Smooth random functions, random ODEs, and Gaussian processes

Abstract : 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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Contributor : Silviu-Ioan Filip <>
Submitted on : Wednesday, December 5, 2018 - 9:38:25 AM
Last modification on : Tuesday, February 25, 2020 - 8:08:10 AM
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Silviu-Ioan Filip, Aurya Javeed, Lloyd Nicholas Trefethen. Smooth random functions, random ODEs, and Gaussian processes. SIAM Review, Society for Industrial and Applied Mathematics, 2019, 61 (1), pp.185-205. ⟨10.1137/17M1161853⟩. ⟨hal-01944992⟩

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