DIFFUSIONS WITH SINGULAR DRIFT RELATED TO WAVE-FUNCTIONS

Schrodinger equations are equivalent to pairs of mutually time-reverse d non-linear diffusion equations. Here the associated diffusion proces ses with singular drift are constructed under assumptions adopted from the theory of Schrodinger operators, expressed in terms of a local sp ace-time Sobolev space. By means of Nagasawa's multiplicative function al N(s)t, a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular d rift is identified by Maruyama's drift transformation. For this a vers ion of Ito's formula for continuous space-time functions with first an d second order derivatives in the sense of distributions satisfying lo cal integrability conditions has to be derived. The equivalence is sho wn between weak solutions of a diffusion equation with singular creati on and killing term and the solutions of a Feynman-Kac integral equati on with a locally integrable potential function.