 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 $p_t=d{P}_t/d\mu$ 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 thatThe 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. The required derivative is in the square-root-density $L^2$ norm.