非齐次平稳分布下Atlas模型的波动
Fluctuations of the Atlas model from inhomogeneous stationary profiles
AI总结：
本文研究非齐次平稳分布下Atlas模型的平衡波动，证明占据测度收敛于SPDE，并揭示下边缘方差发散及最低粒子收敛于非高斯极限。
摘要：
无限Atlas模型描述了实数线上可数布朗粒子集合的演化，其中最低粒子被赋予漂移γ∈[0,∞)。当粒子系统从具有指数增长密度的非齐次平稳分布出发时，我们研究Atlas模型的平衡波动。我们证明，适当中心化和缩放后的粒子位置占据测度（经过适当平移）作为时空随机场，收敛于由某个随机偏微分方程（SPDE）给出的极限。该方程的初始条件由布朗运动给出，方程由加性时空噪声驱动，该噪声在时间上为白噪声、在空间上有色，控制演化的线性算子是几何布朗运动的无穷小生成元。我们利用该SPDE还刻画了经过适当中心化和缩放的排序粒子位置的波动。我们的结果描述了粒子在体相中的行为，并发现描述渐近波动的高斯过程具有与Hurst参数为1/4的分数布朗运动相同的Hölder正则性。我们发现，与齐次分布的情形（Dembo和Tsai (2017)）不同，粒子系统下边缘的行为与体相行为非常不同，事实上，当接近下边缘时，高斯极限的方差发散到∞。确实，我们的结果表明，当粒子间隙由非齐次平稳分布之一给出时，从0出发的最低粒子，经过时间线性平移后，当t→∞时在分布上收敛于一个显式的非高斯极限。
英文摘要：
The infinite Atlas model describes the evolution of a countable collection of Brownian particles on the real line, where the lowest particle is given a drift of $γ\in [0,\infty)$. We study equilibrium fluctuations for the Atlas model when the system of particles starts from an inhomogeneous stationary profile with exponentially growing density. We show that the appropriately centered and scaled occupation measure of the particle positions, with suitable translations, viewed as a space-time random field, converges to a limit given by a certain stochastic partial differential equation (SPDE). The initial condition for this equation is given by a Brownian motion, the equation is driven by an additive space-time noise that is white in time and colored in space, and the linear operator governing the evolution is the infinitesimal generator of a geometric Brownian motion. We use this SPDE to also characterize the fluctuations of the ranked particle positions with a suitable centering and scaling. Our results describe the behavior of the particles in the bulk and one finds that the Gaussian process describing the asymptotic fluctuations has the same Hölder regularity as a fractional Brownian motion with Hurst parameter $1/4$. One finds that, unlike the setting of a homogeneous profile (Dembo and Tsai (2017)), the behavior on the lower edge of the particle system is very different from the bulk behavior and in fact the variance of the Gaussian limit diverges to $\infty$ as one approaches the lower edge. Indeed, our results show that, with the gaps between particles given by one of the inhomogeneous stationary distributions, the lowest particle, started from $0$, with a linear in time translation, converges in distribution to an explicit non-Gaussian limit as $t\to \infty$.
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