Kemeny’s constant for one-dimensional diffusions

Abstract

Let $X(\cdot )$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\mathbb{R} $ with invariant probability density $\mu (x)$, and let $\tau _{y}=\inf \{t\ge 0: X(t)=y\}$ 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.