Brownian Motion, 2nd Edition

2 Brownian motion as a Gaussian process

Recall that a one-dimensional random variable Γ is Gaussian if it has the characteristic function

(2.1)

for some real numbers m ∈ and ϭ ≥ 0. If we differentiate (2.1) two times with respect to ξ and set ξ = 0, we see that

(2.2)

A random vector Γ = ( Γ_1,..., Γ_n) ∈ is Gaussian, if 〈ℓ, Γ〉 is for every ℓ ∈