We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical $α$-stable noises, $α\in (0,2)$, introduced in Priola-Zabczyk "Structural properties of semilinear SPDEs driven by cylindrical stable processes" (PTRF to appear). We first prove that if the nonlinearity is bounded, then the system is ergodic and strong mixing. Then we show that the system is exponentially mixing provided that the nonlinearity, or its Lipschitz constant, are sufficiently small.
We prove $L^{p}$-uniqueness of Dirichlet operators for Gibbs measures on the path space $C(\mathbb R, \mathbb R^{d})$ associated with exponential type interactions in infinite volume by extending an SPDE approach presented in previous work by the last two named authors. We also give an SPDE characterization of the corresponding dynamics. In particular, we prove existence and uniqueness of a strong solution for the SPDE, though the self-interaction potential is not assumed to be differentiable, hence the drift is possibly discontinuous. As examples, to which our results apply, we mention the st…
Ergodicity for SPDEs driven by divergence-free transport noise
Technische Universit\" a t Berlin \& Max-Planck-Institut f\" u r Mathematik in den Naturwissenschaften, Email: black D-MATH, ETH Z\" u rich, Email: black Technische Universit\" a t Berlin, Email: black
We develop a solution theory in Hölder spaces for a quasilinear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE Lemmas which establish a priori Hölder bounds for an equation with irregular right hand side written in divergence form. We apply these deterministic bounds to the case of a noise term which is white in time and trace class in space to obtain stretched exponential bounds for the Hölder semi-norms of the solution for the stochastic equation.
We establish a general criterion which ensures exponential mixing of parabolic Stochastic Partial Differential Equations (SPDE) driven by a non additive noise which is white in time and smooth in space. We apply this criterion on two representative examples: 2D Navier-Stokes (NS) equations and Complex Ginzburg-Landau (CGL) equation with a locally Lipschitz noise. Due to the possible degeneracy of the noise, Doob theorem cannot be applied. Hence a coupling method is used in the spirit of [EMS], [KS3] and [Matt]. Previous results require assumptions on the covariance of the noise which might see…
Quantifying the effect of graph structure on strong Feller property of SPDEs
The Hong Kong Polytechnic University（香港理工大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
We study the mixing properties of a passive scalar advected by an incompressible flow. We consider a class of cellular flows (more general than the class in [Crippa-Schulze M3AS 2017]) and show that, under the constraint that the palenstrophy is bounded uniformly in time, the mixing scale of the passive scalar cannot decay exponentially.
The paper is devoted to studying the stochastic nonlinear wave (NLW) equation in a bounded domain D $\subset$ R3. We show that the Markov process associated with the flow of solution has a unique stationary measure $μ$, and the law of any solution converges to $μ$ with exponential rate in the dual-Lipschitz norm
We prove a version of Bressan's mixing conjecture where the advecting field is constrained to be a shear at each time. Also, inspired by recent work of Blumenthal, Coti Zelati and Gvalani, we construct a particularly simple example of a shear flow which mixes at the optimal rate. The constructed vector field alternates randomly in time between just two distinct shears.
We consider the mixing behaviour of the solutions of the continuity equation associated with a divergence-free velocity field. In this announcement we sketch two explicit examples of exponential decay of the mixing scale of the solution, in case of Sobolev velocity fields, thus showing the optimality of known lower bounds. We also describe how to use such examples to construct solutions to the continuity equation with Sobolev but non-Lipschitz velocity field exhibiting instantaneous loss of any fractional Sobolev regularity.
Exponential mixing and Freidlin--Wentzell large deviation principle for Markov cocycles
School of Mathematics, Sichuan University（四川大学数学学院）；School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学数学科学学院）；Department of Mathematics, New Mexico Institute of Mining and Technology（新墨西哥矿业与技术大学数学系）
In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in $L^\infty$. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in $L^\inf…
We study an infinite white $α$-stable systems with unbounded interactions, proving the existence by Galerkin approximation and exponential mixing property by an $α$-stable version of gradient bounds.
We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes are forced) via Kolmogorov equation approach.
We prove that well posed quasilinear equations of parabolic type, perturbed by bounded nondegenerate random forces, are exponentially mixing for a large class of random forces.
We performed a numerical study of the efficiency of mixing by alternating horizontal and vertical shear ``wedge'' flows on the two-dimensional torus. Our results suggest that except in cases where each individual flow is applied for only a short time, these flows produce exponentially fast mixing. The observed mixing rates are higher when the individual flow times are shorter (but not too short), and randomizing either the flow times or phase shifts of the flows does not appear to enhance mixing (again when the flow times are not too short). In fact, the latter surprisingly seems to inhibit it…
We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mi…
We study the mixing and dissipation properties of the advection-diffusion equation with diffusivity $0 < κ\ll 1$ and advection by a class of random velocity fields on $\mathbb T^d$, $d=\{2,3\}$, including solutions of the 2D Navier-Stokes equations forced by sufficiently regular-in-space, non-degenerate white-in-time noise. We prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity $κ$. Namely, that there is a deterministic, exponential rate (independent of $κ$) such that all mean-zero $H^1$ initial data decays exponentially fast in $H^{-1}$ at this rate wit…
We deduce almost-sure exponentially fast mixing of passive scalars advected by solutions of the stochastically-forced 2D Navier-Stokes equations and 3D hyper-viscous Navier-Stokes equations in $\mathbb T^d$ subjected to non-denegenerate $H^σ$-regular noise for any $σ$ sufficiently large. That is, for all $s > 0$ there is a deterministic exponential decay rate such that all mean-zero $H^s$ passive scalars decay in $H^{-s}$ at this same rate with probability one. This is equivalent to what is known as \emph{quenched correlation decay} for the Lagrangian flow in the dynamical systems literature. …
