Consider a driftless Ornstein-Uhlenbeck process, described by the SDE

$\displaystyle dX_t = -\alpha X_t dt + \epsilon dW_t, \alpha,\epsilon>0.$

What can we say about the sup_norm $\displaystyle \sup_{t\ge0} |X_t| $ ??

Is it bounded in $\displaystyle L^p$, or at least finite a.s. ?

What can we say about the sup-norm of $\displaystyle e^{X_t}$?

Your help will be greatly appreciated.