Consider a scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$\mathrm{d}X_t = b(t,X_t)\mathrm{d}t + \sigma(t,X_t)\mathrm{d}B_t, \quad X_0 = \xi$
for sufficiently regular coefficients $b$ and $\sigma$ and some initial state $\xi$.
Is there a result which guarantees that for every closed interval $I\subset[0,\infty)$ of positive measure, the sample paths of $(X_t)_{t\in I}$ are almost surely non-constant?