Symmetry groups of Markov processes. (English) Zbl 0755.60061

Liao, Ming

Ann. Probab. 20, No. 2, 563-578 (1992).

Let \(X=(\Omega,{\mathcal F},{\mathcal F}_ t,X_ t, x\in M)\) be a standard Markov process on a separable, locally compact Hausdorff state space \(M\). A homeomorphism \(g: M\to M\) is said to be invariant (resp. symmetric) for \(X\) provided that the r.v. \(g(X^ x_ t)\) is identical in law with \(X^{g(x)}_ t\) (resp. via a time change) for any \(x\in M\), where \(X^ x\) denotes \(X\) conditioned to start at \(x\). Let \(\text{Sym}(X)\) be the group of all symmetry transformations for \(X\), then a subgroup \(G\subset\text{Sym}(X)\) is said to be transitive provided that for any couple \(x,y\in M\) there is a \(g\in G\) such that \(g(x)=y\). For any given \(p\in M\), let \(G_ p\) denote the isotropy subgroup of \(G\) admitting \(p\) as a fixed point. The main result of the paper under review is the following extension of Glover’s result:
Theorem 1. If \(G\) is a subgroup of the (time change) symmetry group of a standard Markov process \(X\), which is transitive and has a compact isometry subgroup, then \(X\) becomes \(G\)-invariant via a time change.
Note that the above mentioned result of Glover treats the case where \(G\) is trivial. Furthermore the particular case of symmetry groups of diffusion processes is also studied in details in Section 5.

Reviewer: X.L.Nguyen (Hanoi)

MSC:

60J45	Probabilistic potential theory
58J65	Diffusion processes and stochastic analysis on manifolds

Keywords:

Markov process; Hausdorff state space; group of all symmetry transformations; compact isometry subgroup; time change