I am looking for a (preferably some monograph) reference on the following fact: $$ u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \} $$ satisfies the heat equation $$ \frac{\partial u}{\partial t} = \frac{1}{2} \Delta u \quad \text{in} \quad \mathbb{R}_+ \times A $$ and the following initial-boundary value conditions: $$ u \ |_{t = 0} = 1 \quad \text{and} \quad u \ |_{\partial A} = 0. $$

I'm struggling to find this particular fact among many facts of the same type. Correct me if I'm wrong, but it doesn't seem to follow (at least directly) from Feynman-Kac formula, nor does it follow from the other BM-PDE links I keep finding.

Additionally to the reference request, I'd like to know what's the most straightforward way to see this fact?

P.S. It seems to *formally* follow from Feynman-Kac, which says that
$$
u ( t, x ) = \mathbb{E} \left\{
f ( B_t ) \exp \left( -\int_0^t v(B_s) \, ds \right)
\right\}
$$
solves
$$
\frac{\partial u}{\partial t} = \frac{1}{2} \Delta u - vu
$$
with $u \ |_{t=0} = f$. Letting $f=1$ and formally
$$
v ( x ) =
\begin{cases}
0, & x \in A, \\
\infty, & x \not\in A
\end{cases}
$$
seems to give the result... Maybe it's possible to do it more rigorously by taking $v_n(x) = n$ on $x \not\in A$ and then letting $n \to \infty$.

what should I assume of it... Being compact isn't necessary, but I have no idea how far the result stretches... $\endgroup$