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Stochastic differential equations, and Hoermander form representations of diffusion operators, can determine a linear connection associated to the underlying (sub)-Riemannian structure. This is systematically described, together with its invariants, and then exploited to discuss qualitative properties of stochastic flows, and analysis on path spaces of compact manifolds with diffusion measures. This should be useful to stochastic analysts, especially those with interests in stochastic flows, infinite dimensional analysis, or geometric analysis, and also to researchers in sub-Riemannian geometry. A basic background in differential geometry is assumed, but the construction of the connections is very direct and itself gives an intuitive and concrete introduction. Knowledge of stochastic analysis is also assumed for later chapters.

Stochastic processes --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Diffusieproces --- Diffusion processes --- Flows (Differentiable dynamical systems) --- Flows (Differentieerbare systemen) --- Flows (Systèmes dynamiques différentiables) --- Lineaire operatoren --- Linear operators --- Opérateurs linéaires --- Processus de diffusion --- Probabilities. --- Functional analysis. --- Differential geometry. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Probability Theory and Stochastic Processes. --- Functional Analysis. --- Differential Geometry. --- Global Analysis and Analysis on Manifolds. --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential geometry --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

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Mathematics