Spatial ergodicity for SPDEs via Poincaré-type inequalities
Spatial ergodicity for SPDEs via Poincaré-type inequalities
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DOI：
10.1214/21-ejp690
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发表时间：
2019-07
影响因子：
1.4
通讯作者：
Le Chen;D. Khoshnevisan;D. Nualart;Fei Pu
中科院分区：
文献类型：
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作者：
Le Chen;D. Khoshnevisan;D. Nualart;Fei Pu
考虑形式为 ∂tu = 1 2 Δu+ σ(u)η 的抛物线随机偏微分方程，其中 u = u(t , x)，t ≥ 0 且 x ∈ R，σ : R → R 是 Lipschitz 连续且非随机的，并且 η 是中心高斯噪声，在时间上为白色，在空间上为彩色，具有可能有符号的同质空间相关性 f 。另外，如果 u(0) ≠ 1，那么我们证明，在 f 的温和衰减条件下，过程 x 7→ u(t , x) 在所有时间 t > 0 时都是平稳且遍历的。有人认为，当与矩估计相结合时，u 的空间遍历性告诉我们此类 SPDE 解的间歇性质 [1, 37]。我们的结果为此类讨论提供了严格的理由。我们的方法取决于调和分析和正型函数以及马利亚文微积分和庞加莱不等式的新事实。我们通过以下方式进一步展示这些庞加莱不等式的实用性：(a) 描述确保随机场 u(t) 在每个 t > 0 时混合的条件； (b) 快速证明 Conus 等人 [15] 关于 u 间歇性岛屿“大小”的猜想。本文的遍历性和混合结果非常清晰，因为它们在非线性项 σ 是常数函数的简单设置中包含了 Maruyama [42] 的经典理论（另请参见 Dym 和 McKean [23]）。 MSC 2010主题分类：60H15、37A25、60H07、60G10。
Consider a parabolic stochastic PDE of the form ∂tu = 1 2 ∆u+ σ(u)η, where u = u(t , x) for t ≥ 0 and x ∈ R, σ : R → R is Lipschitz continuous and non random, and η is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation f . If, in addition, u(0) ≡ 1, then we prove that, under a mild decay condition on f , the process x 7→ u(t , x) is stationary and ergodic at all times t > 0. It has been argued that, when coupled with moment estimates, spatial ergodicity of u teaches us about the intermittent nature of the solution to such SPDEs [1, 37]. Our results provide rigorous justification of such discussions. Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field u(t) is mixing for every t > 0; and by (b) giving a quick proof of a conjecture of Conus et al [15] about the “size” of the intermittency islands of u. The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama [42] (see also Dym and McKean [23]) in the simple setting where the nonlinear term σ is a constant function. MSC 2010 subject classification: 60H15, 37A25, 60H07, 60G10.