A stochastic sewing lemma and applications

We introduce a stochastic version of Gubinelli's sewing lemma, providing a sufficient condition for the convergence in moments of some random Riemann sums. Compared with the deterministic sewing lemma, adaptiveness is required and the regularity restriction is improved by a half. The limiting process exhibits a Doob-Meyer-type decomposition. Relations with It\^o calculus are established. To illustrate further potential applications, we use the stochastic sewing lemma in studying stochastic differential equations driven by Brownian motions or fractional Brownian motions with irregulardrifts.Comment: final version, to appear on EJ

A proof of the integral identity conjecture, II

In this note, using Cluckers-Loeser's theory of motivic integration, we prove the integral identity conjecture with framework a localized Grothendieck ring of varieties over an arbitrary base field of characteristic zero.Comment: Changed the content in comparison with the first version. To appear in Comptes Rendus Math\'ematiqu

Geometric structures associated with a simple Cartan 3-form

We introduce the notion of a manifold admitting a simple compact Cartan 3-form \om^3. We study algebraic types of such manifolds specializing on those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form \om^3. We prove the existence of an algebra of multi-symplectic forms $\phi^l$ on these manifolds. Cohomology groups associated with complexes of differential forms on $M^n$ in presence of such a closed multi-symplectic form $\phi^l$ and their relations with the de Rham cohomologies of $M$ are investigated. We show rigidity of a class of strongly associative (resp. strongly coassociative) submanifolds. We include an appendix describing all connected simply connected complete Riemannian manifolds admitting a parallel 3-form.Comment: 32 pages, final version, previous wrong Example 3.5 is remove