= Soft killing limit for Brownian nonnegative survival
{title2=$\lim_{\lambda\to\infty}\mathbb E_xe^{-\lambda\int_0^t\phi(B_s)ds}=\mathbb P_x(\min_{s\leq t}B_s\geq0)$}
Let $\phi$ 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 $1-2\mathbb P_x(B_t\leq0)$ for $x\geq0$ and zero for $x<0$. In particular the limit at $x=0$ is zero for every positive time. The choice $\phi(x)=\min(1,\max(0,-x))$ is a bounded soft killing potential.
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