From homotopy to Ito calculus and Hodge theory

Abstract

We begin with a deformation of a differential graded algebra by adding time and using a homotopy. It is shown that the standard formulae of Itô calculus are an example, with four caveats: First, it says nothing about probability. Second, it assumes smooth functions. Third, it deforms all orders of forms, not just first order. Fourth, it also deforms the product of the DGA. An isomorphism between the deformed and original DGAs may be interpreted as the transformation rule between the Stratonovich and classical calculus (again no probability). The isomorphism can be used to construct covariant derivatives with the deformed calculus. We apply the deformation in noncommutative geometry, to the Podleś sphere $S_q^2$. This involves the Hodge theory of $S_q^2$.

This is an attempt to relate stochastic calculus to deformations of differential graded algebras. Any comments very welcome, as there are quite likely many improvements that could be made, or more references that should be included

References in corpus (2)

• Quantum group covariant (anti)symmetrizers, epsilon-tensors, vielbein, Hodge map and Laplacian
• The weak heat kernel asymptotic expansion and the quantum double suspension

Cited by in corpus (1)

• Emergence of Riemannian geometry and the massive graviton