On Kac functionals of one-dimensional diffusions
In the paper under the review Kac's functionals for a one-dimensional diffusion with killing (only) at the boundaries are studied up to the first exit time from a given open interval. It is seen that the expectation of a functional of this kind is a solution of the corresponding Schrödinger equation. Under the additional assumption of disconjugacy of the equation a probabilistic representation of the expectation of a functional with not necessarily negative potential is obtained. The idea is to construct - using a non-vanishing solution of the Schrödinger equation - a diffusion such that the original law of the Kac's functional is identical with the law of a simple functional for this new diffusion. Some examples are presented. Further, for negative potentials necessary and sufficient conditions for the a.s. convergence and a.s. divergence for the integral in the Kac's functional are given. Some other topics are also discussed.
scientific article; zbMATH DE number 3969816
scientific article