PROBABILISTIC REPRESENTATION OF SOLUTIONS OF SCHRÖDINGER EQUATIONS

In this paper, two equivalent conditions for a solution u in a domain D of the Sohrödinger equation (\Delta+2 q) u=0 to be expressed in the form u(x)=E_x\left(\exp \left(\int_0^\tau_D q(x(s)) d S\right) \varphi\left(x\left(\tau_D\right)\right)\right), \quad x \in D (where x(t) is a Brownian motion, \tau_D is the exit time of D, \varphi is a measurable funotion on \partial D ) are given, and the existenoe and uniqueness of solution of the stochastio Diriohlet problem for the Sohrödinger equation is proved.