On the Forkker-Planck equation in the stochastic theory of mortality: I

This paper, consisting of two parts, gives all the mathematical details that were omitted in a previous work by G. A. Sacher and E. Trucco (“The Stochastic Theory of Mortality.”Ann. N. Y. Acad. Sci.,96, 985–1007, cited here as ST). We assume that the reader is familiar with ST, where the stochastic theory of mortality, originally proposed by Sacher, is discussed at length. We recall that the basic model presented there refers to an ensemble of particles performing Brownian motion in one dimension, with the added constraint of two absorbing barriers. These two points, collectively, are designated as the “lethal bound.” Part I (section 1 to 4) deals with the special case in which the two absorbing barriers are symmetrically located at a finite distance from the origin. The solution of the Fokker-Planck equation is obtained from the theory of eigenvalue problems. Quite generally, the eigenfunctions functions belong to the family of Kummer's confluent hypergeometric functions, but the symmetry condition imposed here results in considerable simplification and makes it possible to estimate the first few eigenvalues by a graphical procedure. In section 3 we show how perturbation theory can be applied in the limiting case of “weak homeostasis,” and section 4 deals with the opposite extreme of “strong homeostasis.” A rigorous proof is given for the result corresponding to equation (28) of ST (asymptotic or quasi-static approximation for the “force of mortality”). This work was performed under the auspices of the U.S. Atomic Energy Commission.