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<p>Stochastic microlensing is a central tool in probing dark matter on galactic scales.
From first principles, we initiate the development of a mathematical theory </p><p>of
stochastic microlensing. We first construct a natural probability space for stochastic
microlensing and characterize the general behaviour of the random time </p><p>delay
functions' random critical sets. Next we study stochastic microlensing in two distinct
random microlensing scenarios: The uniform stars' distribution with</p><p> constant
mass spectrum and the spatial stars' distribution with general mass spectrum. For
each scenario, we determine exact and asymptotic (in the large number</p><p> of point
masses limit) stochastic properties of the random time delay functions and associated
random lensing maps and random shear tensors, including their </p><p>moments and asymptotic
density functions. We use these results to study certain random observables, such
as random fixed lensed images, random bending angles, </p><p>and random magnifications.
These results are relevant to the theory of random </p><p>fields and provide a platform
for further generalizations as well as analytical limits for checking astrophysical
studies of stochastic microlensing.</p><p>Continuing our development of a mathematical
theory of stochastic microlensing, we study the stochastic version of the Image Counting
Problem, first considered </p><p>in the non-random setting by Einstein and generalized
by Petters. In particular, we employ the Kac-Rice formula and Morse theory to deduce
general formulas for </p><p>the expected total number of images and the expected number
of saddle images for a general random lensing scenario. We further </p><p>generalize
these results by considering random sources defined on a countable compact covering
of the light source plane. This is done to introduce the notion of</p><p> global expected
number of positive parity images due to a general lensing map. Applying the result
to the uniform stars' distribution random microlensing </p><p>scenario, we calculate
the asymptotic global expected number of minimum images in the limit of an infinite
number of stars. This global expectation is bounded, </p><p>while the global expected
number of images and the global expected number of saddle images diverge as the order
of the number of stars.</p><p>Finally, we outline a framework for the study of stochastic
microlensing in the neighbourhood of lensed images. This framework is related to the
study of the </p><p>local geometry of a random surface. In our case, the surface is
non-Gaussian, and therefore standard literature on the subject does not apply. We
explore the case</p><p> of a random gravitational field caused by a random star.</p>
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