Starting from a variational formulation, we propose a new method to solve Cauchy problems for parabolic operators $\partial_t + L$. The elliptic part has a divergence structure with possible lower order terms controlled in critical (with respect to scaling) spaces. The approach applies to any order and also to problems with boundary conditions. One new outgrowth is the existence in full generality of a fundamental solution operator (also called propagator) and a new proof of Aronson’s Gaussian upper estimate for the fundamental solution of real operators of order $2$. This is joint work with Moritz Egert.