This axis relates to the use of all methods of stochastic calculus, particularly the detailed analysis of process trajectories, their probabilities, their variation, their couplings, in order to:

• analyse the diffusion semigroups and evolution equations in manifolds (heat equation, mean curvature equation, Ricci flow), and their use in signal and image processing,
• get functional inequalities,
• study the Poisson boundaries,
• computing price sensitivity in financial models,
• get transport inequality,
• design search and optimization algorithms in manifolds for images and signals.

Existence and uniqueness problems are also studied for martingales with given terminal value in manifolds. Several contributions also cover the notion of Fréchet mean which is an extension of the usual Euclidean barycenter to spaces equiped with non-Euclidean distances. In this context, many statistical properties of the Fréchet mean were established for deformable models of signals.