用于顺序相互作用扩散的对数涨落层次结构
A Logarithmic Fluctuation Hierarchy for Sequential Interacting Diffusions
浏览论文内容
中文总结 AI 辅助
研究下三角相互作用扩散系统高斯涨落，引入对数加权涨落场，证明其在加权负索伯列夫路径空间可数乘积中联合收敛，经验涨落场极限不受经典可交换情形支配，方法含条件测度替换等。
中文摘要 AI 辅助
我们研究了一个下三角相互作用扩散系统的高斯涨落，其中粒子\(i\)仅与其前驱粒子相互作用。尽管该系统的经验测度收敛到与相应可交换平均场系统相同的麦克凯恩 - 弗拉索夫极限，但顺序结构在\(N^{-1/2}\)涨落尺度上仍然可见。我们引入对数加权涨落场\[ Y_t^{N,n} = \frac{1}{\sqrt{N}} \sum_{i=1}^N \frac{\bigl(\log(N/i)\bigr)^n}{n!} (\delta_{X_t^i}-\bar\rho_t), \qquad n\ge 0, \]并证明了整个族在加权负索伯列夫路径空间的可数乘积中联合收敛到一个线性层次结构的唯一概率强解，其中\(Y^n\)与\(Y^{n + 1}\)耦合。特别地，经验涨落场\(\sqrt{N}(\mu_t^N-\bar\rho_t)=Y_t^{N,0}\)的极限不受经典可交换情形中出现的封闭涨落随机偏微分方程的支配。证明结合了条件测度替换、对数权重的确定性估计、鞅论证和加权沃尔泰拉估计。
英文摘要
We study Gaussian fluctuations for a lower-triangular system of interacting diffusions in which particle $i$ interacts only with its predecessors. Although the empirical measure of this system converges to the same McKean--Vlasov limit as in the corresponding exchangeable mean-field system, the sequential structure remains visible at the $N^{-1/2}$ fluctuation scale. We introduce the logarithmically weighted fluctuation fields \[ Y_t^{N,n} = \frac{1}{\sqrt{N}} \sum_{i=1}^N \frac{\bigl(\log(N/i)\bigr)^n}{n!} (δ_{X_t^i}-\barρ_t), \qquad n\ge 0, \] and prove joint convergence of the entire family in a countable product of weighted negative Sobolev path spaces to the unique probabilistically strong solution of a linear hierarchy in which $Y^n$ couples to $Y^{n+1}$. In particular, the limit of the empirical fluctuation field $\sqrt{N}(μ_t^N-\barρ_t)=Y_t^{N,0}$ is not governed by the closed fluctuation SPDE arising in the classical exchangeable case. The proof combines conditional-measure replacement, deterministic estimates for the logarithmic weights, a martingale argument, and a weighted Volterra estimate.
