Perturbation of transition functions and a Feynman--Kac formula for the incorporation of mortality
For structured population models, the perturbation approach in terms of output kernels (versus output operator families) was used in [\textit{O. Diekmann, M. Gyllenberg, J. A. J. Metz} and \textit{H. R. Thieme}, J. Math. Biol. 43, No. 4, 349--388 (1998; Zbl 0909.92023); \textit{M. Gyllenberg, T. Lant} and \textit{H. R. Thieme}, Differential Integral Equations 19, No. 4, 401--436 (2006)]. In the present paper, mortality is incorporated on the level of transition kernels \(K\) associated to a Markov process. At first, autonomous kernels (nondegenerate in a sense) are considered; they are in 1-to-1 correspondence with Hille--Yosida operators \(A\) acting on bounded measures. One obtains the perturbed Markov kernel \(K_{\infty}\) as a unique solution of the integral equation involving autonomous output kernels, corresponding to \(K\) describing merely state transitions and the mortality rate. Passing to \(K_{\infty}\) amounts to the additive perturbation of \(A\). The method extends to the nonautonomous case, moreover, \(K_{\infty}\) satisfies then a type of Feynman-Kac formula. The general case may be treated by including time as an extra space variable of the auxiliary autonomous system.
scientific article; zbMATH DE number 5162444
scientific article