Projection unboundedness of a uniformly elliptic diffusion

For an SDE with bounded drift and diffusion coefficients and ${\sum }_k⟨\xi ,{\sigma }_k(x){⟩}^2\ge \lambda |\xi {|}^2$, each fixed nonzero scalar projection of a solution is unbounded in absolute value almost surely. Rescale a slab to $(-1,1)$; its projected drift is controlled by its uniformly positive bracket density. The exit with drift controlled by quadratic variation then forces exit from every slab in a countable exhaustion.

// Target: probability-and-statistics.bigb