  Stochastic microlensing is a central tool in probing dark matter on galactic
scales. From first principles, we initiate the development of a mathematical
theory of stochastic microlensing. Beginning with the random time delay
function and associated lensing map, we determine exact expressions for the
mean and variance of these transformations. We characterize the exact p.d.f. of
a normalized random time delay function at the origin, showing that it is a
shifted gamma distribution, which also holds at leading order in the limit of a
large number of point masses at a general point of the lens plane. For the
large number of point masses limit, we also prove that the asymptotic p.d.f. of
the random lensing map under a specified scaling converges to a bivariate
normal distribution. We show analytically that the p.d.f. of the random scaled
lensing map at leading order depends on the magnitude of the scaled bending
angle due purely to point masses as well as demonstrate explicitly how this
radial symmetry is broken at the next order. Interestingly, we found at leading
order a formula linking the expectation and variance of the normalized random
time delay function to the first Betti number of its domain. We also determine
an asymptotic p.d.f. for the random bending angle vector and find an integral
expression for the probability of a lens plane point being near a fixed point.
Lastly, we show explicitly how the results are affected by location in the lens
plane. The results of this paper are relevant to the theory of random fields
and provide a platform for further generalizations as well as analytical limits
for checking astrophysical studies of stochastic microlensing.
