
224 Modeling and Inverse Problems in the Presence of Uncertainty
It can be shown that the Wiener process is a Gaussian process with mean
and covariance functions respectively given by (e.g., see [7 2, Theorem 3.2])
E(W (t)) = 0, C ov{W (t), W (s)} = min{t, s}. (7 .25)
This implies that for any fixed t, W (t) is Gaussian distributed with zero mean
and variance given by t; tha t is, W (t) ∼ N(0, t). By Theorem 7.1.1 and (7.25)
we know that a Wiener process is also m.s. continuous.
The sample path of the Wiener process has a number of interesting prop-
erties. For example, the sa mple path of a Wiener process is nowhere differ-
entiable (with probability ...