In this paper, we study the generalized heat equation associated with the Heckman-Opdam-Jacobi operator \(\Delta _{\textrm{HJ}}\) on \({\mathbb {R}}^{d+1}\) . Specifically, we show that the extension of this operator on the space of continuous functions on \({\mathbb {R}}^{d+1}\) and which tend towards 0 to infinity, is the generator of a positive strongly continuous contraction semi group. This is ensured by a maximum principle for the Heckman-Opdam-Jacobi operator \(\Delta _{\textrm{HJ}}\) . The explicit solution to the corresponding Cauchy problem incorporates a generalized heat kernel \(h_t\) , which is demonstrated to be nonnegative for real arguments.
