Smooth random functions, random ODEs, and Gaussian processes
Résumé
The 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 length
goes to zero. A Brownian path is continuous but nowhere differentiable, and this non-smoothness 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 prac-
tical 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 1, one obtains the usual
stochastic objects in what is known as their Stratonovich interpretation.
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