Exact moduli of continuity for general chi--square processes and for permanental processes related to the Ornstein--Uhlenbeck process

Abstract

Let $\overline{B}=\{{\overline{B}}_t,t\in R^1\}$ be Brownian motion killed after an independent exponential time with mean $2/{\lambda }^2$. 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 $\alpha$-permanental process $X_{\alpha }=\{X_{\alpha }(t),t\in R^1\}$ for all $\alpha >0$. It is shown that for all $k\ge 1$ and intervals $\Delta \subseteq [0,1]$, \[ \limsup_{h\to 0}\sup_{\stackrel{|u-v|\le h }{ u,v\inΔ}} \frac{|X_{k/2} (u)-X_{k/2} (v)|}{ 2 ( |u-v| \log 1/|u-v|)^{1/2}}= \sqrt 2 \sup_{t\inΔ}X_{k/2}^{1/2}(t)\qquad a.s.\] 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 $\eta =\{\eta (t);t\in [0,1]\}$ is a mean zero Gaussian process and $\{{\eta }_i;i=1,\dots ,k\}$ are independent copies of $\eta .$

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References in corpus (1)

• A sufficient condition for the continuity of permanental processes with applications to local times of Markov processes