论文标题
一般的chi级过程和与Ornstein相关的永久过程的确切连续性模量
Exact moduli of continuity for general chi--square processes and for permanental processes related to the Ornstein--Uhlenbeck process
论文作者
论文摘要
令$ \ Overline b = \ {\ overline b_ {t}，t \ in r^{1} \} $在独立的指数时间后被杀死，平均$ 2/λ^{2} $是布朗尼运动。该过程$ \ overline b $具有潜在的密度，\ [ u（x，y）= {e^{ - λ| y-x |} \avyλ}，\ qquad x，y \ in r^{1}，\]，这也是ornstein-uhlenbeck过程的协方差。令$ f $成为$ \ overline b $的过多功能。然后， \ [ {e^{ - λ| y-x |} \averλ}+f（y），\ qquad x，y \ in r^{1}，\]，\] 是所有$α> 0 $的$α$ - 永久过程的内核$x_α= \ {x_α（t），t \ in r^{1} \} $。结果表明，对于所有$ k \ ge 1 $和间隔$δ\ subseteq [0,1] $，\ [ \ limsup_ {h \ to 0} \ sup _ {\ stackrel {| u-v | \ le h} { 1/| U-V |）^{1/2}} = \ sqrt 2 \ sup_ {t \inδ} x_ {k/2}^{1/2}^{1/2}（t）\ qquad a.s. \] $ x_ {k/2} $的局部连续性，同时$ k { 对于由CHI-Square过程也获得了局部和统一的连续性模量，这些过程由，这些过程定义 \ [ y_ {k/2}（t）= \ sum_ {i = 1}^{k} \ frac {η^2_ {i} {i}（t）} {2}，\ qquad t \ in [0,1]， \] 其中$η= \ {η（t）; t \ in [0,1] \} $是平均零高斯进程，$ \ {η_{i}; i = 1，\ ldots，k \} $是$η。$的独立副本。
Let $ \overline B=\{ \overline B_{t},t\in R^{1} \}$ be Brownian motion killed after an independent exponential time with mean $2/λ^{2}$. The process $\overline B$ has potential densities, \[ u(x,y) ={e^{-λ|y-x|}\over λ},\qquad x,y\in R^{ 1}, \] which is also the covariance of an Ornstein--Uhlenbeck process. Let $f$ be an excessive function for $\overline B$. Then, \[ {e^{-λ|y-x|}\over λ}+f(y),\qquad x,y\in R^{ 1}, \] is the kernel of an $α$-permanental process $ X_α=\{ X_α(t), t\in R^{ 1}\}$ for all $α>0$. It is shown that for all $k\ge 1$ and intervals $Δ\subseteq [0,1] $, \[ \limsup_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}} \frac{|X_{k/2} (u)-X_{k/2} (v)|}{ 2 ( |u-v| \log 1/|u-v|)^{1/2}}= \sqrt 2 \sup_{t\inΔ}X_{k/2}^{1/2}(t)\qquad a.s.\] The local modulus of continuity of $X_{k/2}$ for all $k\ge 1$ is also obtained. Local and uniform moduli of continuity are also obtained for chi--square processes which are defined by, \[ Y_{k/2}(t)=\sum_{i=1}^{k}\frac{η^2_{i}(t)}{2},\qquad t\in [0,1], \] where $η=\{η(t);t\in [0,1]\}$ is a mean zero Gaussian process and $\{η_{i};i=1,\ldots, k\}$ are independent copies of $η.$
