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.