Probabilistic representations of solutions to the heat equation

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if $\varphi$ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition $\varphi$, is given by the convolution of $\varphi$ 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.