  The area enclosed by the two-dimensional Brownian motion in the plane was
studied by L\'evy, who found the characteristic function and probability
density of this random variable. For other planar processes, in particular
ergodic diffusions described by linear stochastic differential equations
(SDEs), only the expected value of the stochastic area is known. Here, we
calculate the generating function of the stochastic area for linear SDEs, which
can be related to the integral of the angular momentum, and extract from the
result the large deviation functions characterising the dominant part of its
probability density in the long-time limit, as well as the effective SDE
describing how large deviations arise in that limit. In addition, we obtain the
asymptotic mean of the stochastic area, which is known to be related to the
probability current, and the asymptotic variance, which is important for
determining from observed trajectories whether or not a diffusion is
reversible. Examples of reversible and irreversible linear SDEs are studied to
illustrate our results.
