  We stochastically formulate the theory of scalar quantum electrodynamics on a
de Sitter background. This reproduces the leading infrared logarithms at each
loop order. It also allows one to sum the series of leading infrared logarithms
to obtain explicit, nonperturbative results about the late time behavior of the
system. One consequence is confirmation of the conjecture by Davis, Dimopoulos,
Prokopec and Tornkvist that super-horizon photons acquire mass during
inflation. We compute a photon mass-suqared of about 3.2991 H^2. The scalar
stays perturbatively light with a mass-squared of about 0.8961 3 e^2 H^2/8pi^2.
Interestingly, the induced change in the cosmological constant is negative, of
about -0.6551 3 G H^4/pi.
