Is there a realtion between rough path theory and functional integration?

Question

This question may be vauge, but still I am asking only to get clear undertanding of the terms and specifically the connections between them if any.

I understand rough path theory https://en.wikipedia.org/wiki/Rough_path as a method to solve stochastic differential equation and later regularity structure https://en.wikipedia.org/wiki/Regularity_structure has extended the method to solve stochastic partial differential equation.

I understand functional integration to an extent as defined in wikipedia. https://en.wikipedia.org/wiki/Functional_integration

Is there a connection between functional integration and regularity structure or rough path.

Edit: Can we say that rough path theory is a method to solve functional integration where the domain of integration is a function?

Answer 1

It's an old question but interesting.

Some interesting series of works on the connection between the two are:

• Regularity structures applied to stochastic quantization, whose goal is to make sense of these functional integrals eg. "Regularity structures and the dynamical $\Phi^4_3$ model"
• Regularity structures applied to random loops where again we end up with a problem of making sense of an invariant measure eg. "Rough Stochastic PDEs" and see main conjecture 4.5 in "Geometric stochastic heat equations"