Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $\mu\ll \mu_0$ on Wiener space so that $B(t)$ under $\mu$ is $F(t)+\tilde B(t)$ by Cameron Martin Girsanov.
What does it mean for $F$ to be deterministic on the level of $\mu$? More precisely, is there a convex function $D$ defined on the space of all Girsanov measures so that $D(\mu)=0$ if and only if $\mu$ corresponds to a deterministic drift?
I know of one such function, but it is not convex (I don't think).
Let $$D(\mu)=D_{KL}(\mu||\mu_0)-\frac12\int_0^T\left(\partial_sE_\mu[\omega(s)]\right)^2ds$$
Then this is simply
$$\int_0^T\operatorname{var}_\mu(F'(s))ds$$
Is there a way of making this convex?
