We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.

中文翻译：

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正定矩阵上的相互作用扩散

我们考虑在正定矩阵空间中的布朗粒子系统，该系统独立于一些简单的相互作用而独立地演化。我们给出了具有可集成结构的此类过程的示例。这些与矩阵自变量的K-贝塞尔函数以及这些函数的多元概括有关。后者是非阿贝尔Toda晶格的特定量化的本征函数。