A probabilistic representation of totally positive matrices
\textit{C. Loewner} [Math. Z. 63, 338-340 (1955; Zbl 0068.250)] has shown that every non-singular totally positive matrix of finite order can be reached from the identity by forward solutions of the control equation \(dR/dt=C(t)R\), when C(t) varies in the class of Jacobi matrices with non- negative off-diagonal entries. We give a probabilistic solution of this equation by associating with it a finite-state non-stationary birth-and- death process and using the Feynman-Kac formula. This yields a probabilistic representation of non-singular totally positive matrices in terms of random evolutions. We then extend the representation to compound matrices by making use of a multi-particle system due to \textit{R. P. Feynman} [Phys. Rev., II. Ser. 76, 749-759 (1949; Zbl 0037.124), ibid. 76, 769-789 (1949; Zbl 0038.133)] and \textit{S. Karlin} and \textit{J. McGregor} [Pac. J. Math. 9, 1141-1164 (1959; Zbl 0092.345)]. As applications, we obtain both the Jacobi-Liouville formula and the inequality of \textit{F. R. Gantmacher} and \textit{M. G. Krein} [Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme. Akademie-Verlag (1964)] for the principal minors of totally positive matrices by simple probabilistic arguments.
finite-state non-stationary birth-and-death process
scientific article; zbMATH DE number 3988396