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I have been trying to understand if we can apply regularity structure to solve differential equations related to Gorini–Kossakowski–Sudarshan–Lindblad or GKSL equations. This is also known as the Lindblad equation. The unknown here is a matrix, for example density matrix. However, as I understand, Regularity structure is developed to find a scalar value at a point in space and time, when the scalar value follows a type of SPDE. The question becomes can we even extend or have a regularity structure (or similar) for matrices? As I understand regularity structure is an applicaton of Taylor expansion type theorem through a complex model.Can we have something for matrix?

The main motivation for this attempt is the assumption that regularity structure would be more tolerent to noise.

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  • $\begingroup$ Regularity structures has been applied in higher dimensions for some examples (as mentioned in the original article for Navier Stokes in 2d and Yang Mills later on). Not sure about matrix-valued but vector valued is discussed in some slides and notes by I. Chevyrev eg. math.ntnu.no/acpms/slides/…. and notes "Hopf and pre-Lie algebras in regularity structures" arxiv.org/pdf/2206.14557.pdf $\endgroup$ Commented May 30, 2023 at 21:33
  • $\begingroup$ For higher dimensions the subcritical condition might fail. So one has to go case by case. For example, that's why the 2d-KPZ has yet to be placed in some version of regularity structures. $\endgroup$ Commented May 30, 2023 at 21:34

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