= Solution
Fix $t>0$ and define $w(s,y)=u(t-s,|y|)$ for $0\leq s\leq t$. The assumed smooth extension and the <Neumann boundary condition> at zero make $w$ a $C^{1,2}$ function. The <heat equation> gives
$$
w_s+\frac12w_{yy}=0.
$$
The <Itô formula> therefore makes $w(s,B_s)$ a local martingale. Stop first when $|B|$ leaves a large compact interval. The exponential growth bound and the finite exponential moments of the maximum of <Brownian motion> on $[0,t]$ give <uniform integrability>, so localization and the <dominated convergence theorem> yield
$$
u(t,x)=w(0,x)=\mathbb E_x[w(t,B_t)]=\mathbb E_x[f(|B_t|)].
$$
This is the <Feynman-Kac formula> for the Neumann heat problem.
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