Multiplicative matrix-valued functionals and the continuity properties of semigroups corresponding to partial differential operators with matrix-valued coefficients
The paper is concerned with probabilistic methods for selfadjoint operators in \(L^2({\mathbb R}^n, {\mathbb C}^d)\) that are formally given by the differential expression \[ \tau(\alpha, V)= -\frac{1}{2}\Delta - \frac{1}{2}\sum_{j=1}^n \alpha^2_j -\frac{1}{2}\sum_{j=1}^n (\partial_j\alpha_j ) -\sum_{j=1}^n \alpha_j\partial_j +V, \] where \(\alpha\) is a 1-form on \(\mathbb R^n\) and \(V\) is a potential. The author proves that the semigroup associated to this partial differential operator maps \(L^2({\mathbb R}^n, {\mathbb C}^d)\) to the space of continuous bounded functions and has a jointly continuous and spatially bounded integral kernel. Differential operators of this type arise in nonrelativistic mechanics when one wants to describe the energy of Yang-Mills particles which live on \(\mathbb R^n\) under the influence of the electrical potential \(V\).
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