On the nonlocal heat equation for certain L\'evy operators and the uniqueness of positive solutions

We develop a Widder-type theory for nonlocal heat equations involving quite general L\'evy operators. Thus, we consider nonnegative solutions and look for conditions on the operator that ensure: (i) uniqueness of nonnegative classical and very weak solutions with a given initial trace; (ii) the existence of an initial trace, belonging to certain admissibility class; and (iii) the existence of a solution, given by a representation formula, for any admissible initial trace. Such results are obtained first for purely nonlocal L\'evy operators defined through positive symmetric L\'evy kernels comparable to radial functions with mixed polynomial growth, and then extended to more general operators, including anisotropic ones and operators that have both a local and a nonlocal part.