Diffusions with singular drift related to wave functions
Schrödinger equations are equivalent to pairs of mutually time-reversed nonlinear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space- time Sobolev space. By means of Nagasawa's multiplicative functional \(N^ t_ s\), a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived. The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solution of a Feynman-Kac integral equation with a locally integrable potential function.
diffusion equation with singular creation and killing term
scientific article; zbMATH DE number 569070
scientific article