2020/07/27 by Jacob Bedrossian, Kyle Liss
Economics, Econometrics and Finance · Engineering · Mathematics · #Applied mathematics #Dissipation #Fluid Dynamics and Turbulent Flows #Geometry #Hypoelliptic operator #Linear differential equation #Mathematical analysis #Mathematics #Multiplicative noise #Partial differential equation #Physics #Quantum mechanics #Resolvent #Scaling #Semigroup #Spectral gap #Stability and Controllability of Differential Equations #Stochastic differential equation #Stochastic processes and financial applications #Upper and lower bounds #math.AP
paper · pdf · doi:10.2140/pmp.2021.2.477
published as Prob. Math. Phys. 2 (2021) 477-532 · 49 pages
arxiv created 2020/07/27 · openalex publication_date 2021/10/15 · arxiv updated 2021/11/03 · openalex created_date 2021/11/08 · openalex updated_date 2026/08/05
We study the convergence rate to equilibrium for a family of Markov semigroups \Ptε\ε > 0 generated by a class of hypoelliptic stochastic differential equations on ℝd, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter ε. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-ε upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a de Giorgi-type iteration (both uniform in ε). The spectral gap estimate on the semigroup is obtained by a weak Poincar'e inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.