On the Continuity of Brownian Motion with a Multidimensional Parameter

Abstract

A stochastic process X(A, ω) is called Brownian motion with an N-dimensional parameter when it satisfies the following conditions:1) For any positive integer n and any set of points A₁, A₂, …, A_n in an N-dimensional Euclidian space E_N, the joint variable is subject to an n-dimensional Gaussian distribution having the vector 0 as its mean vector.