On the total—and strict total—Positivity of the kernels associated with parabolic initial boundary value problems

Abstract

This paper is concerned with the total—and strict total—positivity of the integral kernels which relate the solutions of parabolic initial boundary value problems to the initial and boundary data. Maximum principle arguments, in conjunction with a characterisation of total positivity in terms of variation diminishing properties, are used to prove total positivity of both the initial value and boundary value kernels under mild assumptions; similar methods yield strict total positivity when the coefficients appearing in the equation are analytic in either the time or the space variable. In this way we extend results of Karlin and McGregor on the total positivity of the kernel associated with the initial data; their proofs used the determinantal definition of total positivity and exploited the work of Gohberg and Krein on the Green's functions of Sturm-Liouville operators.