Analytical and probabilistic methods for totally positive matrices

Loewner proved that all non-singular TP matrices can be generated by a differential equation, deriving from the theory of transformation semigroups. We illustrate how his equation, once extended to compounds, can be employed as a tool for the study of TP matrices. It is also related to probability theory. We discuss the problem of stochastic embedding and show how probabilistic methods, applied to his equation, can be used to reveal properties of TP matrices. The paper contains improved versions of older results of the author, together with new proofs of the generalized ''Hadamard-Fischer'' inequalities and the Feynman-Kac formula, as well as a novel derivation of the Frydman-Singer embedding theorem.