Soft killing limit for Brownian nonnegative survival (${\lim }_{\lambda \to \infty }{\mathbb{E}}_x{e}^{-\lambda {\int }_0^t\varphi (B_s)ds}={\mathbb{P}}_x({\min }_{s\le t}{B}_s\ge 0)$)

Let $\varphi$ be continuous, nonnegative, zero on $[0,\infty )$, and strictly positive on $(-\infty ,0)$. By continuity of Brownian paths, its occupation integral vanishes exactly on nonnegative paths. Dominated convergence gives the displayed limit. The Brownian reflection principle makes it $1-2{\mathbb{P}}_x(B_t\le 0)$ for $x\ge 0$ and zero for $x<0$. In particular the limit at $x=0$ is zero for every positive time. The choice $\varphi (x)=\min (1,\max (0,-x))$ is a bounded soft killing potential.