Solution
= Solution
Let $\sigma_x=\inf\{t\geq0:\widetilde B_t=x\}$. Continuity gives $\{S\geq x\}=\{\sigma_x<\infty\}$. On this event, the <Strong Markov property> says that
$$
(\widetilde B_{\sigma_x+t}-x)_{t\geq0}
$$
is an independent Brownian motion with drift $-\mu$. It reaches level $y$ with probability $\mathbb P(S\geq y)$. Therefore
$$
\mathbb P(S\geq x+y)
=\mathbb P(\sigma_x<\infty)\mathbb P(S\geq y)
=\mathbb P(S\geq x)\mathbb P(S\geq y).
$$