  We revise the Levy's construction of Brownian motion as a simple though still
rigorous approach to operate with various Gaussian processes. A Brownian path
is explicitly constructed as a linear combination of wavelet-based "geometrical
features" at multiple length scales with random weights. Such a wavelet
representation gives a closed formula mapping of the unit interval onto the
functional space of Brownian paths. This formula elucidates many classical
results about Brownian motion (e.g., non-differentiability of its path),
providing intuitive feeling for non-mathematicians. The illustrative character
of the wavelet representation, along with the simple structure of the
underlying probability space, is different from the usual presentation of most
classical textbooks. Similar concepts are discussed for fractional Brownian
motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional
Gaussian fields. Wavelet representations and dyadic decompositions form the
basis of many highly efficient numerical methods to simulate Gaussian processes
and fields, including Brownian motion and other diffusive processes in
confining domains.
