arXiv 1903.120052019-03-28math.PR

一维扩散的Kemeny常数

Kemeny's constant for one-dimensional diffusions

Ross G. Pinsky

摘要：

设$X(\cdot)$是$\mathbb{R}$上具有不变概率密度$μ(x)$的非退化正常返一维扩散过程，$τ_y=\inf\{t\ge0: X(t)=y\}$表示$y$的首中时.设$\mathcal{X}$是一个与扩散过程$X(\cdot)$无关的随机变量，并根据过程的不变概率测度$μ(x)dx$分布。用$\mathcal{E}^μ$表示与$\mathcal{X}$相关的期望值。考虑表达式$$ \mathcal{E}^μE_xτ_\mathcal{X}=\int_{-\infty}^\infty (E_xτ_y)μ(y)dy, \ x\in\mathbb{R}. $$换句话说，该表达式是从根据扩散的不变分布随机选择的点的$x$开始的扩散的预期命中时间。我们表明，这个表达式是恒定的$x$，它是有限的当且仅当$\pm\infty$是入口边界的扩散。这一结果推广到扩散过程的相应结果的设置有限马尔可夫链，其中的常数值被称为凯梅尼常数。

英文摘要：

Let $X(\cdot)$ be a non-degenerate, positive recurrent one-dimensional diffusion process on $\mathbb{R}$ with invariant probability density $μ(x)$, and let $τ_y=\inf\{t\ge0: X(t)=y\}$ 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 $μ(x)dx$. Denote by $\mathcal{E}^μ$ the expectation with respect to $\mathcal{X}$. Consider the expression $$ \mathcal{E}^μE_xτ_\mathcal{X}=\int_{-\infty}^\infty (E_xτ_y)μ(y)dy, \ x\in\mathbb{R}. $$ 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.

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