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.


2 Answers 2


Aspects of Brownian Motion (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 for some other references.


The distribution of Y has been studied in great details by Donati-Martin and Yor in their work: Fubini's theorem for double Wiener integrals and the variance of the Brownian motion path


In particular it is proved that in distribution

$Y=^{law}\int_0^1 | W_s |^2 ds$

where W is a standard complex Brownian bridge.


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