| Summary: | 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.
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