Show that if $(B_t)$ is a d-dimensional brownian motion with $B_0\neq 0$ and $d\geq 3$, then $\vert B_t\vert^{2-d}$ is a nonnegative supermartingale.
Using Ito, we have $$\vert B_t\vert^{2-d}=\vert B_0\vert^{2-d}-(d-2)\sum_{i=1}^d\int_0^t \frac{B_s^i}{\vert B_s\vert^d}\mathrm{d}B_s^i$$ so it's a continuous local martingale, but I don't see how to prove that it's a supermartingale.
Any non-negative local martingale with continuous sample paths is a supermartingale.
Let $(M_t)_{t \geq 0}$ be a non-negative local martingale with continuous sample paths and denote by $(\sigma_k)_k$ a localizing sequence of stopping times. Applying Fatou's lemma (for conditional expectations) we find for any $s \leq t$
$$\begin{align*} \mathbb{E}(M_t \mid \mathcal{F}_s)= \mathbb{E} \left( \liminf_{k \to \infty} M_{t \wedge \sigma_k} \mid \mathcal{F}_s \right) &\leq \liminf_{k \to \infty} \mathbb{E}(M_{t \wedge \sigma_k} \mid \mathcal{F}_s) \\ &= \liminf_{k \to \infty} M_{s \wedge \sigma_k} = M_s. \end{align*}$$