= Solution
A <statistical path> through $P$ is a family $t\mapsto P_t\in\mathcal P$, defined on an open interval containing zero, with $P_0=P$. Write $p_t=dP_t/d\mu$ and $p=p_0$. Its <differentiability in quadratic mean> at zero means that there is a <square-integrable function> $g\in L^2(P)$ such that
$$
\boxed{\int\left(\sqrt{p_t}-\sqrt p-\frac t2g\sqrt p\right)^2\,d\mu=o(t^2).}
$$
The <function> $g$ is the <score function> of the <statistical path>. Thus it is the <derivative> of the square-root <probability density function>, multiplied by two and divided by $\sqrt p$ where $p>0$. The <score function> is determined only $P$-almost everywhere. The displayed definition also controls any probability mass entering a region where $p=0$; a pointwise <derivative> of the log <probability density function> alone would not do that. \b[The required <derivative> is in the square-root-density $L^2$ <norm>.]
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