在Lie群作用下不变的Markov过程

Invariant Markov processes under actions of Lie groups

本文的目的是,依照经典的L′evy-Khinchin三参数表示的精神,论述在Lie群作用下不变的Markov过程的表示理论.对不变Markov过程,按其一般性,我们将在三个层面中进行讨论.首先是Lie群里平移不变的Markov过程,然后是一般流形中在可迁群作用下不变的Markov过程.这两类过程都称L′evy过程,具有三参数表示.第三类过程为在不可迁群作用下不变的Markov过程.在一定条件下,这类过程可分解为一横截群轨道的径过程和一沿着群轨道的角过程.后者为时间非齐次的不变Markov过程,具有依赖于时间的三参数表示.

We present a representation theory for invariant Markov processes under Lie group actions,in the spirit of the classic Levy-Khinchin representation.We will study invariant Markov processes at three diffierent levels of generality.First,we consider Markov processes in Lie groups that are invariant under translations,and then Markov processes in manifolds that are invariant under transitive group actions.These two types of processes are called Lévy processes,and they possess a triple representation.The third type of processes are Markov processes that are invariant under non-transitive group actions.Under certain conditions,such a process may be decomposed into a radial part,that is transversal to group orbits,and an angular part,that is along an orbit.The latter is a time inhomogeneous invariant Markov process,and may be represented by a time-dependent triple.