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

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 self-adjoint partial differential operators with matrix-valued coefficients map from L 2 ( R n , C d ) to the space of continuous bounded functions, and that these semigroups have a jointly continuous and spatially bounded integral kernel. These partial differential operators include Yang–Mills type Hamiltonians with “electrical” potentials that are elements of the matrix-valued local Kato class.