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This work studies properties of stochastic processes taking\ud values in a differential manifold M with a linear connection Γ, or\ud in a Riemannian manifold with a metric connection.\ud Part A develops aspects of Ito calculus for semimartingales\ud on M, using stochastic moving frames instead of local co-ordinates.\ud New results include:\ud -a formula for the Ito integral of a differential form along a\ud semimartingale, in terms of stochastic moving frames and the\ud stochastic development (with many useful corollaries);\ud - an expression for such an integral as the limit in probability\ud and in L2 of Riemann sums, constructed using the exponential map;\ud - an intrinsic stochastic integral expression for the 'geodesic\ud deviation', which measures the difference between the stochastic\ud development and the inverse of the exponential map;\ud -a new formulation of 'mean forward derivative' for a wide class\ud of processes on M.\ud Part A also includes an exposition of the construction of non-degenerate\ud diffusions on manifolds from the viewpoint of geometric Ito\ud calculus, and of a Girsanov-type theorem due to Elworthy.\ud Part B applies the methods of Part A to the study of 'Γ-martingales'\ud on M. It begins with six characterizations of Γ-martingales,\ud of which three are new; the simplest is: a process whose image under\ud every local Γ-convex function is (in a certain sense) a submartingale,\ud However to obtain the other characterizations from this one requires\ud a difficult proof. The behaviour of Γ-martingales under harmonic\ud maps, harmonic morphisms and affine maps is also studied.\ud On a Riemannian manifold with a metric connection Γ, a Γ-martingale\ud is said to be L2 if its stochastic development is an L2\ud Γ-martingale. We prove that if M is complete, then every such process\ud has an almost sure limit, taking values in the one-point compactification\ud of M. No curvature conditions are required. (After this\ud result was announced, a simpler proof was obtained by P. A. Meyer,\ud and a partial converse by Zheng Wei-an.)\ud The final chapter consists of a collection of examples of\ud Γ-martingales, e.g. on parallelizable manifolds such as Lie groups,\ud and on surfaces embedded in R3. The final example is of a Γ-martingale\ud on the torus T (Γ is the Levi-Civita connection for the\ud embedded metric) which is also a martingale in R3

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