What is a random function? What is noise? The standard
answers are nonsmooth, defined pointwise via the Wiener process
and Brownian motion. In the Chebfun project, we have found it
more natural to work with smooth random functions defined by
finite Fourier series with random coefficients. The length
of the series is determined by a wavelength parameter
$\lambda$. Integrals give smooth random walks, which approach
Brownian paths as $\lambda\to 0$, and smooth random ODEs,
which approach stochastic DEs of the Stratonovich variety.
Numerical explorations become very easy in this framework.
There are plenty of conceptual challenges in this subject,
starting with the fact that white noise has infinite amplitude
and infinite energy, a paradox that goes back two different
ways to Einstein in 1905. |