## sup-norm of an Ornstein-Uhlenbeck process

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

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

What can we say about the sup_norm $\sup_{t\ge0} |X_t|$ ??
Is it bounded in $L^p$, or at least finite a.s. ?

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

Your help will be greatly appreciated.