X: 14, 216-234, LNM 511 (1976)
WILLIAMS, David
The Q-matrix problem (
Markov processes)
This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched
Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams
Diffusions, Markov Processes and Martingales, vol. 1 (second edition), Wiley 1994. See also
1024Keywords: Markov chains,
Ray compactification,
Local times,
ExcursionsNature: Original
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