  There are a number of reasons to entertain the possibility that locality is
violated on microscopic scales, for example through the presence of an infinite
series of higher derivatives in the fundamental equations of motion. This type
of nonlocality leads to improved UV behaviour, novel cosmological dynamics and
is a generic prediction of string theory. On the other hand, fundamentally
nonlocal models are fraught with complications, including instabilities and
complications in setting up the initial value problem. We study the structure
of the initial value problem in an interesting class of nonlocal models. We
advocate a novel new formulation wherein the Cauchy surface is "smeared out"
over the underlying scale of nonlocality, so that the the usual notion of
initial data at t=0 is replaced with an "initial function" defined over -M^{-1}
\leq t \leq 0 where M is the underlying scale of nonlocality. Focusing on some
specific examples from string theory and cosmology, we show that this
mathematical re-formulation has surprising implications for the well-known
stability problem. For D-brane decay in a linear dilaton background, we are
able to show that the unstable directions in phase space cannot be accessed
starting from a physically sensible initial function. Previous examples of
unstable solutions in this model therefore correspond to unphysical initial
conditions, an observation which is obfuscated in the old formulation of the
initial value problem. We also discuss implication of this approach for
nonlocal cosmological models.
