Smooth Random Functions, Random ODEs, and Gaussian Processes

Abstract

The usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero. A Brownian path is continuous but nowhere differentiable, and this nonsmoothness is associated with technical complications that can be daunting. However, there is another approach to random processes that is more elementary, involving smooth random functions defined by finite Fourier series with random coefficients or, equivalently, by trigonometric polynomial interpolation through random data values. We show here how smooth random functions can provide a very practical way to explore random effects. For example, one can solve smooth random ordinary differential equations using standard mathematical definitions and numerical algorithms, rather than having to develop new definitions and algorithms of stochastic differential equations. In the limit as the number of Fourier coefficients defining a smooth random function goes to $\infty$, one obtains the usual stochastic objects in what is known as their Stratonovich interpretation.