Brownian time change for a variable diffusion speed ($X_t=B_{A^{-1}(t)},A_u={\int }_0^u[2\theta (B_r){]}^{-1}dr$)

Let $\theta :{\mathbb{R}}^d\to (0,\Theta ]$ be a Borel measurable function bounded away from zero on every compact set. If $B$ is standard $d$-dimensional Brownian motion, the displayed clock is finite on compact intervals, continuous, strictly increasing, and unbounded. Its inverse ${\tau }_t=A^{-1}(t)$ is a Brownian stopping time, with ${\tau }_t\le 2\Theta t$. The optional sampling theorem and the change of variables $d{\tau }_t=2\theta (B_{{\tau }_t})dt$ show that $X_t=B_{{\tau }_t}$ solves the martingale problem for $\mathcal{L}=\theta \Delta$ on test functions $C_c^{\infty }({\mathbb{R}}^d)$. The filtration is ${\mathcal{F}}_{{\tau }_t}^B$. These are true martingales: on a finite new-time horizon the sampling times have a deterministic bound. Continuous strictly positive $\theta$ satisfies the compact-set lower bound. Pointwise positivity alone does not; pointwise positivity does not ensure weak existence gives a counterexample.