Global Limit Theorem for Parabolic Equations with a Potential

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