= Solution
Separate $L$ into a <diffusion generator> and the constant killing term $-1/2$. The appropriate <stochastic differential equation> and its explicit <geometric Brownian motion> solution are
$$
dX_s=X_s\,dB_s+\frac12X_s\,ds,\qquad X_0=x,
\qquad \boxed{X_s=xe^{B_s}.}
$$
The second-order term in the <Itô formula> cancels the drift of $\log|X|$ when $x\ne0$; if $x=0$, the solution stays zero.
Fix a terminal time $t$. Apply the <Itô formula> to $e^{-s/2}u(t-s,X_s)$ for $0\leq s\leq t$. Its drift is
$$
e^{-s/2}\left(-u_t+\frac{X_s^2}{2}u_{xx}+\frac{X_s}{2}u_x-\frac12u\right)(t-s,X_s)\,ds=0.
$$
After localization this is a <local martingale>, and it is bounded because $u$ is bounded. It is therefore a true <martingale>. Taking <expectations> at its endpoints gives the <Feynman-Kac formula>
$$
\boxed{u(t,x)=e^{-t/2}\mathbb E[g(xe^{B_t})].}
$$
For $t>0$ and $x\ne0$, the change of variable $y=xe^z$ in the <normal density> produces the <killed geometric Brownian heat kernel> with respect to <Lebesgue measure>:
$$
\boxed{p(t,x,y)=
\begin{cases}
\displaystyle\frac{e^{-t/2}}{|y|\sqrt{2\pi t}}\exp\left[-\frac{\log^2(|y/x|)}{2t}\right],&xy>0,\\
0,&xy\leq0.
\end{cases}}
$$
For $xy>0$, an equivalent form is
$$
p(t,x,y)=\frac1{|x|\sqrt{2\pi t}}\exp\left[-\frac{(\log|y/x|+t)^2}{2t}\right].
$$
In particular,
$$
\boxed{p(t,1,y)=\frac1{\sqrt{2\pi t}}\exp\left[-\frac{(\log y+t)^2}{2t}\right]\quad(y>0).}
$$
The apparent missing $1/y$ in this last expression is absorbed into the completed square, together with the killing factor; it is not a transcription error. The total mass is $e^{-t/2}$, as expected for killing at rate $1/2$. At $x=0$ the fundamental kernel is instead the <measure> $e^{-t/2}\delta_0(dy)$; it has no <Radon-Nikodym derivative> with respect to <Lebesgue measure>. Thus $u(t,0)=e^{-t/2}g(0)$. At $t=0$ the kernel is $\delta_x$, and continuity of bounded $g$ gives the initial condition. This also specifies the fundamental solution at the degenerate point omitted by a density-only formula.
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