Heat kernel for flat generalized Laplacians with anisotropic scaling

We calculate the closed analytic form of the solution of heat kernel equation for the anisotropic generalizations of flat Laplacian. We consider a UV as well as UV/IR interpolating generalizations. In all cases, the result can be expressed in terms of Fox-Wright psi-functions. We perform different consistency checks, analytically reproducing some of the previous numerical or qualitative results, such as spectral dimension flow. Our study should be considered as a first step towards the construction of a heat kernel for curved Ho\v{r}ava-Lifshitz geometries, which is an essential ingredient in the spectral action approach to the construction of the Ho\v{r}ava-Lifshitz gravity.