arXiv:1312.0117v2 [math.PR] 3 Jun 2014 Anatole KHELIF Institut de Mathématique de Jussieu khelif@math.univ-paris-diderot.fr Alain TARICA 8 rue Chausse Coq Genève 1204 alaintarica@bluewin.ch SOMMAIRE 0. Introduction 4 1. D ∞ r -Stochastic Manifold, r ∈ N ∗ 7 1. 1 Definition and charts exchange maps ......................... 7 1. 2 Existence of a tangent linear maps ........................... 8 2. Preliminaries 13 2. 1 Some extensions of continuous linear maps ................... 14 2. 2 D 2 ∞ ∩ L ∞−0 = D ∞ ........................................... 21 2. 3 Existence of a sequence of D ∞ -vector fields converging D ∞ -strongly to wards a derivation ................................. 23 2. 4 D ∞ -Hölderian process and divergence of a derivation ......... 25 2. 5 A generalisation of the Dimi-Lipschitz theorem and interpolation between D P r spaces ................................................ 29 3. D ∞ -stochastic manifolds 29 3. 1 Definition ................................................... 29 3. 2 Canonical σ-algebra associated to a D ∞ -stochastic manifold. . 30 3. 3 D ∞ -morphismes between D ∞ -stochastic manifolds ........... 30 3. 4 Existence of a D ∞ -derivation which is not a vector field ...... 33 3. 5 Derivation field on a D ∞ -stochastic manifold ................. 36 3. 6 Metric and fundamental bilinear form on a D ∞ -stochastic manifold .......................................................... 37 3. 7 Metric, Levi-Civita connection, curvature .................... 46 4. Multiplicators, derivations 48 4. 1 Definition of D ∞ -bounded processes and of multiplicators .... 48 1