Questions tagged [iterated-integral]

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Time ordered integral involving beta function:

Any help on unpacking integrals of the following type, would be helpful: $$ \int_0^1 \int_0^r r^a (1-r)^b t^n (1-t)^m dr dt $$ where $a, b, n, m \in \mathbb{N}$ and $0 \le t \le 1$. Edit/...
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0answers
73 views

Bar construction and space of homotopy invariant functionals

I am introducing myself to the topic of iterated integrals using these notes http://www.ihes.fr/%7Ebrown/ColombiaNotes7.pdf On pag 22 is defined the following operator $$ D : (X^1)^n \to (\mathcal{A}...
1
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0answers
78 views

Iterated integral with a irregular path

For the proof of Fundamental Lemma 3.1 on the page 400 of K.T. Chen's 1957 paper Integration of paths--A faithful representation of paths by noncommutative formal power series, it requires the path $\...
2
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0answers
234 views

How to analytically evaluate this n-dimensional iterated integral?

I would very much appreciate any suggestions and/or pointers to references relevant for the analytic evaluation of the following n-dimensional iterated integral $$\int_{-\infty}^{+\infty}dx_1\int_{-\...
8
votes
1answer
801 views

Understand rough path iterated integral and how to compute it numerically?

The "signature" of rough path theory is defined by iterated integral as $s(k)=\int_{0 \le u_1 \le \cdots \le u_k \le t} \mathrm{d}X_{u_1} \otimes \cdots \otimes \mathrm{d}X_{u_k}$ in witch $X(t)$ is ...
3
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0answers
238 views

Chen's iterated integrals and free loop space

I recently found out about Chen's iterated integrals for paths in a differentiable manifold, and I was wondering if an analogous construction exists for free loops, i.e. a set of variables one ...
4
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0answers
231 views

Restriction of iterated integrals

For a smooth manifold $X$, let $B_s(X)$ denote the space of iterated integrals of length at most $s$. Here we consider iterated integrals as functions on the path space $PX$. Fix a base point $x_0$ ...
17
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3answers
3k views

Understanding iterated integrals

I have encountered iterated integrals on papers dealing with multizeta values, polylogarithms etc.. Since then I am trying to figure out the motivations and purpose of the theory. It seems the ...