  There has been considerable recent interest in solving non-local equations of
motion which contain an infinite number of derivatives. Here, focusing on
inflation, we review how the problem can be reformulated as the question of
finding solutions to a diffusion-like partial differential equation with
non-linear boundary conditions. Moreover, we show that this diffusion-like
equation, and hence the non-local equations, can be solved as an initial value
problem once non-trivial initial data consistent with the boundary conditions
is found. This is done by considering linearised equations about any field
value, for which we show that obtaining solutions using the diffusion-like
equation is equivalent to solving a local but infinite field cosmology. These
local fields are shown to consist of at most two canonically normalized or
phantom fields together with an infinite number of quintoms. We then
numerically solve the diffusion-like equation for the full non-linear case for
two string field theory motivated models.
