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Kemeny’s constant for one-dimensional diffusions. (English) Zbl 07088977
Summary: 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.
##### MSC:
 60J60 Diffusion processes 60J50 Boundary theory for Markov processes
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##### 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. · Zbl 1405.60112 [2] Durrett, R., Probability Theory and Examples, third edition, Brooks/Cole, Belmont, CA (2005). · Zbl 1202.60002 [3] Kemeny, J. and Snell, J. L., Finite Markov chains, Reprinting of the 1960 original, Springer-Verlag, New York-Heidelberg, 1976. · Zbl 0328.60035 [4] Pinsky, R. G., Positive Harmonic Functions and Diffusion, Cambridge Studies in Advanced Mathematics 45, Cambridge University Press, (1995). · Zbl 0858.31001 [5] Pinsky, R. G., Optimizing the drift in a diffusive search for a random stationary target, to appear in Electron. J. Probab. (2019).
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