可积随机粒子系统中的历史重写
Rewriting History in Integrable Stochastic Particle Systems
AI总结：
本研究通过Yang-Baxter方程建立速度参数排列不同的可积粒子系统间的马尔可夫交织关系，导出统一Lax方程并构造“重写历史”耦合，为多点可观测量分析提供新工具。
摘要：
许多一维空间中的可积随机粒子系统（如TASEP——全非对称简单排斥过程——及其$q$-变形，即$q$-TASEP）在给每个粒子配备各自的速度参数后仍保持可积性。在本工作中，我们提出了速度参数排列不同的粒子系统之间的马尔可夫转移算子交织关系。这些关系推广了我们之前的工作（arXiv:1907.09155, arXiv:1912.06067），但此处我们采用了一种基于高自旋随机六顶点模型的Yang-Baxter方程的新方法。我们的交织子是马尔可夫转移算子，这导致了有趣的概率推论。\n首先，我们获得了齐次连续时间版本粒子系统的马尔可夫转移半群的新型Lax型微分方程。我们的Lax方程以统一方式编码了$q$-TASEP和TASEP多点可观测量随时间演化，这可能对这些系统多点可观测量的渐近分析有意义。\n其次，我们证明我们的交织关系导致了速度参数排列不同的粒子系统轨迹上的概率测度之间的耦合。这种耦合的条件分布被实现为一种“重写历史”随机游走，它根据相邻粒子轨迹确定的腔室中随机重新采样粒子的轨迹。作为副产品，我们构造了正实半轴上具有不同速率的标准泊松过程的新耦合。
英文摘要：
Many integrable stochastic particle systems in one space dimension (such as TASEP - Totally Asymmetric Simple Exclusion Process - and its $q$-deformation, the $q$-TASEP) remain integrable if we equip each particle with its own speed parameter. In this work, we present intertwining relations between Markov transition operators of particle systems which differ by a permutation of the speed parameters. These relations generalize our previous works (arXiv:1907.09155, arXiv:1912.06067), but here we employ a novel approach based on the Yang-Baxter equation for the higher spin stochastic six vertex model. Our intertwiners are Markov transition operators, which leads to interesting probabilistic consequences. First, we obtain a new Lax-type differential equation for the Markov transition semigroups of homogeneous, continuous-time versions of our particle systems. Our Lax equation encodes the time evolution of multipoint observables of the $q$-TASEP and TASEP in a unified way, which may be of interest for the asymptotic analysis of multipoint observables of these systems. Second, we show that our intertwining relations lead to couplings between probability measures on trajectories of particle systems which differ by a permutation of the speed parameters. The conditional distribution for such a coupling is realized as a "rewriting history" random walk which randomly resamples the trajectory of a particle in a chamber determined by the trajectories of the neighboring particles. As a byproduct, we construct a new coupling for standard Poisson processes on the positive real half-line with different rates.
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