  Let $X(\cdot)$ be a non-degenerate, positive recurrent one-dimensional
diffusion process on $\mathbb{R}$ with invariant probability density $\mu(x)$,
and let $\tau_y=\inf\{t\ge0: X(t)=y\}$ denote the first hitting time of $y$.
Let $\mathcal{X}$ be a random variable independent of the diffusion process
$X(\cdot)$ and distributed according to the process's invariant probability
measure $\mu(x)dx$. Denote by $\mathcal{E}^\mu$ the expectation with respect to
$\mathcal{X}$. Consider the expression $$ \mathcal{E}^\mu
E_x\tau_\mathcal{X}=\int_{-\infty}^\infty (E_x\tau_y)\mu(y)dy, \
x\in\mathbb{R}. $$ In words, this expression is the expected hitting time of
the diffusion starting from $x$ of a point chosen randomly according to the
diffusion's invariant distribution. We show that this expression is constant in
$x$, and that it is finite if and only if $\pm\infty$ are entrance boundaries
for the diffusion. This result generalizes to diffusion processes the
corresponding result in the setting of finite Markov chains, where the constant
value is known as Kemeny's constant.
