  The Oort Cloud remains one of the most poorly explored regions of the Solar
System. We propose that its properties can be constrained by studying a
population of dust grains produced in collisions of comets in the outer Solar
System. We explore the dynamics of micron-size grains outside the heliosphere
(beyond ~250 AU), which are affected predominantly by the magnetic field of the
interstellar medium (ISM) flow past the Sun. We derive analytic models for the
production and motion of small particles as a function of their birth location
in the Cloud and calculate the particle flux and velocity distribution in the
inner Solar System. These models are verified by direct numerical simulations.
We show that grains originating in the Oort Cloud have a unique distribution of
arrival directions, which should easily distinguish them from both
interplanetary and interstellar dust populations. We also demonstrate that the
distribution of particle arrival velocities is uniquely determined the mass
distribution and dust production rate in the Cloud. Cometary collisions within
the Cloud produce a flux of micron-size grains in the inner Solar System of up
to several per square meter per year. The next-generation dust detectors may be
sensitive enough to detect and constrain this dust population, which will
illuminate us about the Oort Cloud's properties. We also show that the
recently-detected mysterious population of large (micron-size) unbound
particles, which seems to arrive with the ISM flow is unlikely to be generated
by collisions of comets in the Oort Cloud.
