Electronic Communications in Probability

Kemeny’s constant for one-dimensional diffusions

Ross Pinsky

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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
Received: 7 April 2019
Accepted: 16 May 2019
First available in Project Euclid: 22 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1561169052

Digital Object Identifier
doi:10.1214/19-ECP244

Zentralblatt MATH identifier
07088977

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60J50: Boundary theory

Keywords
Kemeny’s constant one-dimensional diffusion entrance boundary

Rights
Creative Commons Attribution 4.0 International License.

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


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References

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