Periodicity and longtime diffusion for mean field systems in $\mathbb{R}^d$

完全正相关共生分支模型的唯一性与长期行为

Uniqueness and Longtime Behavior of the Completely Positively Correlated Symbiotic Branching Model

本文利用共生分支模型与抛物Anderson模型的新对偶性，解决了$ρ=1$时完全正相关共生分支模型的弱唯一性开放问题，并研究了其长期行为，证明了在合适初始条件下其中一个种群最终会灭绝。

Synchronization for KPZ

We study the longtime behavior of KPZ-like equations: $$ \partial_{t}h(t,x) = Δ_{x} h (t, x) + | \nabla_{x}h (t,x)|^{2} + η(t, x), \qquad h(0, x) = h_0(x), \qquad (t, x) \in (0, \infty) \times \mathbb{T}^{d} $$ on the $d-$dimensional torus $\mathbb{T}^{d}$ driven by an ergodic noise $η$ (e.g. space-time white in $d= 1$. The analysis builds on infinite-dimensional extensions of similar results for positive random matrices. We establish a one force, one solution principle and derive almost sure synchronization with exponential deterministic speed in appropriate Hölder spaces.

相互作用粒子系统中$\Phi$-熵的轨迹耗散

Trajectorial dissipation of $Φ$-entropies for interacting particle systems

该研究采用概率方法，证明具有任意几何结构与紧致局部自旋空间的无穷多相互作用粒子空间扩展系统中，$\Phi$-熵的耗散存在于单轨迹层面，将相关经典结果推广至弱假设的无穷维情形。

Optimal scaling for the transient phase of the random walk Metropolis algorithm: The mean-field limit

We consider the random walk Metropolis algorithm on $\mathbb{R}^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one-dimensional law. In the limit $n\to\infty$, it is well known (see [Ann. Appl. Probab. 7 (1997) 110-120]) that, when the variance of the proposal scales inversely proportional to the dimension $n$ whereas time is accelerated by the factor $n$, a diffusive limit is obtained for each component of the Markov chain if this chain starts at equilibrium. This paper extends this result when the initial distribution is not the target probabi…