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.