nLab iterated integral

Contents

Contents

Idea

An iterated integral is a expression involving nested integrals, such as

x=a b( y=g(x) h(x)f(x,y)dy)dx, \int_{x=a}^b \bigg(\int_{y=g(x)}^{h(x)} f(x,y) \,\mathrm{d}y\bigg) \,\mathrm{d}x ,

where the grouping parentheses are usually left out. Iterated integrals are the subject of Fubini theorems.

Iterated integrals include nested integration of differential forms of the kind as it appears in the formulation of parallel transport for nonabelian-value connection form (known as the Dyson formula in physics).

Applied to higher degree forms iterated integrals serve to express generalized transgression of differential forms to loop spaces and other mapping spaces realized as diffeological spaces.

References

The notion of iterated integration of differential forms originates in informal observation like the Dyson formula for parallel transport. It was formalized in the context of differential geometry on diffeological spaces (Chen spaces) (especially loop spaces) in

Essentially these formulas are used in

for expressing higher holonomy of certain flat infinity-connections given by infinity-representations of tangent Lie algebroids.

A review of this and more discussion in the context of a higher Riemann-Hilbert correspondence is in

A relation to algebraic cycles is discussed in

Chen iterated integrals in the context of stochastic analysis

  • Terry J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 2 (1998) 215–310 [pdf]
  • Terry J. Lyons, Michael Caruana, Thierry Lévy, Differential equations driven by rough paths, Springer LNM 1908 (2007)

Kapranov’s work on “noncommutative Fourier transform” uses some ideas from Chen’s work:

Iterated sums and iterated integrals over semirings, where the case of tropical semiring is a central, with applications (including in machine learning):

Last revised on August 30, 2024 at 11:58:02. See the history of this page for a list of all contributions to it.