Assume $X\_t$ is an Ornstein-Uhlenbeck process in the form of
$$
d X\_t = -\alpha X\_t dt + \sigma dB\_t
$$
Is there an exponential bound (large-deviation bound) for
$$
P\left(
\max\_{t\le T} |X\_t| \ge z
\right)
\le
?
$$


P.S. when $\alpha = 0$ above $X\_t$ is a Brownian motion, and hence
$$
P\left( \sup\_{t\le T} |\sqrt{\varepsilon} B\_t | \ge c \right) 
\le 
4d \exp\left( - \frac{c^2 }{2dT \varepsilon} \right)
$$
for Brownian motion (Nonasymptotics result wanted). See e.g. the last display in


<https://en.wikipedia.org/wiki/Schilder%27s_theorem>


Is there an analogous version for the OU process?



Large deviation bound for O-U process
[Fernique's theorem](https://en.wikipedia.org/wiki/Fernique%27s_theorem), valid for all continuous Gaussian processes, implies that there are constants $C,c$ (depending on $T$) such that $P(\max\_{0 \le t \le T} |X\_t| \ge z) \le C e^{-c z^2}$. 


See also [Deviation bound for the maximum of the norm of Wiener process](https://mathoverflow.net/questions/153911/deviation-bound-for-the-maximum-of-the-norm-of-wiener-process).


