  We consider a class of homogeneous partial differential operators on a
finite-dimensional vector space and study their associated heat kernels. The
heat kernels for this general class of operators are seen to arise naturally as
the limiting objects of the convolution powers of complex-valued functions on
the square lattice in the way that the classical heat kernel arises in the
(local) central limit theorem. These so-called positive-homogeneous operators
generalize the class of semi-elliptic operators in the sense that the
definition is coordinate-free. More generally, we introduce a class of
variable-coefficient operators, each of which is uniformly comparable to a
positive-homogeneous operator, and we study the corresponding Cauchy problem
for the heat equation. Under the assumption that such an operator has
H\"{o}lder continuous coefficients, we construct a fundamental solution to its
heat equation by the method of E. E. Levi, adapted to parabolic systems by A.
Friedman and S. D. Eidelman. Though our results in this direction are implied
by the long-known results of S. D. Eidelman for $2\vec{b}$-parabolic systems,
our focus is to highlight the role played by the Legendre-Fenchel transform in
heat kernel estimates. Specifically, we show that the fundamental solution
satisfies an off-diagonal estimate, i.e., a heat kernel estimate, written in
terms of the Legendre-Fenchel transform of the operator's principal symbol--an
estimate which is seen to be sharp in many cases.
