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

Let $\bar{B}=\{{\bar{B}}_t,t\in R^1\}$ be Brownian motion killed after an independent exponential time with mean $2\mathrm{/}{\lambda }^2$. The process $\bar{B}$ has potential densities, $$u(x,y)=\frac{e^{-\lambda \mathrm{\mid }y-x\mathrm{\mid }}}{\lambda },x,y\in R^1,$$ which is also the covariance of an Ornstein–Uhlenbeck process. Let $f$ be an excessive function for $\bar{B}$. Then, $$\frac{e^{-\lambda \mathrm{\mid }y-x\mathrm{\mid }}}{\lambda }+f(y),x,y\in R^1,$$ is the kernel of an $\alpha$-permanental process $ X_{\alpha}={ X_{\alpha}(t), t\in R^{ 1}}$forall$\alpha>0$\mathrm{.}Itisshownthatforall$k\ge 1$andintervals$\Delta \subseteq [0,1] $, $$\underset{h\to 0}{\mathrm{lim\hspace{0.17em}sup}}\underset{}{\sup }\frac{\mathrm{\mid }{X}_{k\mathrm{/}2}(u)-X_{k\mathrm{/}2}(v)\mathrm{\mid }}{2(\mathrm{\mid }u-v\mathrm{\mid }\log 1\mathrm{/}\mathrm{\mid }u-v\mathrm{\mid }{)}^{1\mathrm{/}2}}=\sqrt{2}\underset{t\in \mathrm{\Delta }}{\sup }{X}_{k\mathrm{/}2}^{1\mathrm{/}2}(t)a\mathrm{.}s\mathrm{.}$$ The local modulus of continuity of $X_{k\mathrm{/}2}$ 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, $$Y_{k\mathrm{/}2}(t)=\sum _{i=1}^k\frac{{\eta }_i^2(t)}{2},t\in [0,1],$$ 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 $\eta \mathrm{.}$