English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study the convergence rate to equilibrium for a family of Markovsemigroups $\{\mathcal{P}_t^{\epsilon}\}_{\epsilon > 0}$ generated by a classof hypoelliptic stochastic differential equations on $\mathbb{R}^d$, includingGalerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96,and the shell model SABRA. In the regime of vanishing, balanced noise anddissipation, we obtain a sharp (in terms of scaling) quantitative estimate onthe exponential convergence in terms of the small parameter $\epsilon$. Byscaling, this regime implies corresponding optimal results both for fixeddissipation and large noise limits or fixed noise and vanishing dissipationlimits. As part of the proof, and of independent interest, we obtainuniform-in-$\epsilon$ upper and lower bounds on the density of the stationarymeasure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lowerbounds by a de Giorgi-type iteration (both uniform in $\epsilon$). The spectralgap estimate on the semigroup is obtained by a weak Poincar\'e inequalityargument combined with quantitative hypoelliptic regularization of thetime-dependent problem.

probability and mathematical physics

Quantitative spectral gaps and uniform lower bounds in the small noise  limit for Markov semigroups generated by hypoelliptic stochastic differential  equations