Integrated square of Brownian motion ($Q={\int }_0^1{W}_t^{2}dt$)

The quadratic functional $Q={\int }_0^1{W}_t^{2}dt$ is finite almost surely. The Brownian half-integer sine expansion writes it as a sum of independent scaled squared standard normals. Its mean is $1/2$, and its Laplace transform is determined by a hyperbolic cosine product.

The independent squared-normal series for the integrated square of Brownian motion gives the product ${\prod }_n(1+2\lambda /((n-1/2{)}^2{\pi }^2){)}^{-1/2}$. The half-integer cosine product reduces this to $(\cosh \sqrt{2\lambda }{)}^{-1/2}$. Expanding at zero gives mean $1/2$ and variance $1/3$.