Let us consider $X$ to be an Ornstein–Uhlenbeck process, i.e. the solution of $\text{d}X_t = \text{d}W_t - X_t \text{d}t$. We define $Y$ by

$$\mathbb{P}(Y\in \cdot):= \lim_{t\to+\infty}\mathbb{P}(X\in \cdot|\forall_{s\leq t}X_s \geq 0).$$

I expect that random variable $Y_t$ is well-concentrated, for example

$$\mathbb{P}(Y_t\geq x) \leq C \exp\{-c x^2\},$$ for some $C,c>0$ (independent of $t$).

Is there an "easy proof" of this fact? I can think about a proof using the first eigen-value of the OU operator on the half-line and then applying the $h$-transform but this seems a big gun for an easy problem.