R^n上McKean-Vlasov随机反应扩散方程的状态-律动力学：拉回随机吸引子与零噪声稳定性
State-law dynamics of McKean-Vlasov stochastic reaction-diffusion equations on $\mathbb R^n$: pullback random attractors and zero-noise stability
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中文总结 AI 辅助
针对R^n上一类McKean-Vlasov随机反应扩散方程，构造乘积空间上的连续随机动力系统，得到其唯一拉回随机吸引子，证明零噪声上半连续性及收缩机制下的定量收敛性。
中文摘要 AI 辅助
分布依赖性通常会阻碍McKean-Vlasov状态解映射满足上同调恒等式，而相关的Sobolev嵌入在无界域上是非紧的。对于一类具有耗散多项式反应、单侧单调状态-律耦合及有限维加性噪声的随机反应扩散方程，我们将确定性律半流与路径态演化相结合，在状态空间及其二次Wasserstein律空间的乘积上构造一个连续随机动力系统。一个抽象乘积空间判据、混合能量估计、局部正则性及一致远场估计，得到了唯一的拉回随机吸引子，且无需全局律收缩。其律投影是律半流的全局吸引子，其状态纤维不必为单点。在额外的严格收缩条件下，律吸引子简化为唯一的不变律，对应状态纤维为自洽随机平衡。我们还建立了零噪声上半连续性，且在收缩机制下，不变律与随机平衡的定量收敛界与噪声振幅呈线性关系。
英文摘要
Distribution dependence generally prevents McKean-Vlasov state solution maps from satisfying a cocycle identity, while the relevant Sobolev embedding is noncompact on unbounded domains. For a class of stochastic reaction-diffusion equations with a dissipative polynomial reaction, a one-sided monotone state-law coupling and finite-dimensional additive noise, we combine the deterministic law semiflow with the pathwise state evolution to construct a continuous random dynamical system on the product of the state space and its quadratic Wasserstein law space. An abstract product-space criterion, mixed-energy estimates, local regularity and uniform far-field estimates yield a unique pullback random attractor without global law contraction. Its law projection is the global attractor of the law semiflow, and its state fibers need not be singletons. Under an additional strict contraction condition, the law attractor reduces to the unique invariant law and the corresponding state fiber to a self-consistent random equilibrium. We also establish zero-noise upper semicontinuity and, in the contractive regime, quantitative convergence of the invariant laws and random equilibria with bounds linear in the noise amplitude.
发表机构
- School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
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