Multiplicative matrix-valued functionals and the continuity properties of semigroups correspondings to partial differential operators with matrix-valued coefficients

TL;DR

This paper introduces matrix-valued multiplicative functionals with local Kato potentials and uses probabilistic methods to establish the spatial and joint continuity of semigroups and their kernels for certain matrix-coefficient PDEs, including Yang-Mills and Pauli Hamiltonians.

Contribution

It develops a framework for analyzing semigroups of matrix-valued PDEs with Kato potentials using probabilistic techniques, proving their continuity properties.

Findings

Semigroups are spatially continuous.

Semigroups have jointly continuous integral kernels.

Applicable to Yang-Mills and Pauli Hamiltonians.

Abstract

We define and examine certain matrix-valued multiplicative functionals with local Kato potential terms and use probabilistic techniques to prove that the semigroups of the corresponding partial differential operators with matrix-valued coefficients are spatially continuous and have a jointly continuous integral kernel. These partial differential operators include Yang-Mills type Hamiltonians and Pauli type Hamiltonians, with "electrical" potentials that are elements of the matrix-valued local Kato class.