= Solution
Start the <Bessel process> at $x>0$ and let $\tau_0$ be its first hit of zero. Away from zero its <stochastic differential equation> is
$$
dX_t=dW_t+\frac{\delta-1}{2X_t}\,dt.
$$
Apply the <Itô formula> to $f(x)=x^{2-\delta}$, localizing first to a compact interval in $(0,\infty)$. Its drift cancels:
$$
\frac12f''(x)+\frac{\delta-1}{2x}f'(x)
=\frac{(2-\delta)(1-\delta)}2x^{-\delta}
+\frac{(\delta-1)(2-\delta)}2x^{-\delta}=0.
$$
Thus the <Bessel power local martingale> satisfies
$$
\boxed{d(X_t^{2-\delta})=(2-\delta)X_t^{1-\delta}\,dW_t\qquad(t<\tau_0).}
$$
For $\delta=2$, this power is simply the constant one. For $\delta>2$, it is a <continuous local martingale> for all time, as zero is inaccessible.
Here is a proof of that inaccessibility and of the stronger minimum assertion. For $0<r<x<R$, stop the power at $\tau_r\wedge\tau_R$. The stopped process is bounded, so the <optional stopping theorem> gives
$$
\mathbb P_x(\tau_r<\tau_R)
=\frac{x^{2-\delta}-R^{2-\delta}}{r^{2-\delta}-R^{2-\delta}}.
$$
Exit from the bounded interval occurs almost surely. Taking $r\downarrow0$ with $R$ fixed shows that zero cannot be hit before $R$, hence cannot be hit in finite time. Taking $R\to\infty$ instead gives
$$
\mathbb P_x(\tau_r<\infty)=\left(\frac r x\right)^{\delta-2}.
$$
If the all-time infimum were zero, every level $r>0$ below $x$ would be hit, by <continuity>. Letting $r\downarrow0$ proves the <all-time minimum of a transient Bessel process> assertion:
$$
\boxed{\delta>2,\ X_0>0\quad\Longrightarrow\quad\inf_{t\geq0}X_t>0\quad\text{almost surely}.}
$$
The first claim in the printed question needs a qualification when $\delta<2$: it holds before the first hit of zero, or for the process stopped there, rather than for the usual reflecting continuation. For example, at $\delta=1$ the proposed power is $X$ itself, a <Reflected Brownian motion>; the <Tanaka formula> contains a nonzero <local time of a semimartingale> term, so it is not a <local martingale> after reflection. A positive starting point is also needed for the negative power when $\delta>2$; an entrance process started at zero would give an infinite value at time zero.
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