Suppose that $X_t$ is a diffusion process on $\mathbb{R}$ and define the functionals $F_t$ from $L^2_{\mathbb{P}}(\mathfrak{F}_t)$ to $\mathbb{R}$ by $$ F_t: Y\mapsto \mathbb{E}\left[ (X_{t} - Y)^2 \right], $$ where $\mathfrak{F}_t$ is the $\sigma$-algebra generated by $X_t$.
When does there exits $t_0>0$ for which $(F_t)_{t\in (0,t_0)}$ converge uniformly to $F_0$?