  In this paper we provide a new (probabilistic) proof of a classical result in
partial differential equations, viz. if $\phi$ is a tempered distribution, then
the solution of the heat equation for the Laplacian, with initial condition
$\phi$, is given by the convolution of $\phi$ with the heat kernel (Gaussian
density). Our results also extend the probabilistic representation of solutions
of the heat equation to initial conditions that are arbitrary tempered
distributions.
