20,479 research outputs found
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 on these manifolds. Cohomology groups
associated with complexes of differential forms on in presence of such a
closed multi-symplectic form and their relations with the de Rham
cohomologies of 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
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