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We study geometric stochastic differential equations (SDEs) and theirapproximations on Riemannian manifolds. In particular, we introduce a simplenew construction of geometric SDEs, using which with bounded curvature. Inparticular, we provide the first (to our knowledge) non-asymptotic bound on theerror of the geometric Euler-Murayama discretization. We then bound thedistance between the exact SDE and a discrete geometric random walk, where thenoise can be non-Gaussian; this analysis is useful for using geometric SDEs tomodel naturally occurring discrete non-Gaussian stochastic processes. Ourresults provide convenient tools for studying MCMC algorithms that adoptnon-standard noise distributions.

Theory and Algorithms for Diffusion Processes on Riemannian Manifolds