This article quantifies the asymptotic ε-mixing times, as ε tends to 0, of a multivariate geometric Brownian motion with respect to the Wasserstein-2-distance. We study the cases of commutative, and first order non-commutative drift and diffusion coefficient matrices, respectively, in terms of the nilpotence of the respective iterated Lie commutators.
This article establishes non-asymptotic ergodic bounds in the renormalized, weighted Kantorovich-Wasserstein-Rubinstein distance for a viscous energy shell lattice model of turbulence with random energy injection. The obtained bounds turn out to be asymptotically sharp and establish abrupt thermalization. The types of noise under consideration are Gaussian and symmetric α-stable, white and stationary Ornstein-Uhlenbeck noise, respectively, as well as general Lévy noise with second moments. Furthermore we establish the absence of abrupt thermalization in the inviscid limit case.
This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Lévy noises in the Wasserstein distance. For the additive noise case, we obtain analogous infinite dimensional results to the respective finite dimensional cases obtained recently by Barrera, Högele and Pardo (2021), that is, the (stronger) profile cutoff phenomenon for the stochastic heat equation and the (weaker) window cutoff phenomenon for the stochastic wave equation. For the multiplicative noise case, which is studied in this context for the first time, the stochastic heat equation also exhibits profile cutoff phenomenon, while for the stochastic wave equation the methods break down due to the lack of symmetry. The methods rely strongly on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the stochastic solution flows in terms of stochastic exponentials.
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