Infinite Excessive and Invariant Measures

Abstract

In the paper [10] the following problem was considered. Given a contraction semigroup T_t on a Borel space d and an excessive measure ν, when is it possible find another contraction semigroup T_t such that

$$ \widetilde{{{T_t}}} > {T_t} $$

(1.1.1)

and

$$ \nu \widetilde{{{T_t}}} = \nu . $$

(1.1.2)

The most restrictive condition under which this problem was solved is the finiteness of the excessive measure ν. This condition excludes such an interesting case as the semigroup T_t generated by the transition function of Wiener’s process killed at the origin and the Lebesque measure ν.

This research was sponsored by Office of Naval Research Contract No. 00014-79-C-0685 at the Institute for Mathematical Studies in the Social Sciences, Stanford University.