Firstly, for motivation purposes, we briefly present a few problems mixing notions of probability theory and of partial differential equations (PDE). In discussing the solution to such problems it will become apparent that some stochastic process and differential equations walk together. Next, we introduce a class of stochastic processes called the Ito diffusions, and some of its features such as the Markov property. Each such process has an associated linear operator the, so called, infinitesimal generator. This operator acts as a second-order differential operator on smooth functions, and controls the LOCAL behavior of these diffusions. We discuss these features together with Dynkin s formula a convenient relation derived from the infinitesimal generator, which informs us about the AVERAGE behavior of the diffusion. Finally, we apply these probabilistic tools to find a formula for the solution of the Dirichlet problem for a somewhat general linear elliptic second order PDE. This formula connects the solution of the PDE to the aggregated/average behavior and associated (Ito) diffusion. This type of stochastic representation generalizes the solution method of the problems firstly discussed.