论文标题
无限棕色粒子系统具有等级依赖性漂移的极端不变分布
Extremal Invariant Distributions of Infinite Brownian Particle Systems with Rank Dependent Drifts
论文作者
论文摘要
\ noindent考虑根据独立的布朗动作移动的真实线上的无限粒子集合，因此，从左侧的$ i $ th粒子获得了漂移$ g_ {i-1} $。 $ g_0 = 1 $和$ g_ {i} = 0 $ in \ in \ mathbb {n} $的情况对应于研究良好的无限地图集模型。在漂移矢量$ \ boldsymbol {g} =（g_0，g_1，\ ldots）'$的条件下，众所周知，与关联排名粒子的间隙序列相对应的Markov过程具有产品形式平稳分布的连续性$ \ \ \ {π_a^{π_a^{π_a^{\ BoldSymbol {\ BoldSymbol { s^{\ boldsymbol {g}} \} $其中$ s^{\ boldsymbol {g}} $是真实行的半无限间隔。在这项工作中，我们表明所有这些固定分布都是极端和崇高的。我们还证明，满足轻度集成性条件的Markov进程的任何产品形式的固定分布都必须为$π_a^{\ boldsymbol {g}} $，对于某些$ a \ in s^{\ boldsymbol {g}} $。即使对于无限地图集模型，这些结果也是新的。这项工作在表征通过其相对等级相互作用的一般竞争的布朗粒子系统的所有不变分布的开放问题上取得了进展。证明依赖于无限系统中各个颗粒的当地时间的布朗颗粒的同步和镜像耦合。
\noindent Consider an infinite collection of particles on the real line moving according to independent Brownian motions and such that the $i$-th particle from the left gets the drift $g_{i-1}$. The case where $g_0=1$ and $g_{i}=0$ for all $i \in \mathbb{N}$ corresponds to the well studied infinite Atlas model. Under conditions on the drift vector $\boldsymbol{g} = (g_0, g_1, \ldots)'$ it is known that the Markov process corresponding to the gap sequence of the associated ranked particles has a continuum of product form stationary distributions $\{π_a^{\boldsymbol{g}}, a \in S^{\boldsymbol{g}}\}$ where $S^{\boldsymbol{g}}$ is a semi-infinite interval of the real line. In this work we show that all of these stationary distributions are extremal and ergodic. We also prove that any product form stationary distribution of this Markov process that satisfies a mild integrability condition must be $π_a^{\boldsymbol{g}}$ for some $a \in S^{\boldsymbol{g}}$. These results are new even for the infinite Atlas model. The work makes progress on the open problem of characterizing all the invariant distributions of general competing Brownian particle systems interacting through their relative ranks. Proofs rely on synchronous and mirror coupling of Brownian particles and properties of the intersection local times of the various particles in the infinite system.
