A non-commutative sewing lemma

Abstract

A non-commutative version of the sewing lemma [1] is proved, with some applications.1 Definition : We say that a function V (t) defined on [0, T [ is a control function if it is non decreasing, V (0) = 0 and n≥1 V (1/n) < ∞.As easily seen, this is equivalent to the propertyfor every t ≥ 0. For example, t α and t/(log t −1 ) α with α > 1 are control functions.Observe that we havea control function. Then there exists a unique function ϕ(t) on [0, T [, up to an additive constant, such that |ϕ(b) − ϕ(a) − µ(a, b)| ≤ V (b − a) Proof : Put µ ′ (a, b) = µ(a, c) + µ(c, b) for c = (a + b)/2, and µ (n+1) = µ (n)′ . We easily get for n ≥ 0 a) converges, and the sequence µ (n) (a, b) converges to a limit u(a, b). For c = (a + b)/2 we haveWe say that u is midpoint-additive. Now, we prove that u is the unique midpoint-additive function with the inequalityand by induction |v(a, b) − u(a, b)| ≤ 2 n K.V [2 −n (b − a)] which vanishes as n → ∞ as mentioned above. Let k be an integer k ≥ 3, and take the function

Yet another introduction to rough paths

“…The first proof concerns the general case, and is taken from [Lyo98]. The other proof is a simpler proof in the case ω(s, t) = t − s, which is adapted from [FdLPM08]. For integrals, where x s,t = f (z s )z s,t for some rough path z of finite p-variation, one can find some increasing, continuous function φ : [0, T ] → R + such that z • φ is Hölder continuous (See [CG98] and [CL05] for an example of application in the context of rough paths), so that in many cases, one can consider that ω(s, t) = t − s (as the integral of a differential form along a path in insensitive to change of time).…”

Perturbed linear rough differential equations

“…The theory of rough paths is an extension of the theory of Young integrals [27,49,57]. Let X, Y and Z be Banach spaces with a product (x, y) ∈ X × Y → xy ∈ Z, such that |xy| ≤ |x| · |y|.…”

“…Following the construction proposed by T. Lyons, we construct a rough resolvent from an almost rough resolvent. However, our proof borrows some ideas to the elegant proof of Theorem 10 in [27] where ω(s, t) = V (t − s) provided that for some θ > 2, n≥0 θ n V (t2 −n ) < +∞ for any t > 0, as well as the ones from [3] regarding the composition of flows.…”

“…Let D r,t be its limit. To proof that D r,t is a multiplicative functional is the same as in [27] so that we skip it.…”

“…However, with the exception of the works of D. Feyel, A. de la Pradelle and G. Mokobodzki [27] and S. Aida [1], linear RDE have hardly been considered as objects as such with their own properties, excepted to control the growth of the solution as in [30,40]. Instead, they are generally presented as a special case of RDE.…”