## Electronic Communications in Probability

### Kemeny’s constant for one-dimensional diffusions

Ross Pinsky

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 36, 5 pp.

Dates
Accepted: 16 May 2019
First available in Project Euclid: 22 June 2019

https://projecteuclid.org/euclid.ecp/1561169052

Digital Object Identifier
doi:10.1214/19-ECP244

Zentralblatt MATH identifier
07088977

#### Citation

Pinsky, Ross. Kemeny’s constant for one-dimensional diffusions. Electron. Commun. Probab. 24 (2019), paper no. 36, 5 pp. doi:10.1214/19-ECP244. https://projecteuclid.org/euclid.ecp/1561169052

#### References

• [1] Bini, D., Hunter, J., Latouche, G., Meini, B. and Taylor, P. Why is Kemeny’s Constant a Constant?, J. Appl. Prob. 55 (2018), 1025-1036.
• [2] Durrett, R., Probability Theory and Examples, third edition, Brooks/Cole, Belmont, CA (2005).
• [3] Fitzsimmons, P., private communication.
• [4] Kemeny, J. and Snell, J. L., Finite Markov chains, Reprinting of the 1960 original, Springer-Verlag, New York-Heidelberg, 1976.
• [5] Pinsky, R. G., Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics 45, Cambridge University Press, (1995).
• [6] Pinsky, R. G., Optimizing the drift in a diffusive search for a random stationary target, to appear in Electron. J. Probab. (2019).