Precise regularity estimates on diffusion semigroups are more than a mere theoretical curiosity.
They play a fundamental role in deducing sharp error bounds for higher-order particle methods.
In this thesis error bounds which are of consequence in iterated applications of Wiener
space cubature (Lyons and Victoir [29]) and a related higher-order method by Kusuoka [21]
are considered. Regularity properties for a wide range of diffusion semigroups are deduced. In
particular, semigroups corresponding to solutions of stochastic differential equations (SDEs)
with non-smooth and degenerate coefficients. Precise derivative bounds for these semigroups
are derived as functions of time, and are obtained under a condition, known as the UFG condition,
which is much weaker than Hormander's criterion for hypoellipticity. Moreover, very
relaxed differentiability assumptions on the coefficients are imposed. Proofs of exact error
bounds for the associated higher-order particle methods are deduced, where no such source
already exists. In later chapters, a local version of the UFG condition - `the LFG condition' - is
introduced and is used to obtain local gradient bounds and local smoothness properties of the
semigroup. The condition's generality is demonstrated. In later chapters, it is shown that the
V0 condition, proposed by Crisan and Ghazali [8], may be completely relaxed. Sobolev-type
gradient bounds are established for the semigroup under very general differentiability assumptions
of the vector fields. The problem of considering regularity properties for a semigroup
which has been perturbed by a potential, and a Langrangian term are also considered. These
prove important in the final chapter, in which we discuss existence and uniqueness of solutions
to the Cauchy problem.