Let $B\_t$ be a Brownian motion with variance 1. We know that $\int\_0^1 B(t) \mathrm{d} t \sim \mathcal{N}(0,1/3)$. I am interested to know what we can say about the law of the two random variables 
$X = \int\_{0}^1 B(t)^2 \mathrm{d}t = \langle B,B\rangle\_{L^2([0,1])}$ and $Y = \int\_{0}^{1} \left( B(t) - \int\_0^1 B(s)\mathrm{d}s \right)^2 \mathrm{d}t = X - \left( \int\_0^1 B(s)\mathrm{d}s \right)^2$. 


Is there a well-known probability law hidden behind them? 


Thank you for attention.



Law of the $L^2$ norm of a Brownian motion and related
[Aspects of Brownian Motion](http://www.springer.com/mathematics/probability/book/978-3-540-22347-4) (Mansuy & Yor) give an expression for the joint Laplace transform of $B\_t$ and $\int\_0^t B\_s^2 ds$ (section 2.1). For $\delta$-dimensional Brownian motion,
$$ \mathbb{E}\left[\exp\left(-\alpha|B\_t|-\frac{b^2}{2}\int\_0^t |B\_s|^2 ds \right) \right] $$
is equal to
$$\left(\text{ch}(bt)+2\frac{\alpha}{b}\text{sh}(bt)\right)^{-\delta/2}\exp\left(-\frac{xb}{2}\frac{(1+\frac{2\alpha}{b}\coth(bt)}{(\coth(bt)+\frac{2\alpha}{b}}\right). $$


I don't recognize the generating function, even for $\delta=1$ and $\alpha=0$; and neither do Manuy & Yor give it a name.


See also [Albin 1995](http://www.sciencedirect.com/science/article/pii/030441499400009I) for some other references.


