  We obtain the asymptotics, as $t + |x| \rightarrow \infty$, of the
fundamental solution to the heat equation with a compactly supported potential.
It is assumed that the corresponding stationary operator has at least one
positive eigenvalue. Two regions with different types of behavior are
distinguished: inside a certain conical surface in the $(t,x)$ space, the
asymptotics is determined by the principal eigenvalue and the corresponding
eigenfunction; outside of the conical surface, the main term of the asymptotics
is a product of a bounded function and the fundamental solution of the
unperturbed operator, with the contribution from the potential becoming
negligible if $|x|/t \rightarrow \infty$. A formula for the global asymptotics,
as $t + |x| \rightarrow \infty$, of the solution in the entire half-space $t >
0$ is provided.
  In probabilistic terms, the result describes the asymptotics of the density
of particles in a branching diffusion with compactly supported branching and
killing potentials.
