A Feynman-Kac gauge for solvability of the Schrödinger equation (Q1058679)

Let \(\{X_ t\), \(t\geq 0\}\) be Brownian motion in \({\mathbb{R}}^ d\) (d\(\geq 1)\). Let D be a bounded domain in \({\mathbb{R}}^ d\) with \(C^ 2\) boundary, \(\partial D\), and let q be a continuous (if \(d=1)\), Hölder continuous (if \(d\geq 2)\) function in \(\bar D.\) If the Feynman-Kac ''gauge'' \[ E_ x\{\exp (\int^{\tau_ D}_{0}q(X_ t)dt)\mathbf{1}_ A(X_{\tau_ D})\}, \] where \(\tau_ D\) is the first exit time from D, is finite for some non-empty open set A on \(\partial D\) and some \(x\in D\), then for any \(f\in C^ 0(\partial D)\), \(\phi (x)=E_ x\{\exp (\int^{\tau_ D}_{0}q(X_ t)dt)f(X_{\tau_ D})\}\) is the unique solution in \(C^ 2(D)\cap C^ 0(\bar D)\) of the Schrödinger boundary value problem \((\Delta +q)\phi =0\) in D, \(\phi =f\) on \(\partial D\).