Laws of the iterated logarithm for α-time Brownian motion

Abstract

We introduce a class of iterated processes called $\alpha$-time Brownian motion for $0<\alpha \le 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $\alpha$-stable process. We prove a Chung-type law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in \cite{hu} for iterated Brownian motion. When $\alpha =1$ it takes the following form $$\underset{T\to \infty }{\mathrm{liminf}}{T}^{-1/2}(\log \log T)\underset{0\le t\le T}{\sup }|Z_t|={\pi }^2\sqrt{{\lambda }_1}a.s.$$ where ${\lambda }_1$ is the first eigenvalue for the Cauchy process in the interval $[-1,1].$ We also define the local time $L^{\ast }(x,t)$ and range $R^{\ast }(t)=|\{x:Z(s)=x\text{for some}s\le t\}|$ for these processes for $1<\alpha <2$. We prove that there are universal constants $c_R,c_L\in (0,\infty )$ such that $$\underset{t\to \infty }{\mathrm{limsup}}\frac{R^{\ast }(t)}{(t/\log \log t{)}^{1/2\alpha }\log \log t}=c_{R}a.s.$$ $$ \liminf_{t\to\infty} \frac{\sup_{x\in \RR{R}}L^{*}(x,t)}{(t/\log \log t)^{1-1/2α}}= c_{L} a.s. $$

30 pages

References in corpus (3)

• Iterated Brownian motion in an open set
• Iterated Brownian Motion in Parabola-Shaped Domains
• Iterated Brownian motion in bounded domains in R^n