论文标题
离散KDV-和TODA型系统的详细平衡和不变措施
Detailed balance and invariant measures for discrete KdV- and Toda-type systems
论文作者
论文摘要
为了研究离散KDV-和TODA型系统的不变测量，本文着重于模型，在空间和时间上分散索引，其动力学是确定性的，并且通过晶格方程在本地定义。提出了一个详细的平衡标准，即在描述空间独立且相同/交替分布的配置的措施中，表征了分布时间不变的措施。还给出了建立动力学的恐怖性的条件。这些结果适用于各种可离散集成系统的示例，即超差异和离散的KDV方程，为此表明，相关的不变措施分别是指数/几何和广义的高斯形式，以及高斯的逆向形式，以及相关的toda latte tostrient and Inverriant and Inverriant and Invartiant and Invartiant Anniant Invartiant and Invartiant Anniant Invartiant Anniant Invartiant Anniant Invartiant Anniant Invarianialt Anniant Invartiant to tonemiant伽玛形式。在KDV型模型的情况下，证明了登山性。提出了不同系统的不变度度量之间的联系，与随机集成模型和迭代随机函数的连接也是如此。此外，提出了许多有关标准分布表征的猜想。
In order to study the invariant measures of discrete KdV- and Toda-type systems, this article focusses on models, discretely indexed in space and time, whose dynamics are deterministic and defined locally via lattice equations. A detailed balance criterion is presented that, amongst the measures that describe spatially independent and identically/alternately distributed configurations, characterizes those that are temporally invariant in distribution. A condition for establishing ergodicity of the dynamics is also given. These results are applied to various examples of discrete integrable systems, namely the ultra-discrete and discrete KdV equations, for which it is shown that the relevant invariant measures are of exponential/geometric and generalized inverse Gaussian form, respectively, as well as the ultra-discrete and discrete Toda lattice equations, for which the relevant invariant measures are found to be of exponential/geometric and gamma form. Ergodicity is demonstrated in the case of the KdV-type models. Links between the invariant measures of the different systems are presented, as are connections with stochastic integrable models and iterated random functions. Furthermore, a number of conjectures concerning the characterization of standard distributions are posed.
