Smooth random functions, random ODEs, and Gaussian processes

International audienceThe usual way that mathematicians work with randomness is by a rigorous for-mulation of the idea of Brownian motion, which is the limit of a random walk as the step lengthgoes to zero. A Brownian path is continuous but nowhere differentiable, and this non-smoothness isassociated with technical complications that can be daunting. However, there is another approachto random processes that is more elementary, involving smooth random functions defined by finiteFourier series with random coefficients, or equivalently, by trigonometric polynomial interpolationthrough random data values. We show here how smooth random functions can provide a very prac-tical way to explore random effects. For example, one can solve smooth random ordinary differentialequations using standard mathematical definitions and numerical algorithms, rather than havingto develop new definitions and algorithms of stochastic differential equations. In the limit as thenumber of Fourier coefficients defining a smooth random function goes to 1, one obtains the usualstochastic objects in what is known as their Stratonovich interpretation