Let $W_{t} = (W_{t}^{1}, \ldots, W_{t}^{d}), t \geqslant 0$ be a $d$-dimensional Wiener process, e.i. $W^{1}, \ldots, W^{d}$ are independent 1-dimensional Wiener processes. Let $ d \geqslant 3 $.
Show that $(|W_{t}|^{2-d})_{t \in (0, \infty)}$ is a supermartingale.
I've got a hint that I have to show the existance of a positive constant $C_{d}$, dependent only of the dimension $d$, such that for every $x \in \mathbb{R}^{d}$ we have $|x|^{2-d} = C_{d} \int_{0}^{\infty}{p_{t}(x) dt}$, where $p_t$ is the density of $W_t$.
I also know that there is a theorem which says:
If $X_t$ is a supermartingale and $f$ is a nondecreasing concave function such that $\mathbb{E} |f(X_t)| < \infty$, then also $f(X_t)$ is a supermartingale.
This theorem is true because of Jensen's inequality.
The fact that $\frac 1 {|W_t|}$ is a super-martingale follows easily from the fact that $(W_t)$ is a martingale. Now use the fact that $x \to x^{d-2}$ is an increasing convex function on $[0,\infty)$.