Let
\begin{equation\*}
X\_t=xe^{-\lambda t}+\sigma e^{-\lambda t}\int\_0^t e^{\lambda s} dB\_s
\end{equation\*}


be the solution of Ornstein-Uhlenbeck equation where $B$ is Brownian motion, and $x,\sigma,\lambda$ are all constants. Compute 
\begin{equation\*}
\liminf\_{t\rightarrow \infty}\frac{X\_t}{\sqrt{\log t}}\quad \text{ and } \quad\limsup\_{t\rightarrow \infty}\frac{X\_t}{\sqrt{\log t}}
\end{equation\*} 
Furthermore, What can be said of $\liminf\_{t\rightarrow \infty}X\_t$ and $\limsup\_{t\rightarrow \infty}X\_t$?



Limit distribution of Ornstein-Uhlenbeck equation
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By vertical and horizontal rescaling, without loss of generality (wlog) $\la=\si=1$. It is then known (see e.g. [formula (1.5)](http://www.math.utk.edu/~xchen/limit-LIL.pdf)) that wlog 
\begin{equation}
 X\_t=\frac{B\_{e^t}}{e^{t/2}}. 
\end{equation}
So, 
\begin{equation}
 \limsup\_{t\to\infty}\frac{X\_t}{\sqrt{\ln t}}
 =\lim\_{t\to\infty}\sup\_{s\ge t}\frac{B\_{e^s}}{e^{s/2}\sqrt{\ln s}}
 =\lim\_{t\to\infty}\sup\_{u\ge e^t}\frac{B\_u}{\sqrt{u\ln\ln u}}=\sqrt2
\end{equation}
by the law of the iterated logarithm for the Brownian motion. 
Similarly, 
\begin{equation}
 \liminf\_{t\to\infty}\frac{X\_t}{\sqrt{\ln t}}
=-\sqrt2. 
\end{equation}
For arbitrary positive real $\la$ and $\si$ we then have 
\begin{equation}
 \limsup\_{t\to\infty}\frac{X\_t}{\sqrt{\ln t}}=\si\sqrt2,\quad
 \liminf\_{t\to\infty}\frac{X\_t}{\sqrt{\ln t}}=-\si\sqrt2; 
\end{equation}
the latter limits do not depend on $\la$. 


It then also follows that 
\begin{equation}
 \limsup\_{t\to\infty}X\_t=\si\sqrt2\,\lim\_{t\to\infty}\sqrt{\ln t}=\infty,\quad
 \liminf\_{t\to\infty}X\_t=-\infty. 
\end{equation}


