Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

“…And then, together with Ref. [32,Theorem 4.1] and the convergence of solutions with respect to noise intensity (see Lemma 7), we can obtain that any limit of invariant measures of (1) must be an invariant measure of the corresponding deterministic equations as 𝜀 → 0.…”

“…Proof Taking $$\varepsilon _0=0$$ in Lemma 7, one can obtain that for any bounded subset $$K\subseteq H$$ and $$0\leqslant t\leqslant T_0$$ ($$T_0>0$$), $$$$\begin{equation*} \lim _{\varepsilon \rightarrow 0}\sup _{v\in K} |U^{\varepsilon }(t,\tau -t,\theta _{-t}\omega ,v) -U^{0}(t,\tau -t,v)|=0, \end{equation*}$$$$which, together with Lemma 6 and Ref. [32, Theorems 4.1 and 4.2], shows the desired results.$$\square$$…”

“…Theorem 2 (Ref. 32). Assume that a nonautonomous random dynamical system Φ satisfies that Φ(−𝑡, 𝜏 + 𝑡, 𝜃 𝑡 𝜔, 𝑣) is continuous in (𝑡, 𝑣) ∈ (−∞, 0] × 𝑋 and has a unique -pullback attractor  = {𝐴(𝜏, 𝜔) ∶ 𝜏 ∈ ℝ, 𝜔 ∈ Ω}.…”

“…29, Ref. 32 constructed a family of invariant measures for general nonautonomous random dynamical systems. To apply such abstract theory to system (), we need to verify the system () is jointly continuous in initial time and initial value (see Lemma 4), by which we can obtain the existence of invariant measures supported on the random attractors (see Theorem 3).…”

Invariant measures for first order nonlocal lattice dynamical systems with delays

“…This is because measurements of many important aspects of the turbulent flows are actually the measurements of some time-average quantities. Nowadays, there have been a series of works on invariant measures of evolution equations; see [4,12,5,13,14,24,32,33,34,45,44,50,56,53] for continuous systems. By using the generalized Banach limit, Lukaszewicz, Real and Robinson [33] constructed invariant measures for general continuous dynamical systems on metric spaces.…”