functions is remarkable. Concentration of measure Talagrand inequalities for a distribution P compare Wasserstein distance from P to a test measure Q with relative entropy of Q with respect to P. These inequalities were found for common stochastic processes on [0, T ]; see [18,4,19,21] and references therein. These concentration inequalities are related to other functional inequalities for finite time horizon, which can be obtained using Lyapunov functions, [1].…”

Dimension-free local convergence and perturbations for reflected Brownian motions

“…We also mention the work of [31] who obtained a dimension-free Talagrand type transportation cost-information inequality for reflected Brownian motions. Such inequalities, however, are more useful in dimension-free concentration of measure phenomena as opposed to dimension-free rates of convergence to stationarity.…”