The classical and quantum systems are many times seen as disconnected areas of physics and with totally distinct mathematical methods. This is not entirely true, many methods of the classical mechanics are very near to those used in the bases of quantum mechanics as the Poisson brackets, the Hamilton-Jacobi equation and the path integral formalism. In this presentation we will show an interplay between the classical and the quantum world in the path integral formalism. In the classical theory the path integral formalism is useful to describe the brownian motion, a kind of stochastic process in which positions are randomic variables varying according a parameter, the time; these positions are described by a distribution probability. This type of motion is observed, for exemple, in very small particles suspended on a fluid. In order to determine this distribution probability we can use the diffusion equation used by Einstein in his 1906 article about the brownian motion or use the Jaynes principle, a powerful tool of information theory to obtain probability distributions subjected to constraints. With the knowledge of the position distribution we are able to construct the path probability: we consider that the next step of a brownian particle is independent of the previous steps performed (markovian property), due to this property we can associate successive position probabilities and then obtain the Wiener path integral. This path integral gives the probability of a particle describe a given trajectory. An specific use of the Wiener integral is to treat the brownian movement with absorption and in this case is possible to show that the Wiener path integral satisfies the Feynman-Kac formula, a equation that establishes a connection between the stochastic processes theory and the differential equation theory. Modifying the Feynman-Kac formula to the Schrödinger equation we estabilish a map to modify the Wiener path integral and thus obtain the Feynman path integral to quantum mechanics.