A Piecewise-Deterministic Model for Brownian Motion

In the present paper, the classical Brownian motion of a particle suspended in an homogeneous liquid is modeled as a piecewise–deterministic Markov process with state space inculding position as well as velocity of the particle in motion. This model is less idealized than the classical Wiener or Ornstein–Uhlenbeck processes. It leads to more complex expressions for the transition densities, but also to path properties consistent with Newtonian physics. It is shown that the process introduced in this paper converges to the Wiener process as the intensity of collisions between the particle and the molecules of the liquid tends to infinity. This describes the point of idealization in the classical models. AMS subject classification: 60J65, 70B05, 60J25