Let this O-U equation : $$ dX_t = \alpha X_t dt + \sigma dB_t,\ \ \ X_0=x $$ where $x,\alpha<0,\sigma$ are constants.
I proved that $X_t\xrightarrow{d} X\overset{d}{=}\mathcal N (0, \frac{\sigma^2}{-2\alpha})$.
Is it possible to prove that with probability 1 the limit $\underset{t\rightarrow\infty}{\lim} X_t=X$ exists (a.k.a. convergence a.s) ?
No, recall that the distribution of $X_t$ converges, but the process is ergodic - the measure defined by $X$ can be interpreted as the proportion of time spent by $X_t$ in a set as $t$ gets large. In particular, $\lim_t X_t$ does not exist.