An interdisciplinary approach to Liouville's equation

Résumé

In this paper we propose a new approach for Liouville's equation in the setting of random ordinary differential equations, by employing the mathematical theory of fluid mechanics. The uncertainty in the governing differential equation arises from the initial condition. The timedependent solution is a differentiable stochastic process with a probability density function. The ramdom differential equation is interpreted as the Euler's equations of motion for a particle in a fluid with random initial position. By computing the outflux of probability per unit surface area and unit time across the boundary of any fixed region and by using equation for the probability density function (Liouville's equation). Our approach illustrates a nice and pedagogical connection between probability theory and vector calculus applied to fluis mechanics