Let $V$ be a nonnegative locally bounded function defined in $Q_\infty:=\BBR^n\times(0,\infty)$. In this article we study under what condition on $V$ and on a Radon measure $\gm$ in $\mathbb{R}^d$ it is possible to have a solution to the initial value problem $\partial_t u-\xD u+ Vu=0$ in $Q_\infty$ such that $u(.,0)=\xm.$ We prove the existence of a subcritical case for which any measure is admissible and a supercritical case where capacitary conditions are needed. We prove a general representation theorem of positive solutions when $t V(x,t)$ is bounded and we prove the existence of an initial trace in the class of outer regular Borel measures?