  The Generalized Langevin Equation (GLE) is a Stochastic Integro-Differential
Equation that is commonly used to describe the velocity of microparticles that
move randomly in viscoelastic fluids. Such particles commonly exhibit what is
known as anomalous subdiffusion, which is to say that their position
Mean-Squared Displacement (MSD) scales sublinearly with time. While it is
common in the literature to observe that there is a relationship between the
MSD and the memory structure of the GLE, and there exist special cases where
explicit solutions exist, this connection has never been fully characterized.
Here, we establish a class of memory kernels for which the GLE is well-defined;
we investigate the associated regularity properties of solutions; and we prove
that large-time asymptotic behavior of the particle MSD is entirely determined
by the tail behavior of the GLE's memory kernel.
