Let $D\subset\Bbb R^d$ be an open and bounded set, $f:\partial D\to\Bbb R$ continuous.
We say that $F:\overline D\to\Bbb R$ satisfies the Dirichlet problem with boundary condition given by $f$, if \begin{align*} &F\in\mathcal C(\overline D)\cap\mathcal C^2(D)\\ &\Delta F\stackrel{D}{=}0\;\; \\ &F\stackrel{\partial D}{=}f\;\;. \end{align*}
If now we take a $d$-dimensional browninan motion $B$, we set $B_t^x:=x+B_t$ for fixed $x\in\Bbb R^d$ and $\tau:=\inf\{s\ge0\;:\;B_s^x\in D^C\}$, which is a stopping time, a.s. $<+\infty$.
There is a Theorem, which states that if the solution to the Dirichlet Problem exists, then it's unique and $\forall x\in D$ it is given by $$ F(x)=\Bbb E[f(B_{\tau}^x)]\;\;\;. $$ My problem is: why is this solution well defined? Doesn't $F$ change if we change the BM inside the mean value?