# Kemeny's constant for one-dimensional diffusions

## TL;DR

This paper generalizes Kemeny's constant from finite Markov chains to one-dimensional diffusions, showing that the expected hitting time of a randomly chosen point (according to the invariant measure) is constant in the starting point and finite under certain boundary conditions.

## Contribution

It extends the concept of Kemeny's constant to continuous one-dimensional diffusion processes, establishing conditions for its finiteness and invariance.

## Key findings

- Expected hitting time is constant in the starting point.
- Finiteness of the expected hitting time depends on boundary conditions.
- Generalizes Kemeny's constant from Markov chains to 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 $\tau_y=\inf\{t\ge0: X(t)=y\}$ 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.