Thanks mainly to the relationship between the heat equation, newtonian potential theory and Brownian motion, the laws of a large number of Brownian functionals have been obtained during the last fifty years, at least via explicit expressions of their Laplace and Fourier transforms. Much pioneering work in this area was done by Paul Lévy.
Gradually, with the development of Itô’s stochastic calculus, excursion theory, path decompositions and the technique of enlargement of nitrations, these studies of individual distributions on \( \mathbb{R} \), sometimes exhibiting identities between two laws, which looked a priori to be mere “coincidences”, have been understood in a deeper way, in fact often by showing that two processes are identical in law; see [Biane 1990] for a recent survey in that spirit.
The most elementary examples of Brownian functionals are linear functionals: if
\[ f\in L^2(\mathbb{R}_+,dt) ,\]
and \( (B_t \), \( t\geq 0) \) is a real-valued BM, the Wiener integral
\[ \int_0^{\infty} f(t)\,dB_t \]
is a centered Gaussian variable, with variance
\[ \int_0^{\infty}f^2(t)\,dt .\]
Quadratic functionals of BM represent the next level of complexity; those functionals are of great interest as, somewhat surprisingly, they occur in a number of very different studies of Brownian motion, such as the Ray–Knight theorems for Brownian local times, the Ciesielski–Taylor identities, some limiting laws of planar BM, and principal values of Brownian local times.
We shall take here, as a prototype of a quadratic Brownian functional, the stochastic area of planar BM, and it will be shown how Paul Lévy’s formula for this stochastic area appears again and again in most of the above mentioned studies of Brownian motion.
