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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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Several stochastic processes related to transient Lévy processes with potential densities u(x,y)=u(y-x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures mathcal{V} endowed with a metric d. Sufficient conditions are obtained for the continuity of these processes on (mathcal{V},d). The processes include n-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup n-fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

Gaussian processes. --- Local times (Stochastic processes) --- Loop spaces.

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