Before revision:  We study a "smoothed" version of this problem and ask whether there is a "barrier" b such that \mathbb{E[\exp(-\lambda \int_0^t \psi(B_s - b(s)) \, ds)] } [\exp(-\lambda\int_0^t\psi(B_s-b(s))\,ds)]} =P\{\zeta >t\}, where \lambda is a killing rate parameter  and \psi:R  0,1 is a non-increasing \to[0,1]  function.
After revision:  We study a "smoothed" version of this problem and ask whether there is a "barrier" b such that \mathbb{E[\exp(-\lambda \int_0^t \psi(B_s - b(s)) \, ds)] } \mathbb{E[\exp(-\lambda\int_0^t\psi(B_s-b(s))\,ds)]} =P\{\zeta >t\}, where \lambda is a killing rate parameter and \psi:R  0,1 is a non-increasing \to[0,1] function.